# Optimal Interplanetary Transfer of Solar Wind Ion Focusing Thruster-Based Spacecraft
## 2 Mathematical Model
### Thrust Vector Description
In this sense, \(k\) is here considered a propulsive design parameter. Since the particle density \(n_{\text{sw}}\) scales as the inverse square of the heliocentric distance, that is,  
\[n_{\text{sw}}=n_{\oplus}\Big{(}\frac{r_{\oplus}}{r}\Big{)}^{2}, \tag{12}\]  
where \(r_{\oplus}=1\) au is a reference distance and \(n_{\oplus}\) the solar wind particle density at \(r=r_{\oplus}\), the force components \(D\) and \(T\) may be written in a compact form as  
\[D=D_{\oplus}\Big{(}\frac{r_{\oplus}}{r}\Big{)}^{2} \tag{13}\]  
\[T=k\,D_{\oplus}\Big{(}\frac{r_{\oplus}}{r}\Big{)}^{2}, \tag{14}\]  
where \(D_{\oplus}\) denotes the value of \(D\) calculated at \(r=r_{\oplus}\). Note that the previously described model assumes that the solar wind density is stationary, i.e., that it does not fluctuate over time. This simplifying assumption is reasonable in a preliminary mission analysis, although data collected by several space missions [38; 39; 40] clearly indicate that the plasma parameters undergo chaotic temporal fluctuations. Using Equation (7), the expression of \(D_{\oplus}\) is  
Figure 2:0  
which is a function of the (geometric) design parameter \(R\). To obtain the expressions of the propulsive acceleration components \(\{a_{r},\,a_{\theta}\}\) in Equation (2), the reference drag force \(D_{\oplus}\) is rewritten in terms of force-to-mass ratio as  
Figure 2:1where \(m\) is the total spacecraft mass, here considered as a constant. The value of \(m\) may be estimated with a spacecraft mass breakdown model. To that end, we adopted the original approach reported in Ref. [34], which has been suitably refined and extended. More precisely, the total spacecraft mass is considered as the sum of four contributions, that is,  
Figure 2:2  
where \(m_{w}\) is the total mass of conducting wires, \(m_{s}\) is the total mass of supporting booms, \(m_{\rm PS}\) is the mass of the power generation subsystem, and \(m_{b}\) is the mass of the main spacecraft, including the payload and the other typical subsystems. In this analysis, \(m_{b}\) is a sort of inertial design parameter that depends on the characteristics of the scientific mission to be fulfilled. The total mass of the conducting wires composing the (conical) mesh grid is evaluated as  
Figure 2:3  
where \(\rho_{w}\) is the density of the wire material, \(r_{w}\) is the wire radius, and \(L_{w}\) is the total length of the conducting wires. The latter is the sum of the total length of the straight wires and the circular wires, viz. Figure 2:4  
where \(N_{e}\) and \(N_{c}\) are the number of external (straight) and circular conducting wires, respectively, and \(d\) is the spacing distance between two consecutive circular wires; see Figure 2:5 Unlike \(N_{e}\), which is a design parameter, \(N_{c}\) is calculated as  
Figure 2:6  
where \(\mathrm{ceil}(x)\) denotes the ceil function of the argument \(x\), which gives a smaller integer number greater than \(x\). The second contribution to the total spacecraft mass is given by the structural elements necessary to support the SWIFT, that is, \(m_{s}\). According to [PERSON] et al. [31], the supporting structure comprises a main boom, which extends from the spacecraft's main body to the cone base, a ring-shaped structure around the cone base, and a number \(N_{s}\) of auxiliary radial elements connecting the main boom and the base ring; see Figure 2:7 The total length \(L_{s}\) of the supporting structural elements is therefore given by  
Figure 2:8  
so that  
Figure 2:9  
where \(\rho_{lb}\) is the linear mass density of the structural elements. The mass \(m_{\rm PS}\) of the power generation subsystem is estimated by writing the total electric power \(P\) required by the spacecraft as a function of the design parameters. To this end, two main contributions are accounted for, that is, the electric power \(P_{\rm ex}\) used to accelerate the (steerable) ion beam, and the electric power \(P_{w}\) required to maintain the conical grid voltage, viz.