## 2 BIT-3 Thrust Model for Preliminary Mission Analysis  
In this section, the thrust model of a BIT-3 engine unit is analyzed assuming a TM and considering the admissible operating levels given in the engine's thrust table. To this end, the mathematical model considers the thruster data indicated in Ref. [44], which gives the BIT-3 performances in terms of thrust magnitude \(T\), the Power Processing Unit (PPU) input power \(P\), the specific impulse \(I_{\mathrm{sp}}\), and the screen grid current \(C\). In particular, Tsay [44] indicates the value of the set \(\{T,P,I_{\mathrm{sp}},C\}\) for six possible operating levels (the generic operating level is indicated by the symbol \(\mathrm{Id}\in\mathbb{N}_{0}\)), as reported in Table 1, which coincides with the engine unit's thrust table in this work. Figure 2 shows the bar plot of the BIT-3 performance characteristics \(\{T,P,I_{\mathrm{sp}},C\}\) as a function of the operating level \(\mathrm{Id}=[0,1,2,3,4,5]\). From Table 1 or Figure 1, it can be observed that the operating level \(\mathrm{Id}=0\) substantially corresponds to a zero-thrust condition in which the electric power required for the operation of the solar electric thruster is nevertheless non-zero, amounting to \(42\,\mathrm{W}\). Furthermore, the value of the specific impulse \(I_{\mathrm{sp}}\) relative to the operating level \(\mathrm{Id}=0\) is non-zero, even though it is extremely small (the value is in fact just \(20\,\mathrm{s}\)). This non-zero value of \(I_{\mathrm{sp}}\) indicates, as explicitly observed by [PERSON] [44], that there is propellant consumption \begin{table}
\begin{tabular}{c c c c c} \hline
**Id** & \(T\) **[mN]** & \(P\) **[W]** & \(I_{\mathrm{sp}}\) **[s]** & \(C\) **[mA]** \\ \hline
0 & 0.01 & 42 & 20 & 0 \\
1 & 0.66 & 55 & 1290 & 9.9 \\
2 & 0.78 & 60 & 1530 & 11.4 \\
3 & 0.89 & 65 & 1740 & 12.9 \\
4 & 1.00 & 70 & 1960 & 14.3 \\
5 & 1.10 & 75 & 2150 & 15.6 \\ \hline \end{tabular}
\end{table} 
Table 1: BIT-3 thrust table as indicated in Ref. [44], when the engine unit is in TM. The operating level \(\mathrm{Id}=0\) indicates a condition in which the amount of thrust is substantially zero. Data retrieved from Ref. [44]. and electric power waste even in the condition of \(\text{Id}=0\), which gives in fact practically zero thrust. In accordance with what was discussed in the Introduction Section, \(\text{Id}=0\) indicates the so-called fictitious level which models the zero-thrust condition, and which is in addition to the five effective operating levels \(\text{Id}_{e}\triangleq[1,2,3,4,5]\) which provide an effective condition where the BIT-3 engine unit provides appreciable thrust. The last column of Table 1 summarizes the screen grid current \(C\) required by each operating level. In particular, the thrust magnitude \(T\) exhibits a nearly linear variation with the screen grid current. In fact, as suggested in Ref. [44], the function \(T=T(C)\) when \(\text{Id}\equiv\text{Id}_{e}\) can be accurately approximated by the following linear equation:  
\[T\simeq 0.0764\,(C-0.1)-0.087\qquad\text{with}\qquad C\in[9.9,15.6]\,\text{mA} \tag{1}\]  
where \(C\) is milliampere and \(T\) is in milliNewton. The linear relation in the preceding equation, which is shown in Figure 3, can be used to express in simple analytical form the amount of thrust as a function of \(C\), so that the screen grid current can be considered as a sort of (continuous) scalar control variable that can be used to select the other performance parameters of the BIT-3 engine unit during the flight. In this work, instead, the chosen control variable corresponds to the operating level Id, which provides through Table 1 the effective values of the performance parameters of the electric propulsion system.