# Solar Sail-Based Mars-Synchronous Displaced Orbits for Remote Sensing Applications
## 4 Preliminary Study of the Heliocentric Transfer Trajectory  
The problem of the heliocentric transfer from Earth to the target DNKO is addressed in this section with the simplifying hypotheses that the solar sail-based spacecraft starts from a circular orbit of radius \(r_{\oplus}=1\,\mathrm{au}\) and that the plane of motion of the starting orbit is coplanar to the orbit of Mars, that is, to the \((p,q)\) plane; see the scheme in Figure 1. These two assumptions, in fact, allow both the dynamic model and the optimal control law adopted in [35] to be retrieved while appropriately changing the boundary conditions. Note that these two simplifying hypotheses are reasonable since the orbital plane of Mars is slightly inclined with respect to the ecliptic plane (i.e., about \(1.85\,\mathrm{deg}\)) and the eccentricity of Earth's orbit is very small (i.e., about \(0.0167\)). Practically, such a situation may be achieved by launching the spacecraft at the moment in which the Earth's orbit intersects the orbital plane of Mars and giving it a velocity change (which is of the order of \(1\,\mathrm{km}/\mathrm{s}\)) to make the required orbital plane change in addition to the velocity change necessary to escape from the gravitational field of Earth. During the (interplanetary) transfer phase of the mission, the spacecraft uses the solar sail-induced propulsive acceleration to reach the target DNKO. In this respect, the sail propulsive characteristics are the same required to maintain the target DNKO. According to Ref. [48], the (heliocentric) equations of motion of a spacecraft in the spherical reference frame are  
\[\dot{r} =v_{r} \tag{23}\] \[\dot{\theta} =\frac{v_{\theta}}{r\,\cos\gamma}\] (24) \[\dot{\gamma} =\frac{v_{\gamma}}{r}\] (25) \[\dot{v}_{r} =\frac{v_{\theta}^{2}+v_{\gamma}^{2}}{r}-\frac{\mu_{\odot}}{r^{2 }}+a_{r}\] (26) \[\dot{v}_{\theta} =\frac{v_{\theta}\,v_{\gamma}\,\tan\gamma-v_{r}\,v_{\theta}}{r}+a _{\theta}\] (27) \[\dot{v}_{\gamma} =-\frac{v_{\theta}^{2}\,\tan\gamma+v_{r}\,v_{\gamma}}{r}+a_{\gamma} \tag{28}\]  
where \(\{v_{r},\,v_{\theta},\,v_{\gamma}\}\) are the components of the spacecraft inertial velocity in \(\mathcal{T}_{S}\), while \(\{a_{r},\,a_{\theta},\,a_{\gamma}\}\) are the components of the propulsive acceleration vector \(a_{p}\) in \(\mathcal{T}_{S}\). The latter, with the aid of Equation (1) and Figure 10, may be written as a function of the sail cone angle \(\alpha\) and clock angle \(\delta\) as  
\[a_{r} =\beta\,\frac{\mu_{\odot}}{r^{2}}\,\cos^{3}\alpha \tag{29}\] \[a_{\theta} =\beta\,\frac{\mu_{\odot}}{r^{2}}\,\cos^{2}\alpha\,\sin\alpha\, \cos\delta\] (30) \[a_{\gamma} =\beta\,\frac{\mu_{\odot}}{r^{2}}\,\cos^{2}\alpha\,\sin\alpha\, \sin\delta \tag{31}\]  
where \(\beta\) is given by Equation (9), while angles \(\alpha\) and \(\delta\) are the two typical control parameters in a three-dimensional solar sail transfer. In particular, as it is schematized in Figure 10, \(\delta\in[0,\,2\pi)\,\mathrm{rad}\) is the angle between \(\dot{\mathbf{p}}_{\theta}\) and the projection of \(\dot{\mathbf{n}}\) onto the \((\dot{\mathbf{p}}_{\theta},\dot{\mathbf{p}}_{\gamma})\) plane. Since the spacecraft initially covers a heliocentric circular orbit of radius \(r_{\oplus}\) that is coplanar to Mars orbit, Equations (23)-(28) are completed by the initial (i.e., at time \(t_{i}\triangleq 0\)) conditions  
\[\dot{r} =v_{r} \tag{23}\]0  
For a given control law \(\alpha=\alpha(t)\) and \(\delta=\delta(t)\), with \(t\in[0,\,t_{f}]\) (where \(t_{f}\) represents the total transfer time towards the target DNKO), the time variations of the propulsive acceleration components are given by Equations (29)-(31) and the resulting trajectory is obtained by numerically integrating Equations (23)-(28) with initial conditions (32). The solar sail trajectory was analyzed in an optimal framework by minimizing the flight time \(t_{f}\) required to transfer the vehicle from the circular parking orbit to the final circular DNKO of given characteristics \(\{\rho,\,H\}\).