# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 2 Orbit Dynamics
### Operational Orbit and Coverage Analysis  
A mission devoted to telecommunications requires to cover the target region as long as possible. In particular, if the region of interest is located at high latitudes, a quasi-synchronous, elliptic Mars orbit, similar to an Earth Molinya orbit, is an interesting option. The apoapse must be sufficiently high to increase the duration of the visibility intervals. To do this, quasi-synchronism is enforced at apoapse, and this means that the satellite footprint at apoapse has the same velocity toward East as the corresponding point on the Martian surface. Moreover, desirable features of the orbit are (a) repetitiveness, which implies predictability of the performance, and (b) invariance with respect to the \(J_{2}\) perturbation. For the purpose of constellation design, the solar radiation pressure and the third body gravitational pull due to the Sun are negligible. In contrast, the Mars oblateness (\(J_{2}\) zonal harmonic) produces significant effects on the orbit elements and is included in the model. The quasi-synchronism condition can be expressed as  
\[\frac{v_{E}}{r}R_{M}=R_{M}\omega_{M}\cos\phi \tag{14}\]  
where \(\phi\) denotes the instantaneous latitude, \(\cos\phi=\sqrt{1-\sin^{2}\omega\sin^{2}i}\), whereas \(v_{E}(=v\cos\zeta=v\cos i/\cos\phi)\) is the East component of velocity. Using the expressions for the spacecraft radius \(r\) and velocity magnitude \(v\) at apoapse, Equation (14) leads to the following relation between the semimajor axis \(a\) and the other orbit elements \(e,\ i,\ \omega\):  
\[a=\left\{\frac{\cos i}{\omega_{M}\Big{[}1-(\sin\omega\sin i)^{2}\Big{]}}\sqrt{ \frac{\mu(1-e)}{(1+e)^{3}}}\right\}^{\frac{2}{3}} \tag{15}\]  
Moreover, it is convenient that the orbit has repeating ground track [11]. This means that the spacecraft completes a number \(N_{t}\) of nodal orbital periods every \(m\) nodal days. The repetitiveness property is studied by considering the mean effect of the \(J_{2}\) zonal harmonic. On average, the Martian oblateness yields the following time derivatives [16; 20] of the right ascension of the ascending node (RAAN) \(\Omega\), argument of perigee \(\omega\), and mean anomaly \(M\):  
\[\dot{\Omega}(a,e,i)=-\frac{3}{2}J_{2}\frac{R_{M}^{2}\sqrt{\mu}}{a^{3.5}{(1-e^ {2})}^{2}}\cos i \tag{16}\]  
\[\dot{\omega}(a,e,i)=\frac{3}{2}J_{2}\frac{R_{M}^{2}\sqrt{\mu}}{a^{3.5}{(1-e^{ 2})}^{2}}\Big{(}2-\frac{5}{2}\sin^{2}i\Big{)} \tag{17}\]  
\[\dot{M}(a,e,i)=\frac{3}{2}J_{2}\frac{R_{M}^{2}\sqrt{\mu}}{a^{3.5}{(1-e^{2})}^ {3.5}}\Big{(}1-\frac{3}{2}\sin^{2}i\Big{)}+\sqrt{\frac{\mu}{a^{3}}} \tag{18}\]  
In Equations (16)-(18) \(R_{M}\), \(\mu\), and \(J_{2}\) are respectively the Martian equatorial radius, gravitational parameter, and oblateness coefficient. The parameter \(q\) is introduced, to denote the number of orbits per nodal day,  
\[q=\frac{N_{t}}{m}=\frac{D_{n}}{T_{n}}=\frac{\dot{M}+\dot{\omega}}{\left|\omega _{M}-\dot{\Omega}\right|} \tag{19}\]  
In Equation (19) \(D_{n}\) is the nodal day and \(T_{n}\) is the nodal orbital period. To select the operational orbit, \(q\) is regarded as a function of \(e\). Moreover, the critical inclination is selected, i.e., \(i=i_{cr}=63.4\) deg, to get \(\dot{\omega}(a,e,i)=0\) (cf. Equation (17)), whereas the argument of periapse \(\omega\) is set to \(-90\) deg. The latter value implies that apoapse corresponds to the maximum latitude and is aimed at guaranteeing long-duration coverage of high-latitude regions. The curve \(q(e)\) associated with \(\omega=-90\) deg is illustrated in Figure 1. The constraint on the minimum perigee altitude (200 km) is included in the analysis and reported in the \((q,e)\) plane (denoted with \(q_{\rm lim}\)). The value \(q=1.8\) is chosen, i.e., the vehicle completes nine orbital periods in 5 nodal days, which corresponds to semimajor axis and eccentricity equal to 13,799 km and 0.698.