## 3 Modelling of Three-Phase Orbital Manoeuvring Method
### Orbital Manoeuvre Parsing  
As introduced in Section 3.1, the three-phase manoeuvre is based on perturbations to achieve desired changes in the satellite orbits. Satellite perturbations will directly cause long-term changes in the RAAN and AoL. Therefore, analytical expressions are created by using RAAN and AoL as functions of the total time required for manoeuvring and the total velocity change (\(\Delta V\)). In order to create a complete analytic formula that can be solved quickly, it is necessary to consider the long-term orbit disturbance, which mainly includes four aspects: Earth's non-spherical gravitational field, additional gravitational field, atmospheric drag, and disturbance caused by solar radiation [25]. In order to create a better analytical expression, this paper assumes that the satellite thrust offsets the influence of atmospheric drag in the first phase of manoeuvring, and the influence of atmospheric drag is considered in the third phase of manoeuvring. In addition, other disturbances are ignored since they are far smaller than atmospheric drag disturbances. The analytic expression obtained is established as follows:  
\[\Omega_{total}=\int_{\overline{a}_{0}}^{\overline{a}_{1}}\frac{d\Omega}{da}da+ \int_{t_{1}}^{t_{2}}\frac{d\Omega}{dt}dt+\int_{\overline{a}_{2}}^{\overline{a}_ {3}}\frac{d\Omega}{da}da \tag{14}\]  
\[u_{total}=\int_{\overline{a}_{0}}^{\overline{a}_{1}}\frac{du}{da}da+\int_{t_{1 }}^{t_{2}}\frac{du}{dt}dt+\int_{\overline{a}_{2}}^{\overline{a}_{3}}\frac{du} {da}da \tag{15}\]  
\(\Omega_{total}\) and \(u_{total}\) are representative of the total change in RAAN and AOL:  
\[\frac{d\Omega}{da}=-\frac{3\overline{m}^{\prime}R_{e}^{2}f_{2}}{4\overline{a} ^{2}A}\cos i \tag{16}\]  
\[\frac{d\Omega}{dt}=-\frac{3\overline{n}R_{e}^{2}f_{2}}{2\overline{a}^{2}}\cos(i) \tag{17}\]  
\[\frac{du}{da}=\frac{\overline{m}^{\prime}}{2A}\bigg{(}1+\frac{3R_{e}^{2}f_{2}} {4\overline{a}^{2}}\Big{(}4-5\sin^{2}i\Big{)}\bigg{)} \tag{18}\]  
\[\frac{du}{dt}=\overline{n}+\frac{3\overline{n}R_{e}^{2}f_{2}}{4\overline{a}^{2 }}\Big{(}4-5\sin^{2}(i)\Big{)} \tag{19}\]  
Equations (17) and (19) are the expressions for RAAN and AoL over time for the second stage. Equations (16) and (18) are the expressions of RAAN and AoL as a function of the satellite half-length axis for the first and third phases, respectively, where \(f_{2}\) is the second zonal harmonic coefficient of the gravitational coefficient of the Earth, \(R_{e}\) is the mean radius of the Earth, and \(\overline{a}\) is the mean semi-major axis.  
The first stage manoeuvre time is established as follows:  
\[t_{1}=\frac{\sqrt{\mu}\Big{(}a_{0}^{5/2}\big{\{}20a_{1}^{2}+3f_{2}R_{e}^{2} \big{[}2-3\sin^{2}i\big{]}\big{\}}+3a_{1}^{5/2}f_{2}R_{e}^{2}\big{[}3\sin^{2}i -2\big{]}-20a_{1}^{5/2}a_{0}^{2}\Big{)}}{20a_{1}^{5/2}a_{1}^{5/2}A} \tag{20}\]  
Figure 2: Schematic diagram of the three-phase orbital manoeuvre. a\({}_{0}\): initial orbit. a\({}_{1}\): transfer orbit. a\({}_{2}\): final orbit.  
The speed change of the first stage manoeuvre is established as follows:  
\[\Delta V_{1}=\left|\sqrt{\frac{\mu}{a_{1}}}-\sqrt{\frac{\mu}{a_{0}}}\right| \tag{21}\]  
\(\mu\) is the standard gravity parameter of the Earth. The maneuvering time and velocity change of the third stage are obtained by replacing \(a_{0}\) and \(a_{1}\) with \(a_{2}\) and \(a_{3}\) in the formula of manoeuvring time and velocity change of the first stage.