# Circular Orbit Flip Trajectories Generated by E-Sail
## 3 Numerical Simulations and Discussion  
The numerical solution of the TPBVP, which has been described at the end of the previous section, gives the COFT as a function of a given value of the dimensionless propulsive acceleration \(\beta\). The TPBVP has been numerically solved using a specific procedure based on the multiple shooting method [39; 40; 41] and the classical simplex algorithm. Trajectory optimization was conducted considering both a Direct Transfer (DT) scenario and a single Solar Wind Assist (SWA) transfer. The DT and SWA are two types of transfers introduced by the authors [42] by drawing parallels with similar concepts used in an interstellar mission based on a photonic solar sail [43; 44; 45; 46]. In the recent literature, this two concepts have also been used to analyze the optimal transfer to heliostationary points using a very high-performance E-sail [37]. More precisely, in a DT the radial distance increases continuously with time until the spacecraft reaches the aphelion point (subscript \(a\)), as shown in Figure 2a. Instead, a trajectory with a (single) SWA contains an initial phase where the spacecraft approaches the Sun to increase its propulsive acceleration magnitude, and then rapidly grows its radial distance to reach the aphelion point [37]; see the scheme of Figure 2b. Note that the mathematical model discussed in the previous section does not include a constraint on the minimum solar distance, that is, a constraint on the perihelion (subscript \(p\)) distance, as was considered in the model described in Ref. [37]. Therefore, the (unconstrained) perihelion distance \(r_{p}\) is an output of the optimization process in a transfer with a single SWA. Figure 2: _Cont._  
The numerical simulations show that a generic COFT can ideally be divided into two symmetrical parts: (i) a first half part of the transfer trajectory in which the spacecraft reaches, at time \(\widetilde{t}=\widetilde{t}_{a}=\widetilde{t}_{f}/2\), the aphelion point with a zero inertial velocity, that is, with \(\widetilde{u}(\widetilde{t}_{a})=\widetilde{v}(\widetilde{t}_{a})=0\) (see the green arrows in Figure 2); (ii) a second half-part in which the spacecraft retraces the trajectory obtained in the first half part backwards until it again reaches the starting point with \(\widetilde{u}(\widetilde{t}_{f})=0\) and \(\widetilde{v}(\widetilde{t}_{f})=-1\) (see the orange arrows in Figure 2). In particular, a COFT with a DT has a lenticular shape, while a COFT with a single SWA is characterized by a knot shape that encompasses the Sun. A conceptual sketch of these two possible shapes of a COFT is shown in Figure 2.  
The solution of the optimization process indicates that the spacecraft achieves a heliostationary condition (i.e., a zero value of its inertial velocity) at the aphelion of the generic COFT, both in a DT scenario and in a transfer with a single SWA. Such a heliostationary condition is achieved only at a single point along the transfer trajectory, while in the neighboring of the COFT aphelion the spacecraft experiences a quasi-heliosationary condition as the magnitude of its inertial velocity is very small. In the case of solar sail-based mission [3; 4; 5], such a specific feature of a propelled trajectory has been highlighted by the authors [47] and by [PERSON] et al. [48; 49; 50] in a non-Keplerian orbit [51; 52] with multiple H-reversal points. In particular, an H-reversal point occurs when the magnitude of the spacecraft angular momentum vector is zero but, usually, it also indicates a point where the angular momentum vector of the osculating orbit reverses its direction. The concept of H-reversal trajectory was originally proposed by [PERSON] [53; 54] in the middle of the 1990s as a possible trajectory of a high (or very-high) performance solar sail [55] in an advanced mission application such as an escape from the Solar System [56], or a fast transfer to the outer regions of our planetary system [57].