## 2 Methods
Working with lower values of \(R\) results in an increase in the thrust level, but the lower the value of \(R\) is, the higher the energy of ions that impact the accelerator grid will be, reducing the operative life of the thruster. In this tool, a ratio of 0.8 has been imposed. In Figure 4, there is a representation of the geometrical parameters of the grids. Equation (2) expresses the relation between the total efficiency \(\eta_{T}\)[13], the electrical \(\eta_{e}\), and the mass utilization efficiency \(\eta_{m}\)[13], where \(\gamma\) is a thrust-correction factor used to consider the effects of the beam divergence and the presence of doubly ionized atoms; in this code, it has been considered equal to the typical value of 0.97. Equations (3) and (4) are, respectively, the expression of the specific impulse \(I_{sp}\) and of the thrust \(T\) in ion thrusters; in Equation (3), \(g\) is the gravitational acceleration at sea level. Equation (5) expresses the mass flow rate \(\dot{m}_{p}\); the Equation (6) is an approximated expression of the electrical efficiency \(\eta_{e}\), which is re-written to express the discharge current \(I_{d}\) in the function of the beam power \(I_{b}V_{b}\), the electrical efficiency \(\eta_{e}\), and the discharge voltage \(V_{d}\). The Equation (7) is the definition of the discharge loss \(\eta_{d}\), which is the ratio between the discharge power (given by the product of the discharge current \(I_{d}\) and the discharge voltage \(V_{d}\)) and the beam current \(I_{b}\) and represents the cost of producing ions. Finally, Equation (8) is the definition of the accelerator grid voltage \(V_{acc}\), defined as the difference between the total voltage \(V_{T}\) and the beam voltage \(V_{b}\). To complete the design methodology, analytical equations taken from Goebel and Katz model [13], a 0-D model that describes the physics of the discharge chamber in a ring cusp ion thruster, have been exploited to find quantities such as magnetic field values (maximum transverse magnetic field between the magnetic rings and magnetic field strength at the cusps on the anode wall), useful to size the magnets, and the discharge chamber pressure, useful for injector design. The oA ring cusp magnetic field configuration has been chosen for this tool because empirical studies in the past years showed that this configuration is one of the best solutions for electrons and ions confinement [11; 12]. In this model, a uniform plasma in the volume inside the magnetic confinement is assumed; this assumption is well verified almost everywhere except near the cathode plume, so the model predictions do not deviate so much from the experimental results. The main equations of this model are balances of power and currents inside the chamber. What concerns the power balance is shown in Equation (9). Figure 3:1  
The first term of the equation \(P_{in,c}\), i.e., the input power in the chamber, can be rewritten as a product between the current emitted by the hollow cathode and the voltage to which electrons are subjected in the discharge chamber (Equation (10)). Figure 3:2  
In Equation (10), \(I_{e}\) represents the current emitted by the cathode, \(V_{k}\) the voltage of primaries electrons from the cathode, \(V_{d}\) the discharge voltage, \(V_{c}\) the cathode voltage drop, \(V_{p}\) the potential drop in the plasma, and \(\phi\) the sheath potential relative to the anode wall. This power, from primary electrons (electrons that come from the cathode), is used mainly for producing ions and excited neutrals; some of that is lost at the electrodes. All of these phenomena are taken into account In Equation of the output power in the chamber \(P_{out,c}\) (Equation (11)). Figure 3:3 Representation of the main geometrical parameters of the grids. Figure 3:4 Figure 3:5  
The first term \(I_{p}U^{+}\) is the power needed to produce ions; \(I_{p}\) represents the ions production rate, and \(U^{+}\) represents the ionization potential.