# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 4 Circular Orbit Phasing Maneuver  
This section analyzes the MTC optimal performance results in a heliocentric circular orbit phasing. The latter is a classical mission application in the case of a multiple-impulse transfer trajectory [44; 45]. In fact, this specific scenario has been already studied in the context of a propellantless propulsion system as, for example, a photonic solar sail [46; 47] or [PERSON]'s Electric Solar Wind Sail [48; 49]. More precisely, in this context, the phasing maneuver models the azimuthal repositioning of the MTC along a heliocentric circular orbit of assigned radius. That radius is assumed to be equal to \(r_{\oplus}\) so that the scenario is consistent with a mission in which the MTC moves along the Earth's orbit with the aim of modifying its angular position with respect to the (starting) planet. Without loss of generality, we assume that at the beginning of the transfer, the Earth and the MTC share the same heliocentric position and velocity, with a zero value of the polar angle \(\theta\). This situation models the escape of the CubeSat from the Earth by using a classical parabolic orbit. The transfer starts at time \(t_{0}=0\) and ends at time \(t_{f}>t_{0}\), i.e., at the time instant in which the MTC returns to the circular parking orbit (of radius \(r_{\oplus}\)) with an assigned angular displacement (with respect to the Earth) equal to \(\Delta\theta\
eq 0\). In particular, we assume that \(\Delta\theta>0\) (or \(\Delta\theta<0\)) refers to a phasing ahead (or behind) with respect to the planet. The minimum time transfer trajectory in this specific mission scenario can be obtained using the mathematical model illustrated in the previous section, with the only difference regarding the expressions of the boundary conditions given by Equation (23) being in the case of a circle-to-circle orbit transfer. In a circular orbit phasing maneuver without constraints on the maximum propellant mass, according to ref. [49], Equation (23) is substituted by the following equation  
\[\theta(t_{f})=t_{f}\sqrt{\frac{\mu_{\odot}}{r_{\oplus}^{3}}}+\Delta\theta, \quad\lambda_{m}(t_{f})=0,\quad\mathcal{H}(t_{f})=1+\lambda_{\theta}\sqrt{ \frac{\mu_{\odot}}{r_{\oplus}^{3}}} \tag{32}\]  
in which the term \(t_{f}\sqrt{\mu_{\odot}/r_{\oplus}^{3}}\) indicates the angle swept by the Sun-Earth line in the time interval \(\Delta t\triangleq t_{f}-t_{0}\equiv t_{f}\). The boundary value problem has been solved for a number of mission scenarios defined by the value of the phasing angle in the range \(\Delta\theta\in[-60,\ 60]\deg\), with a step of \(1\deg\) and excluding the trivial case \(\Delta\theta=0\). The numerical results, in terms of the minimum flight time \(t_{f}\) and the required propellant mass \(m_{p}\), are reported in Figure 9, in which the red highlighted zone indicates again the condition \(m_{p}>m_{p_{\max}}\). The optimal transfer trajectory for the case of \(\Delta\theta\in\{-40,\ -20,\,20,\,40\}\) deg is shown in Figure 10. In particular, the left side of the figure clearly indicates that a phasing behind maneuver (i.e., a maneuver in which \(\Delta\theta<0\)) requires a MTC transfer trajectory which \"evolves outside\" the Earth's circular orbit, while in a phasing ahead maneuver (i.e., when \(\Delta\theta>0\)) the Sun-MTC distance is less than or equal to \(r_{\oplus}\) during all the heliocentric transfer; see, for example, the right side of Figure 10.  
According to the graph in Figure 9a (or in Figure 9b), a minimum-time phasing trajectory with \(\Delta\theta\geq-57\deg\) (or \(\Delta\theta\leq 43\deg\)) requires a propellant mass less than the maximum admissible value of \(2.8\deg\). However, in cases where the constraint on the maximum permissible value of the propellant mass is violated, i.e., when \(\Delta\theta\in[-60,\ -58]\deg\) and \(\Delta\theta\in[44,\ 60]\deg\), the circular orbit phasing maneuver of the MTC is still possible considering a flight time greater than the minimum value obtained with the solution of the optimization process.