# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 2 Orbit Dynamics
### Orbit Perturbations  
An accurate dynamical model requires the introduction of the perturbing accelerations that affect the orbit dynamics about Mars. The most relevant perturbation included in the analysis of low-thrust transfers are  
1. harmonics of the areopotential, with coefficient \(\left|J_{l,m}\right|\geq 10^{-5}\) (i.e., \(J_{2}\), \(J_{3}\), \(J_{4}\), \(J_{2,2}\) and \(J_{3,1}\)),
2. third body gravitational pull due to the Sun, and
3. solar radiation pressure. It is worth remarking that the third body attraction due to the two Martian moons is negligible in comparison to that related to the Sun. Correct modeling of these perturbations requires the definition of some useful reference frames. As a preliminary step, the Mars-centered inertial (MCI) frame is introduced. It is associated with the right-hand sequence of unit vectors \((\hat{c}_{1},\hat{c}_{2},\hat{c}_{3})\), where \((\hat{c}_{1},\hat{c}_{2})\) identifies the equatorial plane, whereas \(\hat{c}_{3}\) is aligned with the Mars rotation axis and \(\hat{c}_{1}\) coincides with the so-called International Astronomical Union (IAU) vector of the date, corresponding to the intersection of the Martian equatorial plane with the Earth J2000 equatorial plane. Another useful frame (rotating together with the spacecraft) is associated with \(\left(\hat{r},\hat{E},\hat{N}\right)\), where \(\hat{E}\) is directed along the local East direction, \(\hat{N}\) points toward the local North direction, and \(\hat{r}\) is aligned with the position vector (taken from the center of Mars). The local vertical local horizontal frame (LVLH) is associated with the triad \(\left(\hat{r},\hat{\theta},\hat{h}\right)\), where \(\hat{h}\) is directed along the spacecraft angular momentum. Angles \(\xi\) (absolute longitude), \(\phi\) (latitude), and \(\zeta\) (heading) relate the MCI-frame to the \(\left(\hat{r},\hat{E},\hat{N}\right)\)-frame and to the \(\left(\hat{r},\hat{\theta},\hat{h}\right)\)-frame,  
\[\left[\hat{r}\quad\hat{\theta}\quad\hat{h}\right]^{T}=\text{R}_{1}(\zeta) \left[\hat{r}\quad\hat{E}\quad\hat{N}\right]^{T}=\text{R}_{1}(\zeta)\text{R}_{ 2}(-\phi)\text{R}_{3}(\zeta)\left[\hat{c}_{1}\quad\hat{c}_{2}\quad\hat{c}_{3} \right]^{T} \tag{7}\]  
where \(\text{R}_{j}(\chi)\) denotes the matrix associated with the elementary rotation by angle \(\chi\) about axis \(j\). Moreover, an additional relation between the MCI-frame and the \(\left(\hat{r}\quad\hat{\theta}\quad\hat{h}\right)\)-frame can be written in terms of orbit elements,  
\[\left[\hat{r}\quad\hat{\theta}\quad\hat{h}\right]^{T}=\text{R}_{3}(f+\omega) \text{R}_{1}(i)\text{R}_{3}(\Omega)\left[\hat{c}_{1}\quad\hat{c}_{2}\quad\hat{c }_{3}\right]^{T} \tag{8}\]Mars is not a perfectly spherical body, and its asphericity is modeled through spherical harmonics of the areopotential. In fact, the Martian gravitational field has the following general expression  
\[U=U_{0}+U_{p}=-\frac{\mu}{r}+U_{p} \tag{9}\]  
where \(U_{0}\) is represents the main contribution, while the second term can be written as  
\[U_{p}=-\frac{\mu}{r}\sum_{l=2}^{\infty}J_{l}\bigg{(}\frac{R_{M}}{r}\bigg{)}^{l }P_{l0}(\sin\phi)+\frac{\mu}{r}\sum_{l=2}^{\infty}\sum_{m=1}^{l}J_{lm}\bigg{(} \frac{R_{M}}{r}\bigg{)}^{l}P_{lm}(\sin\phi)\cos[m(\lambda_{g}-\lambda_{lm})] \tag{10}\]  
In Equation (10) \(R_{M}\) is the Martian equatorial radius, \(\{P_{lm}\}\) are Legendre polynomials, \(\{J_{l}\},\ \{J_{lm}\},\ \{\lambda_{lm}\}\) are the coefficients associated with the harmonics, \(\phi\) and \(\lambda_{g}\) are respectively the spacecraft latitude and geographical longitude, and \(r\) is the vehicle distance from the center of Mars.