# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 2 Sail Pitch Angle Constrained Thrust Model of the E-Sail
In order to highlight the dependence of the propulsive acceleration from the sail pitch angle, the vector \(\mathbf{a}\) is projected in a classical radial-transverse-normal reference frame \(\mathcal{T}_{\text{RTN}}\), whose origin coincides with the spacecraft center of mass. In particular, the unit vectors \(\{\mathbf{\hat{R}},\,\hat{T},\,\hat{N}\}\) are related to the inertial position vector \(\mathbf{r}\) and the velocity vector \(\mathbf{v}\) of the E-sail-based spacecraft through the well-known expressions  
\[\mathbf{\hat{R}}=\frac{\mathbf{r}}{\|\mathbf{r}\|}\equiv\mathbf{\hat{r}}\quad,\quad\hat{N}= \frac{\mathbf{r}\times\mathbf{v}}{\|\mathbf{r}\times\mathbf{v}\|}\quad,\quad\hat{T}=\frac{\hat {N}\times\hat{R}}{\|\hat{N}\times\hat{R}\|} \tag{5}\]  
Introducing the clock angle \(\delta\in[0,\,360]\deg\) defined as the angle between the direction of the unit vector \(\mathbf{\hat{T}}\) and the projection of the sail normal unit vector \(\mathbf{\hat{n}}\) into the plane \((\mathbf{\hat{T}},\mathbf{\hat{N}})\), one has  
\[\mathbf{\hat{n}}=\cos\alpha_{n}\,\mathbf{\hat{R}}+\sin\alpha_{n}\,\cos\delta\,\mathbf{ \hat{T}}+\sin\alpha_{n}\,\sin\delta\,\mathbf{\hat{N}} \tag{6}\]  
so that Equation (1) can be rewritten in a compact form as  
\[\mathbf{a}=\tau\,a_{\text{R}}\,\mathbf{\hat{R}}+\tau\,a_{\text{TN}}\,\cos\delta\,\mathbf{ \hat{T}}+\tau\,a_{\text{TN}}\,\sin\delta\,\mathbf{\hat{N}} \tag{7}\]  
where \(a_{\text{R}}\) and \(a_{\text{TN}}\) are two functions of \(r\) and \(\alpha_{n}\) defined as  
\[a_{\text{R}}=\frac{a_{c}\,r_{\oplus}}{2\,r}\left(1+\cos^{2}\alpha_{n}\right) \tag{8}\]  
\[\alpha_{n}\triangleq\arccos(\mathbf{\hat{n}}\cdot\mathbf{\hat{r}}) \tag{2}\]0  
Note that \(a_{\text{R}}\) (or \(a_{\text{TN}}\)) coincides with the radial (or the transversal) component of the local propulsive acceleration vector when the E-sail is turned on. The two components are shown, in a dimensionless form, in Figure 5, in which one can observe that the maximum value of the transversal component is reached when \(\alpha_{n}=45\deg\). \[\alpha_{n}\triangleq\arccos(\mathbf{\hat{n}}\cdot\mathbf{\hat{r}}) \tag{2}\]1 Variation of the dimensionless radial component \(a_{\text{R}}/a_{c}\) and the transversal component \(a_{\text{TN}}/a_{c}\) of the propulsive acceleration vector with \(r\in[0.5,\,2]\,\text{AU}\) and \(\alpha_{n}\in[0,\,90]\deg\). Consequently, at a generic solar distance \(r\) and for a given value of the sail pitch angle \(\alpha_{n}\), which is maintained throughout all the interplanetary flight, the value of the two terms \(\{a_{\rm R},a_{\rm TN}\}\) is given by Equations (8) and (9), so that the control variables are the two dimensionless terms \(\tau\) and \(\delta\) which appear in the right side of Equation (7). For example, in the important case of \(\alpha_{n}=45\deg\), that is, in the case where the transversal component \(a_{\rm TN}\) of the propulsive acceleration vector is maximized, Figure 6 shows the variation of the three components of \(a\) in the radial-transverse-normal reference frame \(\mathcal{T}_{\rm RTN}\) as a function of the clock angle \(\delta\) and the solar distance \(r\), when the E-sail is turned on; see also Equation (7). However, it should be noted that, for a given mission scenario, the (fixed) value of the sail pitch angle must be carefully selected, in order to avoid excessively long flight times or even to try to obtain a physically unfeasible orbital transfer. In this regard, in fact, taking into account Equation (1) or the graphs reported in Figure 5, it is observed that in the two particular cases in which \(\alpha_{n}=\{0,\,90\}\) deg, the direction of the thrust vector is purely radial, that is, \(a_{\rm TN}=0\). In these cases, the presence of a radial outward propulsive acceleration entails the conservation of the semilatus rectum of the spacecraft osculating orbit, that is, of the magnitude of the specific angular momentum vector [38; 39; 40].