# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 4 Circular Orbit Phasing Maneuver
### Case Study: Transfer towards the Sun-Earth Triangular Lagrangian Points  
The mathematical model discussed in the last section can be used to obtain the (constrained) minimum flight time of a MTC in a transfer towards one of the two Sun-Earth triangular Lagrangian points \(\{L_{4},\,L_{5}\}\). The latter are approximately located at a distance of 1 AU from both the Earth and the Sun (see Figure 11) so that a transfer of a MTC towards point \(L_{4}\) (or point \(L_{5}\)) can be approximated with a phasing ahead (or behind) maneuver of an angle \(\Delta\theta=60\) deg (or \(\Delta\theta=-60\) deg) along a circular heliocentric orbit with radius \(r_{\oplus}\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Unconstrained**} & \multicolumn{2}{c}{**Constrained**} \\ \(\Delta\theta\) & \(t_{f}\) **[Years]** & \(m_{p}\) **[kg]** & \(t_{f}\) **[Years]** & \(m_{p}\) **[kg]** \\ \hline \(-60\) & 1.634 & 2.847 & 1.635 & 2.8 \\ \(-59\) & 1.622 & 2.831 & 1.623 & 2.8 \\ \(-58\) & 1.610 & 2.815 & 1.610 & 2.8 \\ \(44\) & 1.284 & 2.820 & 1.284 & 2.8 \\ \(45\) & 1.293 & 2.842 & 1.294 & 2.8 \\ \(46\) & 1.303 & 2.864 & 1.304 & 2.8 \\ \(47\) & 1.312 & 2.886 & 1.314 & 2.8 \\ \(48\) & 1.322 & 2.908 & 1.324 & 2.8 \\ \(49\) & 1.331 & 2.929 & 1.334 & 2.8 \\ \(50\) & 1.340 & 2.951 & 1.343 & 2.8 \\ \(51\) & 1.349 & 2.972 & 1.353 & 2.8 \\ \(52\) & 1.359 & 2.994 & 1.363 & 2.8 \\ \(53\) & 1.368 & 3.015 & 1.373 & 2.8 \\ \(54\) & 1.377 & 3.036 & 1.383 & 2.8 \\ \(55\) & 1.386 & 3.057 & 1.393 & 2.8 \\ \(56\) & 1.395 & 3.079 & 1.403 & 2.8 \\ \(57\) & 1.404 & 3.100 & 1.414 & 2.8 \\ \(58\) & 1.413 & 3.121 & 1.425 & 2.8 \\ \(59\) & 1.422 & 3.142 & 1.436 & 2.8 \\ \(60\) & 1.431 & 3.163 & 1.448 & 2.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation results of the circular orbit optimal phasing maneuver with a constraint on the maximum propellant mass; see also Equation (34).  
In this case, the optimal performance in terms of minimum flight time and required propellant mass can be obtained from the first and the last rows of Table 1. Accordingly, the propellant-constrained optimal transfer towards point \(L_{4}\) (or point \(L_{5}\)) requires a flight time of about 1.45 years (or 1.63 years) and a propellant mass of 2.8 kg. In this scenario, the optimal transfer trajectory is reported in Figure 12, while the time variation in the control terms is shown in Figure 13. Note that the active constraint on the final mass of the MTC introduces a coasting arc almost halfway through the transfer, whose duration is roughly 13 days (or 82 days) for the Earth-\(L_{5}\) (or the Earth-\(L_{4}\)) transfer.  
Figure 11: Approximate position of the triangular Lagrangian points \(\{L_{4},\,L_{5}\}\) of the Sun–Earth system, which has been considered for the study of the MTC transfer trajectory.  
Figure 12: Optimal MTC transfer trajectory towards Sun–Earth triangular Lagrangian points \(\{L_{4},\,L_{5}\}\). 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.  
In particular, Figure 13 shows the radial and transverse components of the thrust unit vector \(\hat{\mathbf{a}}\) as obtained from the optimal guidance law, even when \(\tau=0\) only to facilitate the visualization of the curves. In fact, when \(\tau=0\), the orbital motion of the MTC is purely Keplerian so that its heliocentric trajectory is a conic arc and can be obtained by using a classical approach.