## 2 Methods
The transverse magnetic field reduces the electron mobility and results in a deceleration of the ions and thus a reduction in the ion current to the anode. \[ \begin{split} v_{i}=\frac{1}{2}\sqrt{\left[\frac{el}{M\mu_{e}} \bigg{(}1+\mu_{e}^{2}B_{t}^{2}-\frac{\
u_{ei}}{\
u_{e}}\bigg{)}\right]^{2}+ \frac{4 kT_{e}}{M}}\\ -\left[\frac{el}{M\mu_{e}}\bigg{(}1+\mu_{e}^{2}B_{t}^{2}-\frac{ \
u_{ei}}{\
u_{e}}\bigg{)}\right]\end{split}  \tag{22}\]  
In Equation (22), \(\
u_{ei}\) represents the electrons-ions collision frequency, \(\
u_{e}\) is the sum of the electrons-neutrals collision frequency \(\
u_{en}\) and \(\
u_{ei}\), \(\mu_{e}\) is the electron mobility, \(B_{t}\) is the maximum transverse magnetic field, and \(l\) is the magnetic diffusion length (the transverse location of the maximum magnetic field magnitude between the magnets). In general, the magnetic diffusion length \(l\) can be calculated, with a good approximation as \(0.29d\) where \(d\) is the magnets' distance. The last step of the tool regards the design of the injectors. Typically, two main types of injectors are used in GITs: jet injectors and ring injectors. Ring injectors comprise a ring with multiple small holes; this injector is useful to provide uniform gas injection into the plume of the cathode. Instead, the jet injector is basically a long cylindrical tube that provides a direct flux of neutrals into the cathode plume region. In this work, a jet injector configuration has been considered for the design tool because, according to Ref. [34], it provides the best performance in terms of the life of the cathode and thus the entire engine. The problem faced was the flow of a rarefied gas through a long cylindrical tube; in particular, of sizing a tube that connects two containers at pressures \(P_{1}\) and \(P_{2}\), with a desired mass flow rate. \(P_{1}\) and \(P_{2}\) represent, respectively, the upstream pressure and the chamber pressure. The Knudsen number is higher than 1, and the problem cannot been solved through the fluid dynamics equations but through statistical mechanics equations. Many authors in the literature solved this problem as shown in Refs. [35; 36; 37; 38; 39; 40]. In this work, the model of [PERSON] [40] has been chosen because it is a simple model, and it is valid at any pressure ratio and at any Knudsen number under the hypothesis of high length-to-diameter ratio and small Mach number in the tube; these hypothesis allow to neglect the end effects, to consider the flow 1-D and to linearize the Boltzmann's equation. The pressure in the discharge chamber \(P_{2}\) is calculated by the tool; \(P_{1}\) must be assigned as an input parameter. The flowchart of the developed tool, described previously, with the main outputs, is shown in Figure 6.  
As a result of the model, the values of the magnetic field at the cusps on the anode surface and the maximum transverse magnetic field are obtained. These parameters, once the anode and magnets' material has been chosen in the design phase, are exploited to establish the size of the magnetic rings and the distance between the anode surface and the magnets. This sizing is executed by the software FEMM (finite element method magnetics) [41], a software package for solving low-frequency electromagnetic problems on two-dimensional planar and axisymmetric domains using FEM. The program addresses linear/nonlinear magneto-static problems, linear/nonlinear time harmonic magnetic problems, linear electrostatic problems, and steady-state heat-flow problems. The software, in this article, has been used to solve magneto-static problems. In this kind of problem, the fields are time invariant, so the field intensity \(H\) and the flux density \(B\) must satisfy Equations (23)-(25). \[\
abla\times H=J \tag{23}\]  
\[\
abla\cdot B=0 \tag{24}\]\[B=\mu H \tag{25}\]  
In Equation (23), \(J\) represents the free-current density; in Equation (25), \(\mu\) represents the permeability, that, for non linear material, is a function of \(B\), as shown in Equation (26).