# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 2 Sail Pitch Angle Constrained Thrust Model of the E-Sail  
Bearing in mind that the E-sail is a propellantless propulsion system and, therefore, the mass of the spacecraft hosting this thruster is a constant of motion, it is more convenient to use the propulsive acceleration vector \(\mathbf{a}\) instead of the classical thrust vector during the preliminary phase of the mission design. In this context, the expression of \(\mathbf{a}\) can be obtained by resorting to the elegant approach proposed by [PERSON] et al. (2016), which allows the expression of the acceleration vector induced by the E-sail as a function of the Sun-spacecraft distance \(r\); the radial unit vector \(\mathbf{\hat{r}}\) indicating the direction of the solar wind flow; the unit vector \(\mathbf{\hat{n}}\) normal to the shadowed side of the sail nominal plane; the maximum magnitude \(a_{c}\) of the propulsive acceleration (also referred to as characteristic acceleration) at a reference solar distance \(r_{\oplus}\triangleq 1\) AU; the thruster switching parameter \(\tau\in\{0,\,1\}\) modeling the on/off behavior of the E-sail due to the possibility of zeroing the artificial electric field by changing the tethers' voltage. In particular, according to ref. (2016), the compact form of the propulsive acceleration vector \(\mathbf{a}\) is given by  
\[\mathbf{a}=\frac{\tau\,a_{c}\,r_{\oplus}}{2\,r}\left[\mathbf{\hat{r}}+(\mathbf{\hat{n}} \cdot\mathbf{\hat{r}})\mathbf{\hat{n}}\right] \tag{1}\]  
where it can be observed that, when \(\tau=1\) (i.e., E-sail turned on), \(r=r_{\oplus}\) (i.e., the spacecraft solar distance is one astronomical unit), and \(\mathbf{\hat{r}}=\mathbf{\hat{n}}\) (i.e., the normal direction coincides with the radial one), one has \(\|\mathbf{a}\|\equiv a_{c}\), as expected. In that case, the sail nominal plane is perpendicular to the direction of the solar wind flow, and the E-sail has a Sun-facing attitude. According to their definition, the sail pitch angle \(\alpha_{n}\in[0,\,90]\) deg can be considered as the angle between the two unit vectors \(\mathbf{\hat{r}}\) and \(\mathbf{\hat{n}}\), so that we simply have the following expression  
\[\alpha_{n}\triangleq\arccos(\mathbf{\hat{n}}\cdot\mathbf{\hat{r}}) \tag{2}\]which allows the propulsive acceleration magnitude \(\|\mathbf{a}\|\) to be written as [34]  
\[\|\mathbf{a}\|=\frac{\tau\,a_{c}\,r_{\oplus}}{2\,r}\,\sqrt{3\,\cos^{2}\alpha_{n}+1} \tag{3}\]  
The variation of the dimensionless ratio \(\|\mathbf{a}\|/a_{c}\) with \(r\) and \(\alpha_{n}\), obtained through the previous equation, has been plotted in Figure 3, where it can be observed that, at a distance from the star consistent with the average Sun-Venus (or Sun-Mars) distance, the maximum value of the magnitude \(\|\mathbf{a}\|\) of the propulsive acceleration vector is about 39% greater (or 34.5% less) than \(a_{c}\). Finally, the sail cone angle \(\alpha\in[0,\,90]\,\mathrm{deg}\), that is, the angle between the Sun-spacecraft line and the direction of the propulsive acceleration vector can be obtained from Equation (1) as  
\[\alpha=\arccos\left(\frac{\sin^{2}\alpha_{n}+2\,\cos^{2}\alpha_{n}}{\sqrt{3\, \cos^{2}\alpha_{n}+1}}\right) \tag{4}\]  
whose graph, reported in Figure 4, indicates that the maximum value of the cone angle is slightly less than \(20\,\mathrm{deg}\). This value is reached when the pitch angle of the E-sail is approximately equal to \(55\,\mathrm{deg}\). Note, in passing, that this last value of \(\alpha_{n}\) corrects a typing error in the last part of Equation (21) in ref. [34]. Figure 4: Variation of the sail cone angle \(\alpha\) with the sail pitch angle \(\alpha_{n}\in[0,\,90]\,\mathrm{deg}\), when the E-sail is turned on. Figure 3: Variation of the dimensionless ratio \(\|\mathbf{a}\|/a_{c}\) with \(r\in[0.5,\,2]\,\mathrm{AU}\) and \(\alpha_{n}\in[0,\,90]\,\mathrm{deg}\) for an E-sail with \(\tau=1\), i.e., for a propulsion system turned on.