# Optimal Interplanetary Transfer of Solar Wind Ion Focusing Thruster-Based Spacecraft
## 2 Mathematical Model
### Thrust Vector Description  
According to Refs. [31; 32; 33], the SWIFT propulsion system concept consists of a conically-shaped mesh grid of conducting tethers, which are electrostatically charged with a nominal positive voltage \(\phi_{0}\), and a few auxiliary booms, used as supporting structural elements, as illustrated in Figure 2, where \(\delta\) is the cone aperture angle. The solar wind dynamic pressure impinging on the conically-shaped mesh grid of the SWIFT produces a propulsive force component \(D\), usually referred to as \"drag\" because it opposes the Sun's gravitational pull, which is aligned with the outward radial direction \(\hat{\boldsymbol{i}}_{r}\) (see Figure 1), and given by [34]  
\[D=n_{\text{sw}}\,m_{p}\,v_{\text{sw}}^{2}\,A, \tag{7}\]  
where \(n_{\text{sw}}\) and \(v_{\text{sw}}\) are the local solar wind particle density and bulk velocity, respectively, \(m_{p}\) is the proton mass, \(A\triangleq\pi\,R^{2}\) denotes the SWIFT cone base area, \(R\) being the cone's base radius. The latter is here considered a geometric design parameter. The values of \(m_{p}\) and \(v_{\text{sw}}\), along with other physical parameters later defined and used in the numerical simulations, are reported in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline \(e\) & 1.60217663 \(\times\) 10\({}^{-19}\) & C \\ \hline \(m_{e}\) & 9.1093837 \(\times\) 10\({}^{-31}\) & kg \\ \hline \(m_{p}\) & 1.67262192 \(\times\) 10\({}^{-27}\) & kg \\ \hline \(n_{\odot}\) & 7.3 \(\times\) 10\({}^{6}\) & particles/m\({}^{3}\) \\ \hline \(r_{\odot}\) & 1 & au \\ \hline \(v_{\text{sw}}\) & 400 & km/s \\ \hline \(\mu_{\odot}\) & 132,712,439,935.5 & km\({}^{3}\)/s\({}^{2}\) \\ \hline \(\rho_{\text{sw}}\) & 2700 & kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constants and physical parameters used in the thrust model and in numerical simulations. Figure 2: Conceptual scheme of a SWIFT-based spacecraft (main body not to scale). There exists a second contribution to the SWIFT-induced total force [34], that is, the steerable component \(T\) of Figure 1, obtained by accelerating the collected solar wind positive ions and then ejecting them in a beam that can be suitably oriented by means of a dedicated assembly. In a two-dimensional mission scenario, the orientation of the ion beam (an so the direction of \(T\)) is defined by a single reference angle \(\alpha\), that is, the thrust angle, where \(\alpha\in[-\alpha_{\max},\,\alpha_{\max}]\) (with \(\alpha_{\max}>0\)) is measured counterclockwise from \(\hat{\mathbf{t}}_{r}\) as illustrated in Figure 1. Note that \(\alpha\) is constrained to a maximum (\(\alpha_{\max}\)) and a minimum (\(-\alpha_{\max}\)) value by the presence of the conical structure. Bearing in mind the scheme of Figure 2b, the maximum thrust angle cannot exceed \(\pi\,\text{rad}-\delta/2\), while in the rest of the paper \(\alpha_{\max}\) is calculated as  
\[\alpha_{\max}=\pi\,\text{rad}-\frac{\delta}{2}-\Delta\alpha, \tag{8}\]  
where \(\Delta\alpha\geq 0\) is a contingency factor that is added to the nominal value of \((\pi\,\text{rad}-\delta/2)\) to account for possible uncertainties on the SWIFT cone geometry. According to Ref. [34], the magnitude of \(T\) is given by  
\[T=n_{\text{sw}}\,m_{p}\,v_{\text{sw}}\,v_{\text{ex}}\,A\equiv n_{\text{sw}} \,m_{p}\,v_{\text{sw}}\,v_{\text{ex}}\,\pi\,R^{2}, \tag{9}\]  
where \(v_{\text{ex}}\) is the ion beam exhaust velocity so that, from Equations (7) and (9), it can be observed that the steerable thrust and the drag are related by the simple relation  
\[T=k\,D, \tag{10}\]  
where  
\[k\triangleq\frac{v_{\text{ex}}}{v_{\text{sw}}}. \tag{11}\]  
In a preliminary mission analysis, the solar wind velocity \(v_{\text{sw}}\) can be assumed to be practically constant, that is, nearly independent of both \(r\) and \(t\). This means that the value of \(k\), defined in Equation (11), may be modified by simply varying the ion beam exhaust velocity \(v_{\text{ex}}\).