# GPS Satellites Orbits: Resonance

Authors: Luiz Danilo Damasceno Ferreira; RodolphoVilhena de Moraes
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The effects of perturbations due to resonant geopotential harmonics on the semimajor axis of GPS satellites are analyzed. For some GPS satellites, secular perturbations of about 4 m/day can be obtained by numerical integration of the Lagrange planetary equations considering in the disturbing potential the main secular resonant coefficients. Amplitudes for long-period terms due to resonant coefficients are also exhibited for some hypothetical satellites orbiting in the neighborhood of the GPS satellites orbits. The results are important to perform orbital maneuvers of GPS satellites such that they stay in their nominal orbits. Also, for the GPS satellites that are not active, the long-period effects due to the resonance must be taken into account in the surveillance of the orbital evolutions of such debris.


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## Body

## 1. Introduction

The period of the orbits of the GPS satellites is about 12 hours, and the main perturbations acting on their orbits are caused by the nonuniform distribution of the Earth’s mass, by the lunar and solar gravitational attractions and by the solar radiation pressure. In this paper, it is analyzed just some perturbations due to resonant terms of the geopotential coefficients. The resonance considered here is the 2 : 1 commensurability between the orbital period of the GPS satellites and the period of the Earth’s rotation. As it is pointed out by Hugentobler [1] the resonance leads a daily drift rate in semimajor axis of up to 7 m/day.Lagrange planetary equations, describing the temporal variation of the orbital elements, are used here to analyze the orbital perturbations of the GPS satellites under the influence of resonant coefficients.


## 2. Disturbing Potential

The geopotential acting on Earth’s artificial satellite can be expressed as [2](2.1)R=GMa∑n=2∞∑m=0n(aea)n(Cnmcosmλ+Snmsinmλ)Pnm(sinφ).HereGM is the geogravitational constant, r is geocentric distance of the satellite, ae is the semimajor axis of the adopted Earth’s ellipsoid, a is the orbital semimajor axis, Cnm andSnm are the spherical harmonic coefficients. Pnm denotes the associated Legendre functions, φ,λ are the satellite’s geocentric latitude and the longitude, nand m are, respectively, the degree and order of the harmonic coefficients.The geocentric distance, the latitude, and the longitude of the satellite can be expressed in terms of the orbital elements and (2.1) becomes [2, 3]
(2.2)R=∑Rnmpq=GMa∑n=2∞∑m=0n(aea)n∑p=0nFnmpq(i)∑q=-∞∞Gnpq(e)Snmpq(ω,Ω,M,θT),
where Fnmpq(i) and Gnpq(e) are, respectively, functions of the satellite’s orbital inclination i and eccentricity e, Ωrepresents the right ascension of the orbital ascending node, ω is the argument of perigee, Mis the mean anomaly, and θT is the Greenwich sidereal time. The function Snmpq(ω,Ω,M,θT) can be expressed as
(2.3)Snmpq(ω,Ω,M,θT)=[(CnmSnmn)cosA+(CnmSnmn)sinA]n-m,oddn-m,even,
and the argument A is given by
(2.4)A=(n-2p)ω+(n-2p+q)M+m(Ω-θT).The functionsFnmpq(i)and Gnpq(e) are presented as tables [2] and can be adapted to be used in computers.


## 3. Resonance in GPS Satellites

Resonance is associated with small divisors. For artificial Earth satellites whose orbital periods are in commensurability with the period of the Earth’s rotation resonance can occur when [1, 3, 4](3.1)Ȧ=(n-2p)ω̇+(n-2p+q)Ṁ+m(Ω̇-θ̇T)≅0,
where θ̇T is the Earth’s sidereal rotation.For the GPS satellites where the commensurability is 2 : 1, there are bounds among the parametersn,m,p,q.Taking into account thatω̇,Ω̇≪θ̇T, (3.1) can be put as(3.2)(n-2p+q)Ṁ-mθ̇T=0,
since n̅=Ṁ≅2θ̇T, we get(3.3)(n-2p+q)2≅m.Considering harmonics of order and degree up to order 4, Table1, presents some values for the parameters satisfying the resonance condition for the GPS satellites.Table 1
Resonant parameters.
DegreenOrdermpq3210441022114221Taking into account in the summations the conditions(n-m)even and (n-m)odd and putting [5](3.4)Cnm=Knmcosmλnm,Snm=Knmsinmλnm,
where(3.5)Knm=Cnm2+Snm2,λnm=1mtan-1(SnmCnm),
equation (2.2) can be written in general form as [1](3.6)Rnmpq=GMa(aea)nFnmp(i)Gnpq(e)(cosAsinA)n-m,oddn-m,evenKnm,
where(3.7)A=(n-2p)ω+(n-2p+q)M+m(Ω-θT-λnm).In order to analyze the orbital perturbations of GPS satellites due to the resonant coefficients presented in (Table1), the Lagrange planetary equations will be used. However, here this analysis will be concentrated initially in the secular effects of such perturbation on the orbital semimajor axis. Therefore, we have [2](3.8)dadt=2n̅a∂Rnmpq∂M,
withn̅ being the  mean motion of the satellite.Special care must be taken to compute∂Rnmpq/∂M. In fact, the parameters n,m,p,q given by Table 1 must be considered and each set of them gives a unique solution for (3.8).A numerical integration of (3.8) was performed for a period of one day using IGS/POE (The International GNSS Service Precise Orbital Ephemeredes) given on June, 07, 2007. Transformations were performed to get the corresponding orbital elements.Figures1, 2, 3, and 4 show the daily variations of the semimajor axis a for all GPS satellites.Figure 1
Drift rates in semimajor axis due to the resonant geopotential coefficient 32.Figure 2
Drift rates in semimajor axis due to the resonant geopotential coefficient 44.Figure 3
Drift rates in semimajor axis due to the resonant geopotential coefficient 22.Figure 4
Drift rates in semimajor axis due to the resonant geopotential coefficient 42.Figure1 gives the variation due to the coefficients C32,S32, which are responsible by the greatest variation in the semimajor axis that, in this case, corresponds to the satellite PRN 06. Figures 2, 3, and 4 present, respectively, the daily variations as function of the coefficients C44,S44,C22,S22,C42,S42. Figure 5 represents the total daily variation due to the above considered coefficients.Figure 5
Drift rates in semimajor axis due to all resonant geopotential coefficients, Table1.Table2 shows the maxima and minima values for the daily variation of the semimajor axis according to the resonant coefficients.Table 2
Maximum values.
Resonants coefficientsMaximum drift rates m/dayC32,S32−4.0C44,S441.5C22,S221.5C42,S420.002Figures6 and 7 show the semimajor axis variation for the satellites PRN 02 and PRN 06 for a time interval of 200 days and taking into account the harmonic coefficients C32,S32. During this period and for the considered initial condition, it can be observed variations of about 600 m and 680 m, respectively.Figure 6
Semimajor axis variation (PRN 02) due to the resonant geopotential coefficient 32, for a time interval of 200 days.Figure 7
Semimajor axis variation (PRN 06) due to the resonant geopotential coefficient 32, for a time interval of 200 days.


## 4. Long-Period Perturbations

The effects of the resonance are enhanced when long periods are considered. Table3 and Table 4, Figures 8, 9, 10, 11, 12, and 13 show some simulations using hypothetical satellites and a particular method to study effects of resonance on the orbits of artificial satellites [6–9].Table 3
Amplitude and period of perturbations due to the 2 : 1 resonance:J2+J22.
Orbital elementsAmplitudePeriodao=26561.770kmei∆amax∆emax∆imax(°)t(days)ao + 0.4300.014°12 km0.0148.7× 10-42500ao + 0.7180.014°12.5 km0.01458.6× 10-47000ao + 0.2900.014°4.5 km0.00653.6× 10-42000ao + 0.5000.014°3 km0.00452.8× 10-41000ao −2.50.014°6.5 km0.00854× 10-41700ao −50.014°1.4 km0.00231.3× 10-6500ao0.0555°1.8 km0.0052× 10-33200ao + 0.7180.00555°2.75 km0.0062.8× 10-33900ao −1.7700.00555°1.2 km0.00451.4× 10-32000ao + 5.1290.00555°900 m0.003310-31500ao + 3.7290.00555°1.8 km0.00551.9× 10-32500ao0.0563.4°7.5 km0.00310.8× 10-31000ao −20.0563.4°4 km0.00155.7× 10-31000ao0.0587°7 km0.003713.9×10-32000ao −20.0587°3.5 km0.00210.3× 10-31900Table 4
Amplitude and period of perturbations due to the 2 : 1 resonance:J2 + J32.
Orbital elementsAmplitudePeriodao=26561.770kmei∆amáx∆emáx∆imáx(°)t (days)ao −3.770.014°120 m0.00041.78× 10-3500ao + 0.7180.014°245 m0.00093.9×10-31000ao + 0.9290.014°110 m0.000451.78× 10-3500ao + 50.014°100 m0.00031.15× 10-3250ao −2.50.0054°100 m0.00094.6× 10-5500ao + 0.4290.0054°700 m0.0062.4 x 10-41500ao + 0.9290.00555°1.9 km0.00856.8× 10-416000ao + 1.730.00555°250 m0.00171.13× 10-316000ao + 50.00555°30 m0.000552.3× 10-41500ao −3.770.00555°47 m0.000353.43× 10-5500ao −0.4290.00587°110 m0.000852.29× 10-41800ao −2.230.00587°45 m0.000358.59× 10-5800ao −3.770.00587°20 m0.000144× 10-5500Figure 8
∆aversus time, e = 0.005, i= 55°.Figure 9
Eccentricity versus time,e = 0.005.Figure 10
∆a versus time, e = 0.005, i= 55°, a0 = 26561.18 km.Figure 11
∆aversus time, e = 0.005, i= 55°, considering the harmonics J20 and J32.Figure 12
Time variation of semimajor axis considering the harmonicsJ20, J22, and J3,2.Figure 13
Time variation of the semimajor axis considering the harmonicsJ20 and J22 for e = 0.005, i= 4°.Table3 contains the amplitude and period of the variations of orbital elements for hypothetical satellites considering low eccentricity, small and high inclination, and the influence of the harmonics J20 and J22Figures8 and 9 represent, respectively, the temporary variation of the semimajor axis and of the eccentricity of artificial satellites of the GPS type in the neighborhood of the 2 : 1 resonance region when the harmonics J20 and J22 are considered. By Figure 10, assuming several values for the semimajor axes in the neighborhood of the 2 : 1 resonance, it can be observed distinct behavior for their temporary variations.Figure11 represents the temporary variation for the semimajor axis in the neighborhood of the resonance 2 : 1 considering the influence of the harmonics J20 and J32 for a satellite of the GPS type with inclination of about 55°.Table4 gives the influence of the resonance due to the harmonics J20 and J32 considering different inclinations, including that of the GPS type satellite. It can be observed that the amplitudes of the variations are smaller when compared with those when the harmonics J20 and   J22 are taken into account. The more the satellites approach the region that was defined as a resonant region, the more the variations increase.Figure12 represents the temporary variation of the semimajor axis of a satellite of the GPS type considering resonance due to the harmonics J20, J22, and J32.Figure13 represents the time variation of the orbital semimajor axis of a satellite with eccentricity about 0.01 and inclination about 4° orbiting in a region near the 2 : 1 resonance. It is remarkable the oscillation of the semimajor axis in the region between by a = 26560.0 km and a = 26562.48 km. For instance, a variation of about 10 m in the initial semimajor axis  (a = 26562.48 km) brings up a variation of about more than 10 km in its amplitude in a period of about 2000 days.


## 5. Conclusions

From the obtained results it can be seen that for the GPS satellites no negligible perturbations are provoked by resonant tesseral coefficients. The daily variation of the semimajor axis due to theC32,S32 is less than 4 m/day and those values found for a time interval of 200 days (~680 m) are compatible with the results presented by specialized literature (~7 m/day and ~1200 m) about this subject [1].Taking into account the total effects of daily resonant perturbations, it can be observed that for some GPS satellites these effects are enhanced and for another satellites are attenuated but all of these effects are smaller than the amplitudes mentioned above. It was shown also that the effects of the resonance are very important for the analysis of long-period behavior of the GPS satellites orbits.An important aspect to be considered is the necessity to perform orbital maneuvers of GPS satellites in such way that they stay in their nominal orbits. Also, for the GPS satellites that are not active, the long-term effects due to the resonance must be taken into account in the surveillance of the orbital evolutions of such debris.



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*Source: 347835-2009-12-22.xml*