# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 3 Description of the Mission Scenarios and Notes on the Trajectory Design  
The impact on the transfer performance of a constant value of the sail pitch angle \(\alpha_{n}\) has been studied considering two typical interplanetary transfers, which model an Earth-Venus mission and an Earth-Mars mission. In particular, a three-dimensional orbit-to-orbit transfer has been considered in the study, i.e., a case where the E-sail propelled deep space probe performs the orbital transfer without considering the planetary ephemerides. This allows one to determine, as it is well known, the global optimum performance (in terms of flight time) of the assigned transfer, while simplifying the numerical analysis of the problem since it is not necessary to constrain the initial and final position of the spacecraft on the initial parking orbit (i.e., the orbit of the Earth) and on the final target orbit (i.e., that of Venus or Mars). In these two mission scenarios, the heliocentric orbital characteristics of the planets were derived from JPL's horizons on-line ephemeris system as of 1 March 2025, as can be seen in Figure 7.  
In these two classical interplanetary mission scenarios, the dynamics of the spacecraft equipped with the E-sail with a fixed pitch angle were described using [PERSON]'s equinoctial orbital elements [41; 42; 43], while the transfer trajectory was determined by minimizing the total flight time \(\Delta t\). For this purpose, an indirect method based on the calculus of variations [44] was employed and the procedure recently described by the author in ref. [45] was used to derive the optimal control law and the corresponding mathematical model which allows the rapid transfer trajectory to be calculated. In particular, the general procedure illustrated in ref. [45] can be adapted to the case studied in this paper by simply changing the boundary conditions related to the characteristics of the orbits of planets involved in the transfer, and exclusively considering the optimal control law related to the switching function \(\tau\) and the sail clock angle \(\delta\), being in this case the value of \(\alpha_{n}\) a constant of motion. To be precise, Equations (27)-(28) and (31)-(32) of ref. [45] were used to determine the optimal control law during the transfer, while the boundary value problem was solved via a simple shooting procedure with a precision of \(10^{-6}\) when the equations of motion are expressed in a dimensionless form. These latter, together with the Euler-Lagrange equations [46], were numerically integrated with a precision of \(10^{-10}\) using a routine based on the Adams-Bashforth method [35; 36]. In order to avoid very long flight times, and taking into account the geometric characteristics of the three Keplerian orbits (see Figure 7) involved in the two interplanetary transfers the sail pitch angle has been assumed in the range \(\alpha_{n}\in[35,\,55]\deg\). The latter, according to Equation (4), corresponds to a sail cone angle included in the interval \(\alpha\in[15.7,\,19.5]\deg\), where the upper value of the interval is equal to the maximum admis  
Figure 7: Keplerian orbits of the three planets involved in the numerical analysis. Blue line \(\rightarrow\) Earth; red line \(\rightarrow\) Mars; magenta line \(\rightarrow\) Venus; star \(\rightarrow\) perihelion point; orange dot \(\rightarrow\) the Sun. sible value of the cone angle; see also Figure 4. Moreover, the interval \(\alpha_{n}\in[35,\,55]\) deg contains the value \(\alpha_{n}=45\) deg which maximizes the transverse component of the propulsive acceleration. In other terms, the numerical simulations were obtained by considering a fixed-cone angle with a value greater than roughly 15.7 deg. As for the characteristic acceleration, three possible values of \(a_{c}\) were considered, namely 0.1 mm/s\({}^{2}\), 0.5 mm/s\({}^{2}\), and 1 mm/s\({}^{2}\), which correspond, taking into account current technology, substantially to a low, medium, or high performance E-sail, respectively.