# Optimal Interplanetary Transfer of Solar Wind Ion Focusing Thruster-Based Spacecraft
## 2 Mathematical Model
### Thrust Vector Description
Figure 1.0  
in which the value of \(P_{\rm ex}\) is calculated as  
Figure 1.1while \(P_{w}\) is given by  
Figure 1.2  
where \(e\) is the elementary charge, \(m_{e}\) is the electron mass, and \(L_{w}\) is obtained from Equation (19). Note that the power consumption estimation involves the solar wind ion density at Sun-Earth distance, so it is valid as long as the Sun-spacecraft distance is not too far from 1 au. Finally, the mass of the power generation system \(m_{\rm ps}\) is considered a linear function [31] of \(P\) through the (assigned) power subsystem specific mass \(\beta\), that is,  
Figure 1.3  
Note that, in the right hand side of Equation (26), the dependence on the base radius \(R\) also appears in \(L_{w}\); see Equation (19). Based on the previous discussion, the two propulsive acceleration components \(\{a_{r},a_{\theta}\}\) can be calculated using Equations (7), (9) and (17), recalling that the drag force component \(D\) is radial, while the steerable thrust component \(T\) forms an angle \(\alpha\) with \(\hat{t}_{r}\); see Figure 1.4 The expressions of \(\{a_{r},\,a_{\theta}\}\) are therefore  
Figure 1.5 Figure 1.6  
where \(a_{D_{\oplus}}\) and \(k\) are given by Equations (16) and (11), respectively. The magnitude of the propulsive acceleration vector can be calculated by means of Equations (27) and (28) as  
Figure 1.7  
Its maximum value, for a given solar distance, is obtained when \(\alpha=0\), that is, when \(T\) is aligned with the outward radial direction, so that the expression of \(a_{\rm max}\) depends on the Sun-spacecraft distance as  
Figure 1.8  
The value of \(\|\mathbf{a}\|\) or \(a_{\rm max}\) can be used to obtain the dimensionless version of the propulsive acceleration components as a function of the pair \(\{\alpha,\,k\}\), viz. Figure 1.9  
or  
\[\alpha_{\max}=\pi\,\text{rad}-\frac{\delta}{2}-\Delta\alpha, \tag{8}\]0  
In this regard, Figure 3 shows the propulsive acceleration, normalized with \(a_{\rm max}\), as a function of \(k\) and \(\alpha\). Figure 4, instead, illustrates the dimensionless components of the propulsive acceleration as a function of \(\alpha\) for different values of the propulsive parameter \(k\). While the maximum value of \(a_{r}\) is obtained with \(\alpha=0\), the circumferential component \(a_{\theta}\) is maximized by values of the thrust angle that vary with the design parameter \(k\). Finally, Figure 5 shows the classical force bubble [41] for different values of \(k\), suggesting that a larger value of \(k\) may provide a larger circumferential component of the propulsive acceleration, with the drawback of a corresponding necessary increase of the required power for ion acceleration, as shown by Equation (24). Note that Figure 5 takes into account the presence of a conical grid with aperture angle \(\delta=60\,\deg\) and a contingency \(\Delta\alpha=0\), thus implying that the thrust angle ranges in the interval \(\alpha\in[-150,\,150]\deg\). From the viewpoint of interplanetary transfer performance, the possibility of generating a significant value of circumferential acceleration \(a_{\theta}\) is of primary concern. In this regard, from Equation (28), the maximum value of \(a_{\theta}\) is reached when \(\alpha=90\,\deg\) and is given by  
\[\alpha_{\max}=\pi\,\text{rad}-\frac{\delta}{2}-\Delta\alpha, \tag{8}\]1  
Bearing in mind Equation (31), it is found that  
\[\alpha_{\max}=\pi\,\text{rad}-\frac{\delta}{2}-\Delta\alpha, \tag{8}\]2  
The preceding equation allows us to compare the thrust capabilities of the SWIFT concept and that of an ideal (i.e., perfectly reflecting) flat solar sail in terms of the dimensionless ratio \(\max(a_{\theta})/a_{\max}\). In fact, in an ideal solar sail (subscript ss), the circumferential component of the propulsive acceleration can be written as [16]  
\[\alpha_{\max}=\pi\,\text{rad}-\frac{\delta}{2}-\Delta\alpha, \tag{8}\]3 Dimensionless radial and circumferential components of \(\mathbf{a}\) as functions of \(\alpha\) and \(k\).