# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 3 Spacecraft Dynamics and Trajectory Optimization
### Numerical Results: Classical Case of the Circular Orbit Raising or Lowering  
The optimal transfer trajectory of the MTC has been calculated assuming a circle-to-circle orbit lowering (or raising) in which the radius of the coplanar target orbit is \(r_{f}\in[0.75,\,0.99]\,\mathrm{AU}\) (or \(r_{f}\in[1.01,\,1.25]\,\mathrm{AU}\)) with a step of \(0.01\,\mathrm{AU}\). In other terms, the numerical simulations consider an orbit transfer, in which the radius of the target orbit ranges in the interval \(r_{f}\in[0.75,\,1.25]\,\mathrm{AU}\), which coincides with the validity interval of the original propulsion system surrogate model [18]. The final mass of the spacecraft is left free so that the results of the optimization process give the truly minimum flight time required to complete the circle-to-circle orbit transfer, without any constraints on the maximum value of the propellant mass. The solution of the boundary value problem gives the results which are summarized in Figure 6, in terms of minimum flight time \(t_{f}\) and the corresponding (required) propellant mass \(m_{p}\equiv m_{0}-m(t_{f})\). The red highlighted regions of Figure 6 indicate the zones in which the condition \(m_{p}\leq m_{p_{\mathrm{max}}}\) on the maximum value of the admissible propellant mass is violated. In particular, Figure 6b shows that a circle-to-circle orbit raising is always possible if \(r_{f}\leq 1.25\,\mathrm{AU}\), while a truly rapid orbit lowering with \(r_{f}\in[0.75,\,0.8]\,\mathrm{AU}\) requires a propellant mass greater than \(2.8\,\mathrm{kg}\); see the bottom part of Figure 6a. In that case, however, the orbit lowering is still possible by assuming (a priori) a greater value of the flight time and minimizing the required propellant mass as discussed in ref. [43]. Note that a slightly different procedure, which can be used to obtain a feasible transfer trajectory when the nominal value of the required propellant mass exceeds \(m_{p_{\mathrm{max}}}\), is illustrated in the next section for a circular orbit phasing in a heliocentric mission scenario. Finally, from Figure 6, one has that the value of the optimal flight time is well below the reference value of \(3\,\mathrm{years}\). For example, consider the case in which the radius of the target circular orbit is \(r_{f}\in\{0.8,\,1.2\}\,\mathrm{AU}\), that is, the case in which the minimum (unconstrained) flight time is \(t_{f}\in\{1.228,\,1.22\}\) years. In that case, the optimal transfer trajectories are shown in Figure 7, which shows that the MTC completes a single revolution around the Sun during the transfer  
Figure 6: Results of the optimization process in a circle-to-circle coplanar MTC orbit transfer without constraints on the propellant mass, in terms of the minimum flight time \(t_{f}\) and propellant mass \(m_{p}\) as a function of the target orbit radius \(r_{f}\). The light red region indicates the zone in which the condition \(m_{p}\leq m_{p_{\mathrm{max}}}\) on the maximum value of the admissible propellant mass is violated (\(m_{p_{\mathrm{max}}}=2.8\,\mathrm{kg}\)). (a) Case of orbit lowering; (b) case of orbit raising. in both the orbit raising and lowering. The variation with time \(t\) in the total spacecraft mass \(m\) and the components of the control vector \(\{\tau,\,\hat{\mathbf{a}}\}\) are shown in Figure 8, where one can observe that the propulsion system is switched on (i.e., \(\tau=1\)) during all the transfer as expected. In particular, the green line in Figure 7b indicates that the thrust unit vector \(\hat{\mathbf{a}}\) is substantially aligned with the transverse direction (i.e., aligned with \(\hat{\mathbf{\theta}}\)) during the optimal orbit raising. Finally, the upper part of Figure 8 confirms that when \(\tau=1\) and the solar distance remains close to the radius of the parking orbit, the value of the derivative \(\partial m/\partial t\) is roughly constant during the optimal orbit transfer. Figure 8: Variation with time \(t\) in the spacecraft total mass \(m\) and the control terms \(\{\tau,\,\hat{\mathbf{a}}\}\).