## 4 ESTCube-Luna Stretch Goal: Lunar Escape Manoeuvre
Finally, the E-sail voltage source is turned off when the spacecraft is in eclipse conditions, when the Moon crosses the Earth's magnetosphere, and when the projection of the propulsive acceleration along \(\hat{\mathbf{k}}_{p}\) is negative. More details about the E-sail steering law and the definitions of unit vectors \(\hat{\mathbf{k}}_{p}\) and \(\hat{\mathbf{k}}_{e}\) are given in the work of [PERSON] et al. [42]. The E-sail attitude is conveniently described by considering an inertial reference frame \(\mathcal{T}_{I}(S;\,x_{I},y_{I},z_{I})\) with unit vectors \(\{\hat{\mathbf{t}}_{I},\hat{\mathbf{f}}_{I},\hat{\mathbf{k}}_{I}\}\), whose axes are parallel to those of the International Celestial Reference Frame (ICRF). The E-sail orientation is then defined by the two control angles, \(\zeta\in[0,\pi)\) rad and \(\psi\in[0,2\pi)\) rad, where \(\zeta\) is the angle between the direction of the E-sail spin axis \(\hat{\mathbf{k}}_{B}\) and that of \(\hat{\mathbf{k}}_{I}\), and \(\psi\) is measured counterclockwise from \(\hat{\mathbf{t}}_{I}\) to the projection of \(\hat{\mathbf{k}}_{B}\) on the plane \((\hat{\mathbf{t}}_{I},\hat{\mathbf{f}}_{I})\) (see Figure 9). The two key orbital parameters of the starting orbit that exert the most significant influence on the escape time are the semimajor axis and the inclination. Specifically, when the semimajor axis increases, a reduction in the escape time is observed due to an increase in the specific mechanical energy of the parking orbit around the Moon. As for the inclination, the escape time reaches its minimum when the initial value is on the order of \(1.53^{\circ}\). This particular inclination angle corresponds to the case where the orbital plane is parallel to the ecliptic plane. Consequently, when the inclination is close to \(1.53^{\circ}\), the thrust component lying on the orbital plane becomes more substantial, so that a significant component of the propulsive acceleration can be effectively used for the escape manoeuvre. For this reason, an initial inclination of \(1.53^{\circ}\) was chosen in the simulations presented, assuming that the E-sail starts its manoeuvre in the most favourable condition, in order to reduce the escape time. As an example, three parking orbits with different values of the semimajor axis are considered. All these orbits share a common inclination of 1.53\({}^{\circ}\) and an eccentricity of 0.6. The high eccentricity of the parking orbit ensures that the initial pericentre is close to the Moon surface while the semimajor axis is long enough to allow escape conditions to be reached within a year and a half. Table 2 presents an overview of the results for these three scenarios. For each case, the escape time, average daily changes in the E-sail control angles, and the percentage of time the sail is active are given. The results summarised in Table 2 highlight that a lunar escape manoeuvre is achievable in a relatively short time with an E-sail tether. However, attitude variations pose a major challenge for a spacecraft propelled by an E-sail since, as expected, the attitude control effort is significantly larger than that required by a heliocentric transfer with flight times on the order of years. This is mainly due to the fact that the Sun-spacecraft direction varies with a characteristic timescale comparable to the period of the osculating orbit, so frequent changes of the rotation plane must be made to apply the selected control law in selenocentric scenarios. This consideration also explains why orbits with larger semimajor axes require less significant changes in the rotation plane over time, as observed in Table 2. However, the attitude angle variations reported in Table 2 are relevant for all of the three scenarios and imply large changes of the E-sail rotation plane. Taking into account that the spacecraft moments of inertia would be high due to the presence of the spinning tether, such attitude changes pose a significant technological challenge for the design of the E-sail control system and the remote unit.