Abstract:
A method for calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking, includes: acquiring parameters of gravity satellite system; calculating an effect of satellite loads on the power spectrum of nonspherical perturbation potential, so as to obtain an degree error variance; comparing degree error variance with degree variance given by Kaula Rule, and when degree error variance equals degree variance, considering that the highest degree of gravity field measurement is obtained, calculating geoid degree error and its accumulative error, gravity anomaly degree error and its accumulative error, so as to obtain the performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. The method is capable of evaluating gravity field measurement performance quickly and effectively, obtaining a rule of effects of the gravity satellite system parameters on the gravity field measurement performance, so as to avoid shortcoming caused by numerical simulation.

Description:
CROSS REFERENCE OF RELATED APPLICATION 
     This is a U.S. National Stage under 35 U.S.C 371 of the International Application PCT/CN2013/001264, filed Oct. 17, 2013, which claims priority under 35 U.S.C. 119(a-d) to CN201310454595.3, filed Sep. 29, 2013. 
     BACKGROUND OF THE PRESENT INVENTION 
     Field of Invention 
     The present invention relates to the technical field of satellite gravity field measurement, and more particularly to a method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking, which is applied for parameter design of a gravity field measurement satellite system by low-to-low satellite-to-satellite tracking. 
     Description of Related Arts 
     As a basic physical field of the earth, gravitation field is an important application in earth science research, land resource exploration and geological disaster forecast, and thus the gravitation field has always been an essential issue in geodesy. With the developments of space technology, satellite gravity field measurement attracts more and more attention due to the unique advantages of high coverage, all weather, and without affection by geopolitics and geographical environments. The satellite gravity field measurement has obtained substantial progress in theoretical research and engineering practice and become the most effective way to obtain the global gravity field model [1,2] . 
     Depending on differences in data observed, the gravity field measurement is classified by three principles of orbit perturbation, low-to-low satellite-to-satellite tracking and gravity gradient [3] , wherein the orbit perturbation principle is suitable for low-order gravity field measurement which has main observation data of a gravity gradient orbit; the low-to-low satellite-to-satellite tracking principle is suitable for a medial-to-high order gravity field measurement which has main observation data of a distance between two low earth orbit satellites and a change rate thereof; and the gravity gradient principle is suitable for a high-order gravity field measurement which has main observation data of gravity gradient value. The gravity satellites which are successfully implemented or under developed all adopts the measurement principles mentioned above or a combination thereof. E.g., CHAMP utilizes the orbit perturbation principle to recover low-order gravity field; GRACE Follow-on and NGGM satellites utilize both orbit perturbation principle and low-to-low satellite-to-satellite tracking principle to recover medial-to-high order gravity field; and GOCE respectively utilizes orbit perturbation principle and low-to-low satellite-to-satellite tracking principle to recover the low gravity field and the high gravity field. Though, the three principles are respectively suitable for different frequency band of gravity field measurement, the effective order and accuracy of the gravity field measurement are mainly depends on load indicators of the gravity satellite. Under the medial or high order gravity field measurement by low-to-low satellite tracking and gravity gradient orbit, analyzing from the current load performance indicator, the low-to-low tracking measurement is absolutely capable of reaching or even exceeding measurement level of gravity gradient. Therefore, the symposium named “Measurement of satellite gravity field in the future” held in 2007 in Holland decided that the subsequent international gravity satellite continues adopting the low-to-low satellite-to-satellite tracking manner, and that parameters of the system are considered to be improved to increase the measurement performance of the gravity field. 
     In the conventional design of orbit parameters and loading parameters of gravity field measurement system by low-to-low satellite-to-satellite tracking, a numerical simulation method is mainly utilized to analyze the impact of system parameters on measurement performance of gravity field, so as to further determine design parameters of the system. However, the simulation of gravity field measurement is in high demand of performance of computers, and calculation time lasts very long, which is not conductive to analysis of influence rule of the system parameters on the gravity field measurement, and not convenient for the optimal design of system parameters. In order to overcome the shortcoming, based on the principle of conservation of energy, the present invention establishes an analytical method for gravity field measurement performance by low-to-low satellite-to-satellite tracking, so as to obtain gravity field measurement performances such as an effective order of the gravity field measurement, a geoid error and gravity anomaly error, and determine the influence rule of the system parameters on the gravity field measurement. Thus, the present invention has great guiding significance on parameter design of gravity satellite system by low-to-low satellite-to-satellite tracking. 
     REFERENCES 
     
         
         [1] Jinsheng ning, Following the Development of the World, Devoting to the Study on the Earth Gravity Field. Gemomatics and Information Science of Wuhan University, 2001, 26(6): 471-474. 
         [2] Jinsheng ning, THE SATELLITE GRAVITY SURVEYING TECHNOLOGY AND RESEARCH OF EARTH&#39;S GRAVITY FIELD, JOURNAL OF GEODESY AND GEODYNAMICS, 2002, 22(1): 1-5. 
         [3] Zhenfenggu, Hongwei Liu, Zhaokui Wang, Yulin Zhang, Analysis for satellite gravity field measurement based on relative weight of gravitational coefficients, Progress in Geophysics, 2013, 28(1): 17-23. 
       
    
     SUMMARY OF THE PRESENT INVENTION 
     Targeting at a gravity field measurement satellite system by low-to-low satellite to satellite tracking, parsing relationships between gravity field measurement performances and satellite system parameters; 
     wherein the gravity field measurement performances comprise: a gravity field measurement valid order, a geoid error and a gravity anomaly error; and 
     the satellite system parameters comprise: a satellite orbit height, an inter-satellite distance, an inter-satellite change rate measurement precision, an orbit determination precision and a non-gravitational interference, a measurement data sampling interval and a service period. The method of the present invention is capable of rapidly calculating measurement performances of gravity field measurement performance by the low-to-low satellite-to-satellite, so as to analyzing an effective rule of the system parameters on the gravity field measurement performance and to guide an optimal design of the system parameters of the gravity field measurement performance by low-to-low satellite-to-satellite tracking. 
     Accordingly, in order to achieve the object mentioned above, a technical solution adopted by the present invention is as follows. 
     A method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking, comprises following steps of: 
     step (1): acquiring at least one parameter of a gravity satellite system by low-to-low satellite-to-satellite tracking; 
     step (2): according to the parameter of the gravity satellite system by low-to-low satellite-to-satellite tracking, calculating an effect of a measuring error of a gravity satellite load on a power spectrum of a nonspherical perturbation potential of earth gravity, so as to obtain an degree error variance of a potential coefficient in a gravity field recovery model; 
     step (3): comparing the degree error variance of the potential coefficient in the gravity field recovery model with an degree variance of a potential coefficient given by Kaula Rule, wherein with an increase of an order of the gravity field recovery model, the degree error variance gradually increases and the degree variance gradually decreases; when the degree error variance is equal to the degree variance, considering that a maximum valid order of the gravity field measurement is obtained; 
     step (4): according to the degree error variance of the gravity field recovery model, calculating a geoid-order error and an accumulative error of the geoid-order error, an gravity-anomaly order error and an accumulative error of the gravity-anomaly order error of the gravity field recovery model; and 
     step (5): summarizing the valid order of the gravity field measurement, the geoid-order error and the accumulative error of the geoid-order error, and the gravity-anomaly order error and the accumulative error of the gravity-anomaly order error, so as to obtain the performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. 
     Preferably, in the step (1), the parameter of the gravity satellite system by low-to-low satellite-to-satellite tracking is selected from but not limited to the group consisting of at least one orbit parameter of the gravity satellite system and at least one load indicator of the gravity satellite system; 
     wherein the orbit parameter of the gravity satellite system is selected from the group consisting of a maximum valid order of the gravity field recovery N max , a gravity-anomaly order error Δ n  of an nth order, a geoid-order accumulative error Δ of the nth order, a gravity-anomaly order error Δg n  of the nth order and a gravity-anomaly accumulative error Δg of the nth order; 
     the parameter of the gravity satellite system is selected from the group consisting of a gravity satellite orbit height h and an included angle θ 0  of satellite-to-satellite geocentric vectors, 
     the load indicator of the gravity satellite system is selected from the group consisting of an inter-satellite range change rate measurement error (Δ{dot over (ρ)}) m , a satellite orbit determining position error (Δr) m , a non-gravitational interference ΔF, an inter-satellite range rate data sampling interval (Δt) Δ{dot over (ρ)} , satellite orbit position data sampling interval (Δt) Δr  and a gravity field measurement service life T. 
     Preferably, the step (2) specifically comprises steps of: 
     establishing an analytic formula of a low-to-low satellite-to-satellite tracking gravity field measurement degree error variance δσ n   2  of a potential coefficient: 
               δσ   n   2     =         1       2   ⁢           ⁢   n     +   1       ⁢       ∑     k   =   0     n     ⁢           ⁢     [         (     δ   ⁢       C   _     nk       )     2     +       (     δ   ⁢       S   _     nk       )     2       ]         =       1       ∑     k   =   0     n     ⁢     [               B   1     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )       +                   B   2     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )       +       B   3     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )               ]         ×     1       2   ⁢   n     +   1       ⁢           2   ⁢     (     n   +   1     )     ⁢     r   0     2   ⁢   n             πμ   2     ⁢     Ta   e       2   ⁢   n     -   2           ⁡     [               -   D     ⁢         (               (     Δ   ⁢           ⁢   F     )     2     ⁢     T   arc   4       36     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   F         +         (     Δ   ⁢           ⁢   r     )     m   2     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   r           )     ⁢       f     δ   ⁢           ⁢   r       ⁡     (   n   )             +                   μ     r   0         ⁢         (               (     Δ   ⁢           ⁢   F     )     2     ⁢     T   arc   2       4     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   F         +         (     Δ   ⁢           ⁢     ρ   .       )     m   2     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢     ρ   .             )     ⁢       f     δ   ⁢           ⁢     ρ   .         ⁡     (   n   )                   ]       2               
wherein:
 
     
       
         
           
             
                 
             
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     wherein δσ n   2  is an degree error variance of a geopotential coefficient of the gravity field recovery model, r 0  is a geocentric range of a satellite, θ 0  is a geocentric vector included angle, α e  is an earth average radius, μ is a product of a gravitation constant and an earth mass, h is the gravity satellite orbit height, T arc  is an integral arc length, ΔF is non-gravitational interference, (Δt) ΔF  is non-gravitational interference data interval, (Δr) m  is a satellite orbit determination position error, (Δt) Δr  is a satellite orbit data sampling interval, (Δ{dot over (ρ)}) m  is an inter-satellite range change rate measurement error, (Δt) Δ{dot over (ρ)}  is an inter-satellite range change rate sampling interval, T is the gravity field measurement service life. 
     Preferably, value of a coefficient K and a phase ξ is respectively: 
     
       
         
           
             
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     Preferably, according to a preferred embodiment, if the gravity satellite system measures non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference measurement interval; if the gravity satellite system suppresses non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference suppression interval. 
     Preferably, the step (3) specifically comprises a step of establishing a maximum valid order N max  of the gravity field measurement, which meets a formula of: 
     
       
         
           
             
               
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     Preferably, the step (4) specifically comprises: 
     establishing the geoid-order error corresponding to an nth order:
 
Δ n   =a   e √{square root over ((2 n+ 1)δσ n   2 )},
 
     and/or 
     establishing the geoid accumulative error corresponding to the nth order: 
     
       
         
           
             
               Δ 
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       2 
                     
                     n 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         Δ 
                         k 
                       
                       ) 
                     
                     2 
                   
                 
               
             
             , 
           
         
       
     
     and/or 
     establishing the gravity-anomaly order error corresponding to the nth order: 
     
       
         
           
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   g 
                   n 
                 
               
               = 
               
                 
                   μ 
                   
                     a 
                     e 
                     2 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     n 
                     - 
                     1 
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           n 
                         
                         + 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       δσ 
                       n 
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
     and/or 
     establishing the gravity-anomaly accumulative error corresponding to the nth order. 
     The method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking of the present invention comprises four parts, which are respectively acquiring at least one parameter of a gravity satellite system by low-to-low satellite-to-satellite tracking, calculating a valid order of the gravity field measurement, determining a geoid-order error and a geoid-order accumulative error, and determining a gravity-anomaly order error and a gravity-anomaly order accumulative error. Acquiring at least one parameter of a gravity satellite system by low-to-low satellite-to-satellite tracking is to obtain parameters such as a satellite orbit height, an inter-satellite distance, an inter-satellite change rate measurement precision, an orbit determination precision and a non-gravitational interference, a measurement data sampling interval and a service period, so as to provide input parameters for the gravity field performance calculation. Calculating a valid order of the gravity field measurement means calculating an order geopotential coefficient error variance of the recovery gravity field model by a parsing and analyzing formula of the gravity field measurement performance according to the gravity satellite system parameters by low-to-low satellite-to-satellite tracking; and then comparing the degree error variance with a geopotential coefficient order error provided by Kaula rule. When the degree error variance is equal to the degree variance, considering that a maximum valid order of the gravity field measurement is obtained. Determining the geoid order error and the accumulative error thereof means obtaining the geoid order error and the accumulative error in a valid range unitizing a geoid order formula according to an error variance of the recovery gravity field model in a previous step. Determining an error of gravity anomaly and an accumulative error of gravity anomaly refers to obtaining the error of gravity anomaly and the accumulative error of gravity anomaly in a valid range utilizing a gravity anomaly calculating formula according to an degree error variance of geopotential coefficient of the recovery gravity field model. 
     An implement step of the method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking comprises following steps of: 
     step (1): acquiring at least one parameter of a gravity satellite system by low-to-low satellite-to-satellite tracking comprising: a satellite orbit height, an inter-satellite distance, an inter-satellite change rate measurement precision, an orbit determination precision and a non-gravitational interference, a measurement data sampling interval and a service period; 
     step (2): according to the parameter of the gravity satellite system by low-to-low satellite-to-satellite tracking, calculating an effect of a measuring error of a gravity satellite load on a power spectrum of a nonspherical perturbation potential of earth gravity, so as to obtain an degree error variance of a potential coefficient in a gravity field recovery model; 
     step (3): comparing the degree error variance of the potential coefficient in the gravity field recovery model with an degree variance of a potential coefficient given by Kaula Rule, wherein with an increase of an order of the gravity field recovery model, the degree error variance gradually increases and the degree variance gradually decreases; when the degree error variance is equal to the degree variance, considering that a maximum valid order of the gravity field measurement is obtained; 
     step (4): according to the degree error variance of the gravity field recovery model, calculating a geoid-order error and an accumulative error of the geoid-order error, an gravity-anomaly order error and an accumulative error of the gravity-anomaly order error of the gravity field recovery model; and 
     step (5): summarizing the valid order of the gravity field measurement, the geoid-order error and the accumulative error of the geoid-order error, and the gravity-anomaly order error and the accumulative error of the gravity-anomaly order error, so as to obtain the performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. 
     The advantages of the present invention are as follows. 
     The present invention establishes parsing relationships between a measurement performance of the gravity field performance by low-to-low satellite-to-satellite tracking and the system parameters. The method of the present invention is capable of evaluating a measurement effect of the gravity field quickly and quantitatively, obtaining a rule of effects of the gravity satellite system parameters on the gravity field measurement performance, so as to avoid shortcoming caused by numerical simulation of the satellite gravity field measurement. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a sketch view of measuring a satellite formation of gravity field by low-to-low satellite-to-satellite tracking. 
         FIG. 2  is a sketch view showing relationships among an inter-satellite amplitude and r 0  and θ 0 . 
         FIG. 3  is a geopotential coefficient degree error variance curve measured in a gravity field under a GRACE system parameter. 
         FIG. 4  is a GRACE gravity field measurement geoid order error and a geoid accumulative error. 
         FIG. 5  is a GRACE gravity field measurement degree error of a gravity anomaly error and an accumulative error of gravity anomaly error. 
     
    
    
     Table 1 shows an inter-satellite range change rate amplitude (m/s) under different orbit height and an inter-satellite geocentric vector included angle. 
     Table 2 shows physical parameters of gravity field measurement by low-to-low satellite-to-satellite tracking. 
     Table 3 shows parameters of the GRACE satellite system. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     These and other objectives, features, and advantages of the present invention will become apparent from the following detailed description, the accompanying drawings, and the appended claims. 
     One skilled in the art will understand that the embodiment of the present invention as shown in the drawings and described above is exemplary only and not intended to be limiting. 
     As shown in  FIGS. 1-5 , a method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking, comprises following steps of: 
     step (1): acquiring at least one parameter of a gravity satellite system by low-to-low satellite-to-satellite tracking; 
     step (2): according to the parameter of the gravity satellite system by low-to-low satellite-to-satellite tracking, calculating an effect of a measuring error of a gravity satellite load on a power spectrum of a nonspherical perturbation potential of earth gravity, so as to obtain an degree error variance of a potential coefficient in a gravity field recovery model; 
     step (3): comparing the degree error variance of the potential coefficient in the gravity field recovery model with an degree variance of a potential coefficient given by Kaula Rule, wherein with an increase of an order of the gravity field recovery model, the degree error variance gradually increases and the degree variance gradually decreases; when the degree error variance is equal to the degree variance, considering that a maximum valid order of the gravity field measurement is obtained; 
     step (4): according to the degree error variance of the gravity field recovery model, calculating a geoid-order error and an accumulative error of the geoid-order error, an gravity-anomaly order error and an accumulative error of the gravity-anomaly order error of the gravity field recovery model; and 
     step (5): summarizing the valid order of the gravity field measurement, the geoid-order error and the accumulative error of the geoid-order error, and the gravity-anomaly order error and the accumulative error of the gravity-anomaly order error, so as to obtain the performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. 
     In the step (1), the parameter of the gravity satellite system by low-to-low satellite-to-satellite tracking is selected from but not limited to the group consisting of at least one orbit parameter of the gravity satellite system and at least one load indicator of the gravity satellite system; 
     wherein the orbit parameter of the gravity satellite system is selected from the group consisting of a maximum valid order of the gravity field recovery N max , a gravity-anomaly order error Δ n  of an nth order, a geoid-order accumulative error Δ of the nth order, a gravity-anomaly order error Δg n  of the nth order and a gravity-anomaly accumulative error Δg of the nth order; 
     the parameter of the gravity satellite system is selected from the group consisting of a gravity satellite orbit height h and an included angle θ 0  of satellite-to-satellite geocentric vectors, 
     the load indicator of the gravity satellite system is selected from the group consisting of an inter-satellite range change rate measurement error (Δ{dot over (ρ)}) m , a satellite orbit determining position error (Δr) m , a non-gravitational interference ΔF, an inter-satellite range rate data sampling interval (Δt) Δ{dot over (ρ)} , satellite orbit position data sampling interval (Δt) Δr  and a gravity field measurement service life T. 
     The step (2) specifically comprises steps of: 
     establishing an analytic formula of a low-to-low satellite-to-satellite tracking gravity field measurement degree error variance δσ n   2  of a potential coefficient: 
               δσ   n   2     =         1       2   ⁢           ⁢   n     +   1       ⁢       ∑     k   =   0     n     ⁢           ⁢     [         (     δ   ⁢       C   _     nk       )     2     +       (     δ   ⁢       S   _     nk       )     2       ]         =       1       ∑     k   =   0     n     ⁢     [               B   1     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )       +                   B   2     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )       +       B   3     ⁡     (       r   0     ,   n   ,   k   ,     θ   0       )               ]         ×     1       2   ⁢   n     +   1       ⁢           2   ⁢     (     n   +   1     )     ⁢     r   0     2   ⁢   n             πμ   2     ⁢     Ta   e       2   ⁢   n     -   2           ⁡     [               -   D     ⁢         (               (     Δ   ⁢           ⁢   F     )     2     ⁢     T   arc   4       36     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   F         +         (     Δ   ⁢           ⁢   r     )     m   2     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   r           )     ⁢       f     δ   ⁢           ⁢   r       ⁡     (   n   )             +                   μ     r   0         ⁢         (               (     Δ   ⁢           ⁢   F     )     2     ⁢     T   arc   2       4     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢   F         +         (     Δ   ⁢           ⁢     ρ   .       )     m   2     ⁢       (     Δ   ⁢           ⁢   t     )       Δ   ⁢           ⁢     ρ   .             )     ⁢       f     δ   ⁢               ⁢   ρ     .         ⁡     (   n   )                   ]       2               
wherein:
 
     
       
         
           
             
                 
             
             ⁢ 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               B 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 A 
                                 n 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   , 
                                   k 
                                   , 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                              
                             
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       n 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               B 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       a 
                                       e 
                                     
                                     
                                       r 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   n 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
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                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
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                                
                               
                                 
                                   
                                     A 
                                     n 
                                   
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                                     ( 
                                     
                                       
                                         n 
                                         - 
                                         1 
                                       
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                                       k 
                                       , 
                                       
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                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                           
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                               ( 
                               
                                 
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                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                           ⁢ 
                           
                              
                             
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       n 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               B 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       a 
                                       e 
                                     
                                     
                                       r 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 4 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   n 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       n 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     e 
                                   
                                   
                                     r 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         + 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                           ⁢ 
                           
                              
                             
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       n 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       A 
                       n 
                     
                     ⁡ 
                     
                       ( 
                       
                         l 
                         , 
                         k 
                         , 
                         
                           θ 
                           0 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       δ 
                       k 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         π 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 lk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 lk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             
                               P 
                               _ 
                             
                             nk 
                           
                           ⁡ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           δ 
                           k 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 2 
                                 , 
                               
                             
                             
                               
                                 k 
                                 = 
                                 0 
                               
                             
                           
                           
                             
                               
                                 1 
                                 , 
                               
                             
                             
                               
                                 k 
                                 ≠ 
                                 0 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                       
                       = 
                       
                         
                           
                             K 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       
                                         θ 
                                         0 
                                       
                                       2 
                                     
                                   
                                   2 
                                 
                                 - 
                                 
                                   4 
                                   3 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 0 
                               
                             
                             
                               r 
                               0 
                               4 
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             
                               f 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                               
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 π 
                               
                               ⁢ 
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           P 
                                           _ 
                                         
                                         nk 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   ⅆ 
                                   θ 
                                 
                               
                             
                           
                           = 
                           
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 
 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   f 
                                   
                                     δ 
                                     ⁢ 
                                     
                                       ρ 
                                       . 
                                     
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     
                                       1 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   
 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                 
                               
                               = 
                               
                                 
                                   a 
                                   e 
                                 
                                 + 
                                 h 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein δσ n   2  is an degree error variance of a geopotential coefficient of the gravity field recovery model, r 0  is a geocentric range of a satellite, θ 0  is a geocentric vector included angle, a e  is an earth average radius, μ is a product of a gravitation constant and an earth mass, h is the gravity satellite orbit height, T arc  is an integral arc length, ΔF is non-gravitational interference, (Δt) ΔF  is non-gravitational interference data interval, (Δr) m  is a satellite orbit determination position error, (Δt) Δr  is a satellite orbit data sampling interval, (Δ{dot over (ρ)}) m  is an inter-satellite range change rate measurement error, (Δt) Δρ  is an inter-satellite range change rate sampling interval, T is the gravity field measurement service life. 
     Value of a coefficient K and a phase ξ is respectively: 
     
       
         
           
             
               K 
               = 
               
                 1.3476 
                 × 
                 
                   10 
                   11 
                 
                 ⁢ 
                 
                   m 
                   2 
                 
               
             
             , 
             
               ξ 
               = 
               
                 - 
                 
                   
                     π 
                     2 
                   
                   . 
                 
               
             
           
         
       
     
     The step (3) specifically comprises a step of establishing a maximum valid order N max  of the gravity field measurement, which meets a formula of: 
     
       
         
           
             
               
                 δσ 
                 N 
                 2 
               
               max 
             
             = 
             
               
                 1 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       N 
                       max 
                     
                   
                   + 
                   1 
                 
               
               ⁢ 
               
                 
                   
                     1.6 
                     × 
                     
                       10 
                       
                         - 
                         10 
                       
                     
                   
                   
                     N 
                     max 
                     3 
                   
                 
                 . 
               
             
           
         
       
     
     The step (4) specifically comprises: 
     establishing the geoid-order error corresponding to an nth order:
 
Δ n   =a   e √{square root over ((2 n+ 1)δσ n   2 )},
 
     and/or 
     establishing the geoid accumulative error corresponding to the nth order: 
     
       
         
           
             
               Δ 
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       2 
                     
                     n 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         Δ 
                         k 
                       
                       ) 
                     
                     2 
                   
                 
               
             
             , 
           
         
       
     
     and/or 
     establishing the gravity-anomaly order error corresponding to the nth order: 
     
       
         
           
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   g 
                   n 
                 
               
               = 
               
                 
                   μ 
                   
                     a 
                     e 
                     2 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     n 
                     - 
                     1 
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                         
                         + 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       δσ 
                       n 
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
     and/or 
     establishing the gravity-anomaly accumulative error corresponding to the nth order. 
     According to a preferred embodiment, if the gravity satellite system measures non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference measurement interval; if the gravity satellite system suppresses non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference suppression interval. 
     Further description of the present invention is illustrated combining with the preferred embodiment. 
     A low-to-low satellite-to-satellite tracking gravity field satellite system comprises two low-orbit satellites to form a following formation along a track. By measuring a range between the two satellites and a change rate thereof, a gravity field of the earth is recovered. In order to make a measurement accuracy of the data as consist as possible, and facilitate controlling satellite attitude and orbit, the orbit of the satellite is generally selected as a circular orbit or a near-circular orbit. And further, in order to meet the requirement of measuring all over the world, the satellite orbit is usually selected as a polar orbit or a near-polar orbit. Thus, in the method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking, an eccentricity of the satellite orbit is supposed to be 0, and an orbit inclination is 90°. 
     1. Energy Conservation Equation of the Gravity Field Measurement by Low-to-Low Tracking 
     As showing in  FIG. 1 , two satellites which form the formation of the gravity field measurement by low-to-low satellite-to-satellite tracking are respectively denoted as A and B. B is a guiding satellite and A is a tracking satellite. An energy conservation exists between A and B as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       A 
                     
                     - 
                     
                       V 
                       B 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               r 
                               . 
                             
                             A 
                             2 
                           
                           - 
                           
                             
                               r 
                               . 
                             
                             B 
                             2 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       ω 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     x 
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       . 
                                     
                                     y 
                                   
                                 
                                 - 
                                 
                                   
                                     r 
                                     y 
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       . 
                                     
                                     x 
                                   
                                 
                               
                               ) 
                             
                             A 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     x 
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       . 
                                     
                                     y 
                                   
                                 
                                 - 
                                 
                                   
                                     r 
                                     y 
                                   
                                   ⁢ 
                                   
                                     
                                       r 
                                       . 
                                     
                                     x 
                                   
                                 
                               
                               ) 
                             
                             B 
                           
                         
                         ] 
                       
                     
                     - 
                     
                       
                         ∫ 
                         
                           t 
                           0 
                         
                         t 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 a 
                                 A 
                               
                               · 
                               
                                 
                                   r 
                                   . 
                                 
                                 A 
                               
                             
                             - 
                             
                               
                                 a 
                                 B 
                               
                               · 
                               
                                 
                                   r 
                                   . 
                                 
                                 B 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                     - 
                     
                       V 
                       
                         t 
                         , 
                         AB 
                       
                     
                     - 
                     
                       E 
                       
                         0 
                         , 
                         AB 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In formula (1), a left of an equal sign is gravitational potential difference of the satellite A and the satellite B. A known gravitational potential spheric harmonics expansion is 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       r 
                     
                     ⁡ 
                     
                       [ 
                       
                         1 
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               2 
                             
                             ∞ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       a 
                                       e 
                                     
                                     r 
                                   
                                   ) 
                                 
                                 n 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         C 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                   + 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     λ 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     wherein (r, θ, λ) is a spherical coordinate of a Earth-Centered Earth-Fixed coordinate system, μ is a gravitational constant, a e  is an earth diameter,  C   nk  and  S   nk  are respectively a cosine value and a sine value of a nth-order k-gravitational coefficient,  P   nk (cos θ) is a fully normalized Associated Legendre polynomials. Gravitational potential V(r, θ, λ) is divided into two parts of a central gravitational potential μ/r and a nonspherical perturbation potential. 
     
       
         
           
             
               
                 
                   
                     V 
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       r 
                     
                     + 
                     
                       R 
                       ⁡ 
                       
                         ( 
                         
                           r 
                           , 
                           θ 
                           , 
                           λ 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       r 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           2 
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 r 
                               
                               ) 
                             
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     C 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                               + 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               
                                 P 
                                 _ 
                               
                               nk 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Thus, a left hand of the formula (1) is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               V 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               
                                 r 
                                 , 
                                 θ 
                                 , 
                                 λ 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               V 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               
                                 r 
                                 , 
                                 θ 
                                 , 
                                 λ 
                               
                               ) 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             μ 
                             
                               r 
                               A 
                             
                           
                           + 
                           
                             
                               R 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               
                                 r 
                                 , 
                                 θ 
                                 , 
                                 λ 
                               
                               ) 
                             
                           
                           - 
                           
                             μ 
                             
                               r 
                               B 
                             
                           
                           - 
                           
                             
                               R 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               
                                 r 
                                 , 
                                 θ 
                                 , 
                                 λ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 μ 
                                 
                                   r 
                                   A 
                                 
                               
                               - 
                               
                                 μ 
                                 
                                   r 
                                   B 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     R 
                                     A 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       r 
                                       , 
                                       θ 
                                       , 
                                       λ 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     B 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       r 
                                       , 
                                       θ 
                                       , 
                                       λ 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     In the formula (1), a first item is a difference of kinetic energy of the satellite A and the satellite B. Setting e is a unit vector of the satellite A directing the satellite B, 
     
       
         
           
             
               
                 
                   e 
                   = 
                   
                     
                       
                         r 
                         AB 
                       
                       
                          
                         
                           r 
                           AB 
                         
                          
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     A difference of kinetic energy of the satellite A and the satellite B is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             r 
                             . 
                           
                           A 
                           2 
                         
                         - 
                         
                           
                             r 
                             . 
                           
                           B 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               B 
                             
                           
                           ) 
                         
                         · 
                         
                           
                             r 
                             . 
                           
                           AB 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               B 
                             
                           
                           ) 
                         
                         · 
                         
                           
                             { 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         . 
                                       
                                       AB 
                                     
                                     · 
                                     e 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 e 
                               
                               + 
                               
                                 [ 
                                 
                                   
                                     
                                       r 
                                       . 
                                     
                                     AB 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             . 
                                           
                                           AB 
                                         
                                         · 
                                         e 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     e 
                                   
                                 
                                 ] 
                               
                             
                             } 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Considering a range between the satellite A and the satellite B is very small, an average velocity thereof is along a line segment of the satellite A and the satellite B, 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               B 
                             
                           
                           ) 
                         
                         · 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     . 
                                   
                                   AB 
                                 
                                 · 
                                 e 
                               
                               ) 
                             
                             ⁢ 
                             e 
                           
                           ] 
                         
                       
                     
                     ⪢ 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               B 
                             
                           
                           ) 
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 r 
                                 . 
                               
                               AB 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       . 
                                     
                                     AB 
                                   
                                   · 
                                   e 
                                 
                                 ) 
                               
                               ⁢ 
                               e 
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     wherein the expression (8) has an approximate expression of: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                               2 
                             
                             - 
                             
                               
                                 r 
                                 . 
                               
                               B 
                               2 
                             
                           
                           ) 
                         
                       
                       ≈ 
                       
                         
                           - 
                           
                             1 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   r 
                                   . 
                                 
                                 A 
                               
                               + 
                               
                                 
                                   r 
                                   . 
                                 
                                 B 
                               
                             
                             ) 
                           
                           · 
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       . 
                                     
                                     AB 
                                   
                                   · 
                                   e 
                                 
                                 ) 
                               
                               ⁢ 
                               e 
                             
                             ] 
                           
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 . 
                               
                               A 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               B 
                             
                           
                           ) 
                         
                         · 
                         
                           ( 
                           
                             
                               ρ 
                               . 
                             
                             ⁢ 
                             e 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     wherein {dot over (ρ)} is a range change rate between the satellite A and the satellite B, which is equal to a projection of {dot over (r)} AB  on a unit vector e. 
     Since orbit of the satellite A and the satellite B are assumed to be circular orbit, and geocentric distances of the satellite A and the satellite B are basically equal, an average value is set to be r 0 , the formula (9) is further expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             r 
                             . 
                           
                           A 
                           2 
                         
                         - 
                         
                           
                             r 
                             . 
                           
                           B 
                           2 
                         
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       - 
                       
                         
                           μ 
                           
                             r 
                             0 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         ρ 
                         . 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In the formula (1), a second item of a right hand of the formula is an energy difference of the satellite A and the satellite B caused by earth rotation, wherein calculation of value indicates that the second item thereof is 4 magnitudes less than a kinetic energy difference, and thus can be ignored in parsing and analyzing. A third item of a right hand of the formula (1) is an energy difference between the satellite A and the satellite B caused by non-gravitational disturbance. In the gravity field measurement, time accumulation of the non-gravitational disturbance causes a result that the satellite orbit shifts a pure gravity orbit, and meanwhile an inter-satellite distance range change rate shifts an inter-satellite range change rate under an effect of pure gravity. An offset of both a pure gravity orbit and a measurement error determine acquiring ability of change rate of the range between the pure-gravity satellite, in such a manner that an effect of the non-gravity disturbance can be introduced in analyzing a pure gravity orbit error and a pure gravity inter-satellite range change rate. A fourth item of the right hand of the item 4 is an energy difference of the satellite A and the satellite B caused by a third body gravitation and a tidal perturbation and etc. The third body gravitation and the tidal perturbation have high-precision model, and are capable of meeting measurement requirements of a static gravity field, and an effect of the third body gravitation and the tidal perturbation can be neglected. A fifth item on a right hand of the formula (1) is an integral constant which has no effect on measurement of the gravity field measurement. 
     When formula (5) and formula (6) are substituted into the formula (1), it can be obtained that 
     
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         A 
                       
                       ⁡ 
                       
                         ( 
                         
                           r 
                           , 
                           θ 
                           , 
                           λ 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         R 
                         B 
                       
                       ⁡ 
                       
                         ( 
                         
                           r 
                           , 
                           θ 
                           , 
                           λ 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             μ 
                             
                               r 
                               0 
                             
                           
                         
                       
                       ⁢ 
                       
                         ρ 
                         . 
                       
                     
                     - 
                     
                       ( 
                       
                         
                           μ 
                           
                             r 
                             A 
                           
                         
                         - 
                         
                           μ 
                           
                             r 
                             B 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       
                         E 
                         
                           0 
                           , 
                           AB 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Performing variation on the formula (11), it is obtained that 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               A 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               B 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                   3 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ρ 
                                 . 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   r 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                               ρ 
                               . 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   μ 
                                   
                                     r 
                                     A 
                                     2 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   
                                     r 
                                     B 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ≈ 
                             
                           ⁢ 
                           
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   μ 
                                   
                                     r 
                                     0 
                                     3 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   ρ 
                                   . 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     r 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   μ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                 ρ 
                                 . 
                               
                             
                             + 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   μ 
                                 
                                 
                                   r 
                                   0 
                                   3 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     r 
                                     B 
                                   
                                   - 
                                   
                                     r 
                                     A 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   r 
                                 
                                 ) 
                               
                             
                           
                         
                         ; 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     wherein δr is a pure gravity orbit position error, comprising an orbit determination error and a pure gravity orbit offset caused by non-gravitational interference; δ{dot over (ρ)} is a pure-gravity orbit inter-satellite range change rate, comprising an inter-satellite range measurement error and an offset of an inter-satellite range change rate caused by non-gravitational interference. A relationship is established between the pure gravity orbit position error δr and the pure-gravity orbit inter-satellite range change rate δ{dot over (ρ)} in the following disclosure. 
     2. Geometrical Relationship of the Difference of an Inter-Satellite Geocentric Range and an Inter-Satellite Range Change Rate. 
     As known that an inter-satellite range change rate of the satellite A and the satellite B is:
 
{dot over (ρ)}= {dot over (r)}   AB   ·e=|{dot over (r)}   AB | cos              {dot over (r)}   AB   ,e                 (13)

     Polar coordinates are adopted for calculating a geometrical relationship between the difference of the inter-satellite geocentric range r B −r A  and the inter-satellite range change rate {dot over (ρ)}. As shown in  FIG. 1 , in a plane determined by r A  and r B , L 2  is set to equally divide an included angle of r A  and r B  and a straight line L 1  is perpendicular to L 2 . The core of the earth is selected as a pole, and the straight line is a polar axis, an anti-clockwise direction is an angle positive position, so a position vector of the satellite A and the satellite B is: 
     
       
         
           
             
               
                 
                   
                     r 
                     A 
                   
                   = 
                   
                     
                       r 
                       A 
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         ⅈ 
                         ⁢ 
                         
                           
                             π 
                             - 
                             
                               θ 
                               0 
                             
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       B 
                     
                     = 
                     
                       
                         r 
                         B 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           ⅈ 
                           ⁢ 
                           
                             
                               π 
                               + 
                               
                                 θ 
                                 0 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     wherein θ 0  is a geocentric vector included angle of the satellite A and the satellite B. 
     A velocity vector of the satellite A and the satellite B is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         r 
                         . 
                       
                       A 
                     
                     = 
                     
                       
                         
                           r 
                           . 
                         
                         A 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           ⅈ 
                           ⁢ 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               - 
                               
                                 θ 
                                 0 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       . 
                     
                     B 
                   
                   = 
                   
                     
                       
                         r 
                         . 
                       
                       B 
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           ⅈ 
                           ⁢ 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               + 
                               
                                 θ 
                                 0 
                               
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Thus, a position vector difference and a velocity vector difference of the satellite A and the satellite B is obtained: 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       AB 
                     
                     = 
                     
                       
                         
                           r 
                           B 
                         
                         - 
                         
                           r 
                           A 
                         
                       
                       = 
                       
                         
                           
                             r 
                             B 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               ⅈ 
                               ⁢ 
                               
                                 
                                   π 
                                   + 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 2 
                               
                             
                           
                         
                         - 
                         
                           
                             r 
                             A 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               ⅈ 
                               ⁢ 
                               
                                 
                                   π 
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       . 
                     
                     AB 
                   
                   = 
                   
                     
                       
                         
                           r 
                           . 
                         
                         B 
                       
                       - 
                       
                         
                           r 
                           . 
                         
                         A 
                       
                     
                     = 
                     
                       
                         
                           
                             r 
                             . 
                           
                           B 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             ⅈ 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 + 
                                 
                                   θ 
                                   0 
                                 
                               
                               2 
                             
                           
                         
                       
                       - 
                       
                         
                           
                             r 
                             . 
                           
                           A 
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               ⅈ 
                               ⁢ 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 2 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Under the non-spherical gravitational perturbation of the earth, a geocentric distance between the satellite A and the satellite B has periodic change. Setting a geocentric distance of the satellite A at a t moment is r 0 , a geocentric distance of the satellite B is r 0 +Δr 0 , an argument of a vector e is calculated as follows. The formula (18) is substituted into the formula (6) to obtain: 
     
       
         
           
             
               
                 
                   
                     e 
                     = 
                     
                       
                         
                           r 
                           AB 
                         
                         
                            
                           
                             r 
                             AB 
                           
                            
                         
                       
                       = 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 ⅈ 
                                 ⁢ 
                                 
                                   
                                     π 
                                     + 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   2 
                                 
                               
                             
                           
                           - 
                           
                             
                               r 
                               0 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 ⅈ 
                                 ⁢ 
                                 
                                   
                                     π 
                                     + 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   2 
                                 
                               
                             
                           
                         
                         ρ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Wherein ρ is an inter-satellite distance of the satellite A and the satellite B. Rearranging the formula (20) to obtain; 
     
       
         
           
             
               
                 
                   
                     e 
                     = 
                     
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                             
                               θ 
                               0 
                             
                             2 
                           
                         
                         + 
                         
                           ⅈ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             0 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                             
                               θ 
                               0 
                             
                             2 
                           
                         
                       
                       ρ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     In such a manner that an argument of e is obtained: 
     
       
         
           
             
               
                 
                   
                     arg 
                     ⁡ 
                     
                       ( 
                       e 
                       ) 
                     
                   
                   = 
                   
                     π 
                     + 
                     
                       
                         arctan 
                         ( 
                         
                           - 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 0 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               tan 
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     An argument of {dot over (r)} AB  is obtained, under a consumption of a circular orbit, a velocity value of the satellite A and the satellite B is approximately: 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       . 
                     
                     A 
                   
                   = 
                   
                     
                       μ 
                       
                         r 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       . 
                     
                     B 
                   
                   = 
                   
                     
                       
                         μ 
                         
                           
                             r 
                             0 
                           
                           + 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               0 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     The formula (23) and the formula (24) are substituted into the formula (19) to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       . 
                     
                     AB 
                   
                   = 
                   
                     
                       
                         
                           r 
                           . 
                         
                         B 
                       
                       - 
                       
                         
                           r 
                           . 
                         
                         A 
                       
                     
                     = 
                     
                       
                         
                           
                             μ 
                             
                               
                                 r 
                                 0 
                               
                               + 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             ⅈ 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                                 + 
                                 
                                   θ 
                                   0 
                                 
                               
                               2 
                             
                           
                         
                       
                       - 
                       
                         
                           
                             μ 
                             
                               r 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               ⅈ 
                               ⁢ 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 2 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Rearrange the formula (25) and obtains: 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       . 
                     
                     AB 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               μ 
                               
                                 r 
                                 0 
                               
                             
                           
                           - 
                           
                             
                               μ 
                               
                                 
                                   r 
                                   0 
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                         
                           θ 
                           0 
                         
                         2 
                       
                     
                     - 
                     
                       
                         ⅈ 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 μ 
                                 
                                   
                                     r 
                                     0 
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                         
                           
                             θ 
                             0 
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Thus, a ratio of an inscriber to a real part of {dot over (r)} AB  is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Im 
                         ⁡ 
                         
                           ( 
                           
                             
                               r 
                               . 
                             
                             AB 
                           
                           ) 
                         
                       
                       
                         Re 
                         ⁡ 
                         
                           ( 
                           
                             
                               r 
                               . 
                             
                             AB 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     μ 
                                     
                                       
                                         r 
                                         0 
                                       
                                       + 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           r 
                                           0 
                                         
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     μ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                               
                                 θ 
                                 0 
                               
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     μ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     μ 
                                     
                                       
                                         r 
                                         0 
                                       
                                       + 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           r 
                                           0 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                               
                                 θ 
                                 0 
                               
                               2 
                             
                           
                         
                       
                       ≈ 
                       
                         - 
                         
                           
                             
                               ( 
                               
                                 
                                   4 
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             tan 
                             ⁢ 
                             
                               
                                 θ 
                                 0 
                               
                               2 
                             
                           
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               0 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Thus, an argument of {dot over (r)} AB  is: 
     
       
         
           
             
               
                 
                   
                     arg 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           . 
                         
                         AB 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       arctan 
                       ( 
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             0 
                           
                         
                         
                           
                             ( 
                             
                               
                                 4 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                 
                               
                               + 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           tan 
                           ⁢ 
                           
                             
                               θ 
                               0 
                             
                             2 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       π 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     From the formula (22) and (28), an included angle between e and {dot over (r)} AB  is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           〈 
                           
                             
                               
                                 r 
                                 . 
                               
                               AB 
                             
                             , 
                             e 
                           
                           〉 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             arg 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   . 
                                 
                                 AB 
                               
                               ) 
                             
                           
                           - 
                           
                             arg 
                             ⁡ 
                             
                               ( 
                               e 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             arctan 
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       4 
                                       ⁢ 
                                       
                                         r 
                                         0 
                                       
                                     
                                     + 
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         r 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 tan 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     0 
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               arctan 
                               ( 
                               
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     0 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           r 
                                           0 
                                         
                                       
                                       + 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           r 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   tan 
                                   ⁢ 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     2 
                                   
                                 
                               
                               ) 
                             
                             - 
                             
                               
                                 3 
                                 ⁢ 
                                 π 
                               
                               2 
                             
                           
                           ; 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     so 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                       〈 
                       
                         
                           
                             r 
                             . 
                           
                           AB 
                         
                         , 
                         e 
                       
                       〉 
                     
                   
                   = 
                   
                     
                       cos 
                       [ 
                       
                         
                           arctan 
                           ( 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 0 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               tan 
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           arctan 
                           ( 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 0 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               tan 
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           
                             3 
                             ⁢ 
                             π 
                           
                           2 
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     Simplifying the formula (30) to obtain: 
     
       
         
           
             
               
                 
                   
                     cos 
                     ⁢ 
                     
                       〈 
                       
                         
                           
                             r 
                             . 
                           
                           AB 
                         
                         , 
                         e 
                       
                       〉 
                     
                   
                   ≈ 
                   
                     
                       
                         3 
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           r 
                           0 
                         
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           r 
                           0 
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                           
                             θ 
                             0 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     From the formula (13) and the formula (31), it is obtained that: 
     
       
         
           
             
               
                 
                   
                      
                     
                       
                         r 
                         . 
                       
                       AB 
                     
                      
                   
                   = 
                   
                     
                       
                         ρ 
                         . 
                       
                       
                         cos 
                         ⁢ 
                         
                           〈 
                           
                             
                               
                                 r 
                                 . 
                               
                               AB 
                             
                             , 
                             e 
                           
                           〉 
                         
                       
                     
                     = 
                     
                       
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               r 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               
                                 tan 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     0 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ρ 
                             . 
                           
                         
                         
                           3 
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             0 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     From the formula (26), it is obtained that: 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         
                           r 
                           . 
                         
                         AB 
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             - 
                             
                               
                                 μ 
                                 
                                   
                                     r 
                                     0 
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       
                         
                           θ 
                           0 
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               
                                 μ 
                                 
                                   
                                     r 
                                     0 
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             θ 
                             0 
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     From the formula (32) and the formula (33), a relationship between an inter-satellite geocentric range difference Δ 0  and an inter-satellite distance change rate {dot over (ρ)} is: 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       B 
                     
                     - 
                     
                       r 
                       A 
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         0 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             2 
                             ⁢ 
                             
                               r 
                               0 
                             
                           
                           
                             3 
                             ⁢ 
                             cos 
                             ⁢ 
                             
                               
                                 θ 
                                 0 
                               
                               2 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             r 
                             0 
                           
                           μ 
                         
                       
                       ⁢ 
                       
                         
                           ρ 
                           . 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     When the formula (34) is substituted into the formula (12), an energy conservation of the gravity field measurement by low-to-low satellite-to-satellite is: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                         A 
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                         B 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             4 
                             
                               3 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           μ 
                           
                             r 
                             0 
                             3 
                           
                         
                       
                       ⁢ 
                       
                         ρ 
                         . 
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       r 
                     
                     - 
                     
                       
                         
                           μ 
                           
                             r 
                             0 
                           
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                         
                           ρ 
                           . 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     In order to calculate a power spectral density of items in a right hand of the formula (35), a change rule of {dot over (ρ)} with time is needed. 
     3. Mathematical Expression of the Inter-Satellite Distance Change Rate 
     A semi-major of the satellite orbit changes periodically due to perturbation affection of the earth gravity, which causes a result that the inter-satellite change rate changes periodically accordingly, wherein J2 perturbation is a main perturbation item, and an effect thereof is classified as a long term item and a short term item. The long term item of the J2 perturbation has no effect on semi-major of the orbit, and an effect of the short term item on the semi-major is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         a 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         
                           J 
                           2 
                         
                         a 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               2 
                               3 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       3 
                                       2 
                                     
                                     ⁢ 
                                     
                                       sin 
                                       2 
                                     
                                     ⁢ 
                                     i 
                                   
                                 
                                 ) 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         a 
                                         r 
                                       
                                       ) 
                                     
                                     3 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           e 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         - 
                                         3 
                                       
                                       / 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               sin 
                               2 
                             
                             ⁢ 
                             
                               
                                 i 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     a 
                                     r 
                                   
                                   ) 
                                 
                               
                               3 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 f 
                                 + 
                                 ω 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     wherein a is a semi-major, i is an orbit inclination, r is an geocentric distance, e is an orbit eccentricity, f is a true anomaly, ω is a perigee argument. Regarding to a low-to-low satellite-to-satellite tracking gravity field satellite, 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           i 
                           ≈ 
                           
                             90 
                             ⁢ 
                             ° 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           a 
                           ≈ 
                           r 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     so the formula (36) is approximately expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         r 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         
                           J 
                           2 
                         
                         r 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               θ 
                             
                             + 
                             ξ 
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     wherein θ is colatitude, ξ is an initial phase angle, so a difference of an instantaneous geocentric distance between two satellites of low-to-low satellite-to-satellite tracking is: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       0 
                     
                   
                   = 
                   
                     
                       3 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           J 
                           2 
                         
                         r 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     So it is known that an amplitude of change of the difference of the geocentric distance of the two satellites is in direct proportion to 
                 sin   ⁢           ⁢     θ   0         r   0       ,         
wherein θ 0  is geocentric vector included angle of the two satellites. Considering the formula (34), it is known that:
 
     
       
         
           
             
               
                 
                   
                     
                       Am 
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           . 
                         
                         ) 
                       
                     
                     ∝ 
                     
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               θ 
                               0 
                             
                             2 
                           
                         
                         
                           r 
                           0 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           μ 
                           
                             r 
                             0 
                           
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         0 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     wherein Am(•) represents an amplitude, a change period of the inter-satellite distance change rate is identical to an orbit period, so: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ρ 
                         . 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       K 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               θ 
                               0 
                             
                             2 
                           
                         
                         
                           r 
                           0 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           μ 
                           
                             r 
                             0 
                           
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         0 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               n 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                             + 
                             ξ 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     wherein K is an undetermined coefficient, n is an angular velocity of the orbit, ξ=−π/2 is an initial phase. Since an orbit of the satellite is a polar orbit, so: 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       . 
                     
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     K 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             θ 
                             0 
                           
                           2 
                         
                       
                       
                         r 
                         0 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         μ 
                         
                           r 
                           0 
                         
                       
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       0 
                     
                     ⁢ 
                     
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             + 
                             ξ 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     In order to obtain a coefficient K, integral of pure gravity orbit is calculated under different orbit height and inter-satellite distance. An amplitude of vibration of the inter-satellite distance change rate is obtained according to a integral orbit, as shown in Table. 1. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
             
             
               
                   
                 0.2° 
                 0.4° 
                 0.6° 
                 0.8° 
                 1.0° 
               
               
                   
               
               
                 200 km 
                 8.52384717 × 10 −2   
                 1.70054302 × 10 −1   
                 2.54878559 × 10 −1   
                 3.39375389 × 10 −1   
                 4.24052520 × 10 −1   
               
               
                 250 km 
                 8.39648169 × 10 −2   
                 1.66955167 × 10 −1   
                 2.49980172 × 10 −1   
                 3.32888860 × 10 −1   
                 4.15999957 × 10 −1   
               
               
                 300 km 
                 8.22614068 × 10 −2   
                 1.63879328 × 10 −1   
                 2.45575028 × 10 −1   
                 3.26890250 × 10 −1   
                 4.08533929 × 10 −1   
               
               
                 350 km 
                 8.07749317 × 10 −2   
                 1.60715522 × 10 −1   
                 2.40566235 × 10 −1   
                 3.20445000 × 10 −1   
                 4.00451832 × 10 −1   
               
               
                 400 km 
                 7.90730919 × 10 −2   
                 1.57561129 × 10 −1   
                 2.35965947 × 10 −1   
                 3.14446876 × 10 −1   
                 3.92894479 × 10 −1   
               
               
                   
               
               
                   
                 1.2° 
                 1.4° 
                 1.6° 
                 1.8° 
                 2.0° 
               
               
                   
               
               
                 200 km 
                 5.08769221 × 10 −1   
                 5.93621975 × 10 −1   
                 6.78081254 × 10 −1   
                 7.62879389 × 10 −1   
                 8.47098944 × 10 −1   
               
               
                 250 km 
                 4.99688820 × 10 −1   
                 5.82412568 × 10 −1   
                 6.65497074 × 10 −1   
                 7.48424917 × 10 −1   
                 8.31446183 × 10 −1   
               
               
                 300 km 
                 4.90153499 × 10 −1   
                 5.71442505 × 10 −1   
                 6.53303618 × 10 −1   
                 7.34316987 × 10 −1   
                 8.15750088 × 10 −1   
               
               
                 350 km 
                 4.80516090 × 10 −1   
                 5.60716101 × 10 −1   
                 6.40541874 × 10 −1   
                 7.20322030 × 10 −1   
                 8.00195000 × 10 −1   
               
               
                 400 km 
                 4.71103131 × 10 −1   
                 5.49539036 × 10 −1   
                 6.28048672 × 10 −1   
                 7.06380954 × 10 −1   
                 7.84346464 × 10 −1   
               
               
                   
               
             
          
         
       
     
                 cos   ⁢           ⁢       θ   0     2         r   0   2       ⁢       μ     r   0         ⁢   sin   ⁢           ⁢     θ   0           
is a horizontal ordinate, an amplitude of the inter-satellite distance change rate in the Table 1 is an ordinate, so as to obtain a straight line shown in  FIG. 2 , which improves that the inter-satellite change rate formula in the expression (42) is rational. Fitting to obtain the coefficient K;
 
 K= 1.3476×10 11  m 2   (43)
 
     The formula (42) is substituted into the formula (35), so as to obtain an energy change equation by low-to-low satellite-to-satellite tracking: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                         A 
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                         B 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         ( 
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                             2 
                           
                           - 
                           
                             4 
                             3 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           μsin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             0 
                           
                         
                         
                           r 
                           0 
                           4 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             + 
                             ξ 
                           
                           ) 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       r 
                     
                     - 
                     
                       
                         
                           μ 
                           
                             r 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                             
                           
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ρ 
                           . 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Based on the formula (44), an degree error variance of the gravity field measurement by low-to-low satellite-to-satellite tracking is established, in such a manner that a valid order of the gravity field measurement, a geoid error and the gravity anomaly error is obtained. 
     4. Degree Error Variance of the Gravity Field Measurement by Low-to-Low Satellite-to-Satellite Tracking 
     From the formula (44), an degree error variance of an nth order is: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             A 
                           
                         
                         - 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             B 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         P 
                         n 
                         2 
                       
                       [ 
                       
                         
                           
                             K 
                             ( 
                             
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     2 
                                   
                                 
                                 2 
                               
                               - 
                               
                                 4 
                                 3 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               μ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 0 
                               
                             
                             
                               r 
                               0 
                               4 
                             
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           r 
                         
                         - 
                         
                           
                             
                               μ 
                               
                                 r 
                                 0 
                               
                             
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             . 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Regarding to the formula (45), it is obtained from a power spectrum definition that: 
     
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         n 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               A 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               B 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             n 
                           
                         
                         n 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   π 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           R 
                                           A 
                                         
                                       
                                       - 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           R 
                                           B 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         , 
                                         λ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                     ⅆ 
                                     θ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     λ 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         _ 
                       
                       nk 
                     
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                           _ 
                         
                         
                           n 
                           ⁢ 
                           
                              
                             k 
                              
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         Q 
                         k 
                       
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Q 
                       k 
                     
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               λ 
                             
                           
                           
                             
                               k 
                               ≥ 
                               0 
                             
                           
                         
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 k 
                                  
                               
                               ⁢ 
                               λ 
                             
                           
                           
                             
                               k 
                               &lt; 
                               0 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     Since the two satellites of low-to-low satellite-to-satellite tracking are both operated along a polar orbit and are provide in an identical orbit plane, so a nonspherical perturbation potential of the two satellites are respectively expressed: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       A 
                     
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       
                         r 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           2 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     C 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                               + 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               
                                 P 
                                 _ 
                               
                               nk 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     θ 
                                     + 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       R 
                       B 
                     
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                         , 
                         λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       
                         r 
                         
                           0 
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           2 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   e 
                                 
                                 
                                   r 
                                   0 
                                 
                               
                               ) 
                             
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     C 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                               + 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 λ 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 nk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
     The formula (49) and the formula (50) are substituted into the formula (46) to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         n 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               A 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               R 
                               B 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               - 
                               n 
                             
                           
                           n 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         4 
                                         ⁢ 
                                         π 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ∫ 
                                         0 
                                         
                                           2 
                                           ⁢ 
                                           π 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           ∫ 
                                           0 
                                           π 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     μ 
                                                     
                                                       r 
                                                       0 
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     
                                                       ∑ 
                                                       
                                                         l 
                                                         = 
                                                         2 
                                                       
                                                       ∞ 
                                                     
                                                     ⁢ 
                                                     
                                                       
                                                         ∑ 
                                                         
                                                           m 
                                                           = 
                                                           0 
                                                         
                                                         l 
                                                       
                                                       ⁢ 
                                                       
                                                         
                                                           
                                                             ( 
                                                             
                                                             
                                                             a 
                                                             e 
                                                             
                                                             
                                                             r 
                                                             0 
                                                             
                                                             
                                                             ) 
                                                           
                                                           l 
                                                         
                                                         ⁢ 
                                                         
                                                           ( 
                                                           
                                                             
                                                             δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             
                                                             
                                                             C 
                                                             _ 
                                                             
                                                             
                                                             l 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             m 
                                                             
                                                             
                                                             ⁢ 
                                                             cos 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             m 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             λ 
                                                             
                                                             + 
                                                             
                                                             δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             
                                                             
                                                             S 
                                                             _ 
                                                             
                                                             
                                                             l 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             m 
                                                             
                                                             
                                                             ⁢ 
                                                             sin 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             m 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             λ 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   [ 
                                                   
                                                     
                                                       
                                                         
                                                           P 
                                                           _ 
                                                         
                                                         
                                                           l 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           m 
                                                         
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           
                                                             ( 
                                                             
                                                             θ 
                                                             + 
                                                             
                                                             θ 
                                                             0 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     
                                                       
                                                         
                                                           P 
                                                           _ 
                                                         
                                                         
                                                           l 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           m 
                                                         
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           θ 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   ] 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           Y 
                                           _ 
                                         
                                         nk 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           θ 
                                           , 
                                           λ 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       θ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       λ 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                       
                       ≈ 
                       
                         
                           
                             1 
                             16 
                           
                           ⁢ 
                           
                             
                               μ 
                               2 
                             
                             
                               r 
                               0 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           ∑ 
                                           
                                             
                                               l 
                                               = 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                             , 
                                             n 
                                             , 
                                             
                                               n 
                                               + 
                                               1 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             δ 
                                             ⁡ 
                                             
                                               ( 
                                               k 
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             ( 
                                             
                                               δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   C 
                                                   _ 
                                                 
                                                 lk 
                                               
                                             
                                             ) 
                                           
                                           ⁢ 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   a 
                                                   e 
                                                 
                                                 
                                                   r 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                             l 
                                           
                                           ⁢ 
                                           
                                             
                                               ∫ 
                                               0 
                                               π 
                                             
                                             ⁢ 
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             P 
                                                             _ 
                                                           
                                                           lk 
                                                         
                                                         ⁡ 
                                                         
                                                           ( 
                                                           
                                                             cos 
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             θ 
                                                             + 
                                                             
                                                             θ 
                                                             0 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       - 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           P 
                                                           _ 
                                                         
                                                         lk 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           θ 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             
                                               P 
                                               _ 
                                             
                                             nk 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               cos 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               θ 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ⅆ 
                                           θ 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             16 
                           
                           ⁢ 
                           
                             
                               μ 
                               2 
                             
                             
                               r 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                               
                               2 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               n 
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 l 
                                                 = 
                                                 
                                                   n 
                                                   - 
                                                   1 
                                                 
                                               
                                               , 
                                               n 
                                               , 
                                               
                                                 n 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                           ∞ 
                                         
                                         ⁢ 
                                         
                                           
                                             ( 
                                             
                                               δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   S 
                                                   _ 
                                                 
                                                 lk 
                                               
                                             
                                             ) 
                                           
                                           ⁢ 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   a 
                                                   e 
                                                 
                                                 
                                                   r 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                             l 
                                           
                                           ⁢ 
                                           
                                             
                                               ∫ 
                                               0 
                                               π 
                                             
                                             ⁢ 
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             P 
                                                             _ 
                                                           
                                                           lk 
                                                         
                                                         ⁡ 
                                                         
                                                           ( 
                                                           
                                                             cos 
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             θ 
                                                             + 
                                                             
                                                             θ 
                                                             0 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       - 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           P 
                                                           _ 
                                                         
                                                         lk 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           θ 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             
                                               P 
                                               _ 
                                             
                                             nk 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               cos 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               θ 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                         ⁢ 
                                         
                                           ⅆ 
                                           θ 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               
                 
                   
                     δ 
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               , 
                             
                           
                           
                             
                               k 
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               k 
                               ≠ 
                               0 
                             
                           
                         
                       
                       ; 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     setting: 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         n 
                       
                       ⁡ 
                       
                         ( 
                         
                           l 
                           , 
                           k 
                           , 
                           
                             θ 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         δ 
                         k 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           π 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   lk 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         + 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   lk 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
                               
                                 P 
                                 _ 
                               
                               nk 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                           ⁢ 
                           
                             ⅆ 
                             θ 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     Then the formula (51) becomes: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             A 
                           
                         
                         - 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             B 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         16 
                       
                       ⁢ 
                       
                         
                           μ 
                           2 
                         
                         
                           r 
                           0 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               a 
                               e 
                             
                             
                               r 
                               0 
                             
                           
                           ) 
                         
                         
                           2 
                           ⁢ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               C 
                                               _ 
                                             
                                             
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                               , 
                                               k 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               C 
                                               _ 
                                             
                                             nk 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             C 
                                             _ 
                                           
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                         1 
                         16 
                       
                       ⁢ 
                       
                         
                           μ 
                           2 
                         
                         
                           r 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                           
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               a 
                               e 
                             
                             
                               r 
                               0 
                             
                           
                           ) 
                         
                         
                           2 
                           ⁢ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               
                                                 S 
                                                 _ 
                                               
                                               
                                                 
                                                   n 
                                                   - 
                                                   1 
                                                 
                                                 , 
                                                 k 
                                               
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               
                                                 S 
                                                 _ 
                                               
                                               nk 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               a 
                                               e 
                                             
                                             
                                               r 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               n 
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               S 
                                               _ 
                                             
                                             
                                               
                                                 n 
                                                 + 
                                                 1 
                                               
                                               , 
                                               k 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           ( 
                                           
                                             
                                               a 
                                               e 
                                             
                                             
                                               r 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             2 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     An expression in a first bracket is calculated, and a quadratic term is launched and an inequality expression is utilized to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               C 
                                               _ 
                                             
                                             
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                               , 
                                               k 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               C 
                                               _ 
                                             
                                             nk 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             C 
                                             _ 
                                           
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           2 
                         
                       
                       ≤ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         C 
                                         _ 
                                       
                                       
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               [ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     A 
                                     n 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                               
                               ] 
                             
                             + 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           _ 
                                         
                                         nk 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 [ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               n 
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       3 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               n 
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 + 
                                                 1 
                                               
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           _ 
                                         
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               a 
                                               e 
                                             
                                             
                                               r 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         4 
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               a 
                                               e 
                                             
                                             
                                               r 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               A 
                                               n 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   - 
                                                   1 
                                                 
                                                 , 
                                                 k 
                                                 , 
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               A 
                                               n 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                                 , 
                                                 k 
                                                 , 
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               a 
                                               e 
                                             
                                             
                                               r 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         3 
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               A 
                                               n 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 n 
                                                 , 
                                                 k 
                                                 , 
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               A 
                                               n 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                                 , 
                                                 k 
                                                 , 
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     if 
                   
                   ⁢ 
                   
                     
 
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       A 
                                       n 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             A 
                                             n 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               n 
                                               , 
                                               k 
                                               , 
                                               
                                                 θ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             - 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     3 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       4 
                                     
                                     ⁢ 
                                     
                                       A 
                                       n 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           + 
                                           1 
                                         
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     3 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     wherein the expression (55) is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           _ 
                                         
                                         
                                           
                                             n 
                                             - 
                                             1 
                                           
                                           , 
                                           k 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       n 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           _ 
                                         
                                         nk 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     A 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         C 
                                         _ 
                                       
                                       
                                         
                                           n 
                                           + 
                                           1 
                                         
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         + 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                   ≤ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     _ 
                                   
                                   
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               B 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     _ 
                                   
                                   nk 
                                 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               B 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     _ 
                                   
                                   
                                     
                                       n 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               B 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   0 
                                 
                                 , 
                                 n 
                                 , 
                                 k 
                                 , 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   ≈ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         
                           
                             ( 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   C 
                                   _ 
                                 
                                 nk 
                               
                             
                             ) 
                           
                           2 
                         
                         · 
                         
                           1 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   B 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   B 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   B 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     Similarly, an expression in a second bracket in the formula (54) is: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           S 
                                           _ 
                                         
                                         
                                           
                                             n 
                                             - 
                                             1 
                                           
                                           , 
                                           k 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       n 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           S 
                                           _ 
                                         
                                         nk 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       A 
                                       n 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         n 
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       
                                         
                                           n 
                                           + 
                                           1 
                                         
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         n 
                                         + 
                                         1 
                                       
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                   ≈ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         
                           
                             ( 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   S 
                                   _ 
                                 
                                 nk 
                               
                             
                             ) 
                           
                           2 
                         
                         · 
                         
                           1 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   B 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   B 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   B 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       0 
                                     
                                     , 
                                     n 
                                     , 
                                     k 
                                     , 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
     The formula (57) and the formula (58) is substituted into the formula (54): 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             A 
                           
                         
                         - 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                             B 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         16 
                         ⁢ 
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         μ 
                         2 
                       
                       
                         r 
                         0 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             a 
                             e 
                           
                           
                             r 
                             0 
                           
                         
                         ) 
                       
                       
                         2 
                         ⁢ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         n 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               
                                 B 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     r 
                                     0 
                                   
                                   , 
                                   n 
                                   , 
                                   k 
                                   , 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 B 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     r 
                                     0 
                                   
                                   , 
                                   n 
                                   , 
                                   k 
                                   , 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 B 
                                 3 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     r 
                                     0 
                                   
                                   , 
                                   n 
                                   , 
                                   k 
                                   , 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         · 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             n 
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           _ 
                                         
                                         nk 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           S 
                                           _ 
                                         
                                         nk 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               ] 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
     A right hand part of the formula (45) is calculated: 
     
       
         
           
             
               
                 
                   
                     D 
                     = 
                     
                       
                         K 
                         ( 
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   θ 
                                   0 
                                 
                                 2 
                               
                             
                             2 
                           
                           - 
                           
                             4 
                             3 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             0 
                           
                         
                         
                           r 
                           0 
                           4 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     according to a definition of power spectral density, a right hand part of the formula (45) is: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                       2 
                     
                     [ 
                     
                       
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               θ 
                               + 
                               ξ 
                             
                             ) 
                           
                         
                         ⁢ 
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         r 
                       
                       - 
                       
                         
                           
                             μ 
                             
                               r 
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                         
                         ⁢ 
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ρ 
                           . 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             n 
                           
                         
                         n 
                       
                       ⁢ 
                       
                         
                           { 
                           
                             
                               D 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   π 
                                 
                                 ⁢ 
                                 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         + 
                                         ξ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   r 
                                   ⁢ 
                                   
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         , 
                                         λ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                     ⅆ 
                                     θ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     λ 
                                   
                                 
                               
                             
                           
                           } 
                         
                         2 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             n 
                           
                         
                         n 
                       
                       ⁢ 
                       
                         
                           { 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   
                                     0 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   π 
                                 
                                 ⁢ 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ρ 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         , 
                                         λ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                     ⅆ 
                                     θ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     λ 
                                   
                                 
                               
                             
                           
                           } 
                         
                         2 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             n 
                           
                         
                         n 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             2 
                             · 
                             
                               D 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 π 
                               
                               ⁢ 
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                                 ⁢ 
                                 
                                   
                                     
                                       Y 
                                       _ 
                                     
                                     nk 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       , 
                                       λ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   ⅆ 
                                   θ 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     λ 
                                   
                                   · 
                                   
                                     1 
                                     
                                       4 
                                       ⁢ 
                                       π 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     μ 
                                     
                                       r 
                                       0 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ∫ 
                                     0 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ∫ 
                                       0 
                                       π 
                                     
                                     ⁢ 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ρ 
                                         . 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             Y 
                                             _ 
                                           
                                           nk 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             θ 
                                             , 
                                             λ 
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θ 
                                       ⁢ 
                                       
                                         ⅆ 
                                         θ 
                                       
                                       ⁢ 
                                       
                                         ⅆ 
                                         λ 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     wherein each item in the formula (61) is respectively: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         
                           - 
                           n 
                         
                       
                       n 
                     
                     ⁢ 
                     
                       
                         { 
                         
                           
                             D 
                             
                               4 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 π 
                               
                               ⁢ 
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                                 ⁢ 
                                 
                                   
                                     
                                       Y 
                                       _ 
                                     
                                     nk 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       , 
                                       λ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   ⅆ 
                                   θ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   λ 
                                 
                               
                             
                           
                         
                         } 
                       
                       2 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           D 
                           
                             4 
                             ⁢ 
                             π 
                           
                         
                         ) 
                       
                       2 
                     
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             r 
                           
                         
                       
                       T 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         n 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           π 
                         
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   nk 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               cos 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                           ⁢ 
                           
                             ⅆ 
                             θ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             n 
                           
                         
                         n 
                       
                       ⁢ 
                       
                         
                           { 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   π 
                                 
                                 ⁢ 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ρ 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       nk 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         θ 
                                         , 
                                         λ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     θ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     λ 
                                   
                                 
                               
                             
                           
                           } 
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             S 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ρ 
                                 . 
                               
                             
                           
                         
                         T 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             π 
                           
                           ⁢ 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       P 
                                       _ 
                                     
                                     nk 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θ 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                               
                               2 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                               ⅆ 
                               θ 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
     wherein S δr  is a power spectral density of a δr of a pure gravity orbit position, S δ{dot over (ρ)}  is a power spectral density of orbit inter-satellite distance change rate error δ{dot over (ρ)} under a pure gravity, T is a total measurement time of the gravity field, in order to express in a convenient way, in the formula (62) and the formula (63), setting: 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         r 
                       
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         π 
                       
                       ⁢ 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 nk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 ) 
                               
                             
                             ] 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             cos 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               θ 
                               + 
                               ξ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         θ 
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ρ 
                           . 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         π 
                       
                       ⁢ 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 nk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 ) 
                               
                             
                             ] 
                           
                           2 
                         
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         θ 
                         ⁢ 
                         
                           
                             ⅆ 
                             θ 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     By (61)-(65) to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           D 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 + 
                                 ξ 
                               
                               ) 
                             
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           r 
                         
                         - 
                         
                           
                             
                               μ 
                               
                                 r 
                                 0 
                               
                             
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             . 
                           
                         
                       
                       ] 
                     
                   
                   ≤ 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 D 
                               
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         r 
                                       
                                     
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       r 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   
                                     0 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         δ 
                                         ⁢ 
                                         
                                           ρ 
                                           . 
                                         
                                       
                                     
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     
                                       δ 
                                       ⁢ 
                                       
                                         ρ 
                                         . 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     It is known that the relationship between a variance of a pure gravity satellite position error βr and the pure gravity satellite distance change rate error β{dot over (ρ)} and the power spectral density is: 
     
       
         
           
             
               
                 
                   
                     σ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       r 
                     
                     2 
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           - 
                           
                             f 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                               
                               , 
                               
                                 ma 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         
                           f 
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               r 
                             
                             , 
                             
                               ma 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             S 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               r 
                             
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           f 
                         
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                         S 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           r 
                         
                       
                       ⁢ 
                       
                         f 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             r 
                           
                           , 
                           
                             ma 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       σ 
                       
                         δ 
                         ⁢ 
                         
                           ρ 
                           . 
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           ∫ 
                           
                             - 
                             
                               f 
                               
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ρ 
                                 
                                 , 
                                 
                                   ma 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   x 
                                 
                               
                             
                           
                           
                             f 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ρ 
                               
                               , 
                               
                                 ma 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               S 
                               
                                 δ 
                                 ⁢ 
                                 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     ρ 
                                   
                                   . 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             f 
                           
                         
                       
                       = 
                       
                         2 
                         ⁢ 
                         
                           S 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ρ 
                               . 
                             
                           
                         
                         ⁢ 
                         
                           f 
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ρ 
                                 . 
                               
                             
                             , 
                             
                               ma 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     wherein the maximum frequency f δr,max  and f δ{dot over (ρ)},max  usually takes a Nyquist frequency, 
     
       
         
           
             
               
                 
                   
                     f 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         r 
                       
                       , 
                       
                         ma 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                     
                   
                   = 
                   
                     1 
                     
                       2 
                       ⁢ 
                       
                         
                           ( 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           r 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             . 
                           
                         
                         , 
                         
                           ma 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           x 
                         
                       
                     
                     = 
                     
                       1 
                       
                         2 
                         ⁢ 
                         
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ρ 
                                 . 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
     
     wherein (Δt) δr  and (Δt) δ{dot over (ρ)}  are respectively data sampling intervals of the pure gravity position error and the pure inter-satellite distance change rate. The formulas (67)-(70) are substituted into the formula (66) to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         n 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   ξ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             r 
                           
                           - 
                           
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ρ 
                               . 
                             
                           
                         
                         ] 
                       
                     
                     ≤ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 D 
                               
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           σ 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             r 
                                           
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             t 
                                           
                                           ) 
                                         
                                       
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         r 
                                       
                                     
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       r 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               
                                 4 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   r 
                                   0 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           σ 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ρ 
                                               . 
                                             
                                           
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             t 
                                           
                                           ) 
                                         
                                       
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ρ 
                                           . 
                                         
                                       
                                     
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     f 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ρ 
                                         . 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           1 
                           
                             8 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             T 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 - 
                                 D 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       
                                         σ 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           r 
                                         
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                     
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       r 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       f 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         r 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 
                                   μ 
                                   
                                     r 
                                     0 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       
                                         σ 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ρ 
                                             . 
                                           
                                         
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                     
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ρ 
                                         . 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       f 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ρ 
                                           . 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
           
         
       
     
     From the formula (45), (59) and (71), an degree error variance of the gravity field measurement by low-to-low satellite-to-satellite tracking is obtained: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         n 
                         2 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             n 
                           
                           ⁢ 
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         C 
                                         _ 
                                       
                                       nk 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       nk 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                               [ 
                               
                                 
                                   
                                     B 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     B 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     B 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         × 
                         
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 n 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     72 
                     ) 
                   
                 
               
             
             ⁢ 
             
               
                 
                   
                     
                       r 
                       0 
                       
                         2 
                         ⁢ 
                         n 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         μ 
                         2 
                       
                       ⁢ 
                       
                         Ta 
                         e 
                         
                           
                             2 
                             ⁢ 
                             n 
                           
                           - 
                           2 
                         
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           - 
                           D 
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   σ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     r 
                                   
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                               
                             
                             ⁢ 
                             
                               
                                 f 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   r 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             μ 
                             
                               r 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   σ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       . 
                                     
                                   
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ρ 
                                   . 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 f 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ρ 
                                     . 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 2 
               
               . 
             
           
         
       
     
     In the formula (72), the pure gravity position error σ δr   2  comprises a pure gravity orbit determination error (Δr) m   2  and a pure gravity orbit offset (Δr) ΔF   2  caused by non-gravitational interference. Similarly, an inter-satellite distance change rate error σ δ{dot over (ρ)}   2  under the pure gravity effect comprises: an inter-satellite distance change rate measurement error (Δρ&amp;) m   2  and an inter-satellite distance change rate measurement error (Δ{dot over (ρ)}) m   2  and an inter-satellite distance change rate offset (Δ{dot over (ρ)}) ΔF   2  caused by non-gravitational interference:
 
σ δr   2 (Δ t ) δr =(Δ r ) ΔF   2 (Δ t ) ΔF +(Δ r ) m   2 (Δ t ) Δr   (73)
 
σ δ{dot over (ρ)}   2 (Δ t ) δ{dot over (ρ)} =(Δ{dot over (ρ)}) ΔF   2 (Δ t ) ΔF +(Δ{dot over (ρ)}) m   2 (Δ t ) Δ{dot over (ρ)}   (74),
 
     wherein (Δt) ΔF  is non-gravitational interference data interface, (Δt) Δr  is a sampling interval of the satellite orbit data, (Δt) Δ{dot over (ρ)}  is a sampling interval of an inter-satellite distance change rate data. It is worth mentioning that: 
     if the gravity satellite system measures non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference measurement interval; if the gravity satellite system suppresses non-gravitational interference by low-to-low satellite-to-satellite tracking, (Δt) ΔF  is a non-gravitational interference suppression interval. 
     A maximum value of the pure gravity orbit position error caused by non-gravitational interference δF and the inter-satellite distance change rate accumulative error, and a maximum value thereof are corresponded to an accumulative error under a uniformly accelerated rectilinear motion, and an average accumulative error is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           r 
                         
                         ) 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         F 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             T 
                             
                               ar 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               T 
                               
                                 ar 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                               
                             
                           
                           ⁢ 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 F 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               F 
                             
                             ) 
                           
                           ⁢ 
                           
                             T 
                             
                               ar 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                             
                             2 
                           
                         
                         6 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ( 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             . 
                           
                         
                         ) 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         F 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             T 
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               rc 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               T 
                               
                                 ar 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                               
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 F 
                               
                               ) 
                             
                             ⁢ 
                             t 
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               F 
                             
                             ) 
                           
                           ⁢ 
                           
                             T 
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               rc 
                             
                           
                         
                         2 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
     
     wherein Tarc is an integral arc length in the recovery gravity field recovery. The formulas (73)-(76) are substituted into the formula (72) to obtain an error variance: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         n 
                         2 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             n 
                           
                           ⁢ 
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         C 
                                         _ 
                                       
                                       nk 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       nk 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                               [ 
                               
                                 
                                   
                                     B 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     B 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     B 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         × 
                         
                           1 
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   r 
                                   0 
                                   
                                     2 
                                     ⁢ 
                                     n 
                                   
                                 
                               
                               
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   μ 
                                   2 
                                 
                                 ⁢ 
                                 
                                   Ta 
                                   e 
                                   
                                     
                                       2 
                                       ⁢ 
                                       n 
                                     
                                     - 
                                     2 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           D 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             
                                                             ( 
                                                             
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             F 
                                                             
                                                             ) 
                                                             
                                                             2 
                                                             
                                                             ⁢ 
                                                             
                                                             T 
                                                             
                                                             ar 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             c 
                                                             
                                                             2 
                                                             
                                                           
                                                           4 
                                                         
                                                         ⁢ 
                                                         
                                                           
                                                             ( 
                                                             
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             t 
                                                             
                                                             ) 
                                                           
                                                           
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             F 
                                                           
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           ( 
                                                           
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             r 
                                                           
                                                           ) 
                                                         
                                                         m 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         
                                                           ( 
                                                           
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             t 
                                                           
                                                           ) 
                                                         
                                                         
                                                           Δ 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           r 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             
                                               
                                                 f 
                                                 
                                                   δ 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   r 
                                                 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 n 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           μ 
                                           
                                             r 
                                             0 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             ( 
                                                             
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             F 
                                                             
                                                             ) 
                                                             
                                                             2 
                                                           
                                                           ⁢ 
                                                           
                                                             T 
                                                             
                                                             ar 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             c 
                                                             
                                                             2 
                                                           
                                                         
                                                         4 
                                                       
                                                       ⁢ 
                                                       
                                                         
                                                           ( 
                                                           
                                                             Δ 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             t 
                                                           
                                                           ) 
                                                         
                                                         
                                                           Δ 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           F 
                                                         
                                                       
                                                     
                                                     + 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     
                                                       
                                                         ( 
                                                         
                                                           Δ 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           
                                                             ρ 
                                                             . 
                                                           
                                                         
                                                         ) 
                                                       
                                                       m 
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       
                                                         ( 
                                                         
                                                           Δ 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           t 
                                                         
                                                         ) 
                                                       
                                                       
                                                         Δ 
                                                         ⁢ 
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         
                                                           ρ 
                                                           . 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                           ⁢ 
                                           
                                             
                                               f 
                                               
                                                 δ 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   ρ 
                                                   . 
                                                 
                                               
                                             
                                             ⁡ 
                                             
                                               ( 
                                               n 
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           2 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       A 
                                       n 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         , 
                                         k 
                                         , 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         A 
                                         n 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             - 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     3 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     B 
                                     3 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         0 
                                       
                                       , 
                                       n 
                                       , 
                                       k 
                                       , 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             e 
                                           
                                           
                                             r 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       4 
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           e 
                                         
                                         
                                           r 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               - 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           A 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               n 
                                               + 
                                               1 
                                             
                                             , 
                                             k 
                                             , 
                                             
                                               θ 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         e 
                                       
                                       
                                         r 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   3 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         A 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             n 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                           , 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       n 
                     
                     ⁡ 
                     
                       ( 
                       
                         l 
                         , 
                         k 
                         , 
                         
                           θ 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       δ 
                       k 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         π 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 lk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 lk 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             
                               P 
                               _ 
                             
                             nk 
                           
                           ⁡ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       δ 
                       k 
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               2 
                               , 
                             
                           
                           
                             
                               k 
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               k 
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     D 
                     = 
                     
                       
                         
                           K 
                           ( 
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     0 
                                   
                                   2 
                                 
                               
                               2 
                             
                             - 
                             
                               4 
                               3 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             μ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               0 
                             
                           
                           
                             r 
                             0 
                             4 
                           
                         
                       
                       &lt; 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               f 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 r 
                               
                             
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               n 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 π 
                               
                               ⁢ 
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           P 
                                           _ 
                                         
                                         nk 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       θ 
                                       + 
                                       ξ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   ⅆ 
                                   θ 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               2.3562 
                               ⁢ 
                               n 
                             
                             + 
                             1.1781 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               3 
                               8 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             π 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         f 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ρ 
                         
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             π 
                           
                           ⁢ 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       P 
                                       _ 
                                     
                                     nk 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θ 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                               2 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                               ⅆ 
                               θ 
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             n 
                             + 
                             
                               1 
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         π 
                       
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       r 
                       0 
                     
                     = 
                     
                       
                         a 
                         e 
                       
                       + 
                       h 
                     
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       K 
                       = 
                       
                         1.3476 
                         × 
                         
                           10 
                           11 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           m 
                           2 
                         
                       
                     
                     , 
                     
                       ξ 
                       = 
                       
                         - 
                         
                           
                             π 
                             2 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
     
     Physical parameters in the formulas mentioned above are shown in Table 2. 
     
       
         
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                 δσ n   2   
                 degree error variance 
                 (δ C   nk , δ S   nk ) 
                 geopotential coefficient 
               
               
                   
                   
                   
                 error 
               
               
                 r 0   
                 Satellite geocentric 
                 θ 0   
                 geocentric vector included 
               
               
                   
                 range 
                   
                 angle of two satellite 
               
               
                 a e   
                 earth average radius 
                 μ 
                 gravitation constant of earth 
               
               
                 h 
                 satellite orbit height 
                 T arc   
                 integral arc length 
               
               
                 ΔF 
                 non-gravitational 
                 (Δt) ΔF   
                 non-gravitational 
               
               
                   
                 interference 
                   
                 interference data interval 
               
               
                 (Δr) m   
                 satellite orbit 
                 (Δt) Δr   
                 satellite orbit data sampling 
               
               
                   
                 determining position 
                   
                 interval 
               
               
                   
                 error 
                   
                   
               
               
                 (Δρ) m   
                 inter-satellite range 
                 (Δt) Δρ   
                 inter-satellite distance 
               
               
                   
                 change rate 
                   
                 change rate sampling 
               
               
                   
                 measurement error 
                   
                 interval 
               
               
                 T 
                 gravity field 
                   
                   
               
               
                   
                 measurement service 
                   
                   
               
               
                   
                 life 
               
               
                   
               
             
          
         
       
     
     5. Performance Calculation of the Gravity Field Measurement by Low-t-Low Satellite-to-Satellite Tracking 
     From the error variance, a valid order of the gravity field measurement, a geoid error and a gravity anomaly error is determined. According to Kaula standard, an degree variance of the earth gravity field model is: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       n 
                       2 
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   C 
                                   _ 
                                 
                                 nk 
                                 2 
                               
                               + 
                               
                                 
                                   S 
                                   _ 
                                 
                                 nk 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       ≈ 
                       
                         
                           1 
                           
                             
                               2 
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             1.6 
                             × 
                             
                               10 
                               
                                 - 
                                 10 
                               
                             
                           
                           
                             n 
                             3 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
           
         
       
     
     Wherein the degree error variance δσ n   2  is an increasing function of an nth order gravity field model, degree variance σ n   2  is a decreasing function of n. With the increase of n, when δσ n   2  is equal to σ n   2 , a maximum valid order N max  of the gravity field recovery is considered to be reached, 
     
       
         
           
             
               
                 
                   
                     i 
                     . 
                     e 
                     . 
                   
                   , 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         
                           N 
                           
                             ma 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                         2 
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             2 
                             ⁢ 
                             
                               N 
                               
                                 ma 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                             
                           
                           + 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             1.6 
                             × 
                             
                               10 
                               
                                 - 
                                 10 
                               
                             
                           
                           
                             N 
                             
                               ma 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                             3 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
           
         
       
     
     From the formula (87), a geoid-order error corresponding to an nth order is 
     obtained:
 
Δ n   =a   e √{square root over ((2 n+ 1)δσ n   2 )}  (88);
 
     so a gravity-anomaly order error corresponding to the nth order is: 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             2 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               Δ 
                               k 
                             
                             ) 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
     From the formula (87), a gravity-anomaly order error corresponding to the nth order is: 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         n 
                       
                     
                     = 
                     
                       
                         μ 
                         
                           a 
                           e 
                           2 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           n 
                           - 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 n 
                               
                               + 
                               1 
                             
                             ) 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             σ 
                             n 
                             2 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
           
         
       
     
     and further, a gravity-anomaly accumulative error corresponding to the nth order is: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     g 
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             2 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 g 
                                 k 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
     Since the GRACE gravity satellite adopts a gravity field measurement principle by low-to-low satellite-to-satellite tracking, according to system parameters of the GRACE satellite, measurement performances of the GRACE gravity field can be calculated by the method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. And then the measurement performance of GRACE is compared with a GRACE gravity measurement value simulated result, so as to verify validity of the present invention. 
     It is known that parameters of GRACE gravity satellite system are shown in Table. 3. The degree error variance, geoid error and gravity anomaly of the gravity field measurement are obtained by the method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking. As shown in  FIGS. 3-5 , a valid order of GRACE gravity field measurement by the method of the present invention is 146, which is corresponding to a geoid accumulative error of 11.73 cm and a gravity anomaly accumulative error is 2.51 mGal, which is basically consist with a simulated result of a GRACE gravity field measurement performance value, in such a manner that the method for parsing and calculating performance of satellite gravity field measurement by low-to-low satellite-to-satellite tracking of the present invention is valid. 
     
       
         
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
             
             
               
                 satellite orbit height 
                 480 
                 km 
                 inter-satellite 
                 220 km 
               
               
                   
                   
                   
                 distance 
                   
               
               
                 satellite orbit 
                 5 
                 cm 
                 satellite orbit data 
                  60 s 
               
               
                 determination error 
                   
                   
                 sampling interval 
                   
               
               
                 inter-satellite 
                 1.0 × 10 −6   
                 m/s 
                 measurement data 
                  60 s 
               
               
                 distance 
                   
                   
                 sampling interval 
                   
               
               
                 change rate 
                   
                   
                   
                   
               
               
                 measurement error 
                   
                   
                   
                   
               
               
                 non-gravitational 
                 1.0 × 0 −10   
                 m/s 2   
                 service life designed 
                  5 year 
               
               
                 interference 
                   
                   
                   
                   
               
               
                 measurement error 
               
               
                   
               
             
          
         
       
     
     It will thus be seen that the objects of the present invention have been fully and effectively accomplished. Its embodiments have been shown and described for the purposes of illustrating the functional and structural principles of the present invention and is subject to change without departure from such principles. Therefore, this invention includes all modifications encompassed within the spirit and scope of the following claims.