# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 4 Circular Orbit Phasing Maneuver
In this regard, a possible approach to the study of a feasible phasing trajectory when \(\Delta\theta\in[-60,\ -58]\deg\) and \(\Delta\theta\in[44,\ 60]\deg\) is to constrain the required propellant mass to its maximum value \(m_{p_{\max}}\). This corresponds to removing the constraint \(\lambda_{m}(t_{f})=0\) and to enforcing the following condition at the (free) final time  
\[m(t_{f})=m_{0}-m_{p_{\max}}\quad\text{if}\quad\Delta\theta\in[-60,\ -58]\deg\quad\text{or}\quad\Delta\theta\in[44,\ 60]\deg \tag{33}\]Figure 10: Optimal circular orbit phasing trajectory when \(\Delta\theta\in\{-40,\,-20,\,20,\,40\}\) deg. Radial distances are in astronomical units. Black line \(\rightarrow\) MTC transfer trajectory; blue line \(\rightarrow\) Earth’s circular orbit with \(r=r_{\oplus}=1\,\mathrm{AU}\); black circle \(\rightarrow\) start point; red square \(\rightarrow\) MTC arrival point; blue circle \(\rightarrow\) Earth at the end of the transfer; orange circle \(\rightarrow\) the Sun. Figure 9: Results of the optimization process in a circular orbit phasing maneuver (with radius equal to \(r_{\oplus}\)), as a function of the desired phasing angle \(\Delta\theta\in[-60,\,60]\) deg of the MTC with respect to the Earth, with \(\Delta\theta\
eq 0\). The numerical results are in terms of the minimum flight time \(t_{f}\) and propellant mass \(m_{p}\). The light red region indicates the zone in which the constraint \(m_{p}\leq m_{p_{\mathrm{max}}}\) on the maximum value of the admissible propellant mass is violated (\(m_{p_{\mathrm{max}}}=2.8\,\mathrm{kg}\)). (**a**) Case of phasing behind (\(\Delta\theta<0\)); (**b**) case of phasing ahead (\(\Delta\theta>0\)). Therefore, in such a suboptimal transfer problem, Equation (32) is substituted by  
\[\theta(t_{f})=t_{f}\sqrt{\frac{\mu_{\odot}}{r_{\oplus}^{3}}}+\Delta\theta,\quad m (t_{f})=m_{0}-m_{p_{\max}},\quad\mathcal{H}(t_{f})=1+\lambda_{\theta}\sqrt{ \frac{\mu_{\odot}}{r_{\oplus}^{3}}} \tag{34}\]  
which is then used to complete the associated two-point boundary value problem. In this case, the numerical solution of the optimization problem when \(\Delta\theta\in[-60,\,-58]\) deg and \(\Delta\theta\in[44,\,60]\) deg gives the results summarized in Table 1. In particular, for comparative purposes, the table also shows the results of the unconstrained optimization problem (i.e., an optimization procedure with a free value of the propellant mass). As emerges from Table 1, the presence of a constraint on the mass of propellant required to complete the transfer leads to a small increase in the flight time compared to the unconstrained case. For example, when \(\Delta\theta=55\) deg, the \"constrained\" case gives a flight time of about 508.6 days, while the truly minimum transfer time is 506 days, that is, an increase of roughly 2.5 days. This small rise in flight time allows a propellant saving of approximately 0.25 kg, which allows the constraint on the maximum allowable propellant mass \(m_{p_{\max}}\) to be satisfied. This aspect will also be evident in the next paragraph, which will analyze the transfer trajectories towards the triangular Lagrangian points \(\{L_{4},\,L_{5}\}\) of the Sun-Earth system, taking into account the constraint on the maximum available propellant mass.