## 3 Results
### Validation
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **NSTAR (Throttle & **Model** & **Error** \\  & **Level 15) [23]** & & \\ \hline \(T\) (mN) & 92.4 & 92.4 & 0.0\% \\ \(P_{in}\) (kW) & 2.29 & 2.32 & 1.3\% \\ \(I_{sp}\) (s) & 3120 & 3064 & 1.7\% \\ \(\eta_{T}\) & 0.62 & 0.60 & 3.2\% \\ \(\eta_{m}\) & 0.78 & 0.79 & 1.3\% \\ \(\eta_{e}\) & 0.84 & 0.80 & 4.7\% \\ \(V_{b}\) (V) & 1100 & 1039 & 5.5\% \\ \(I_{b}\) (A) & 1.76 & 1.79 & 1.7\% \\ \(V_{T}\) (V) & 1280 & 1299 & 1.5\% \\ \(|V_{accl}\) (V) & 180 & 260 & 44.4\% \\ \(\textit{n}_{fp}\) (mg/s) & 3.02 & 3.07 & 1.6\% \\ \(|B_{cusp}|\) (G) & Not available & 1842 & / \\ \(|B_{i}|\) (G) & Not available & 10 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the characteristics and performances of NSTAR ion thruster (throttle level 15) [23] and the results of the developed model. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **XIPS-13 cm** & & \\  & **(Nominal Operating** & **Model** & **Error** \\  & **Condition) [24]** & & \\ \hline \(T\) (mN) & 17.8 & 17.8 & 0.0\% \\ \(P_{in}\) (kW) & 0.44 & 0.45 & 2.3\% \\ \(I_{sp}\) (s) & 2585 & 2567 & 0.7\% \\ \(\eta_{T}\) & 0.51 & 0.50 & 2.0\% \\ \(V_{b}\) (V) & 751 & 1064 & 41.6\% \\ \(I_{b}\) (A) & 0.40 & 0.35 & 12.5\% \\ \(V_{T}\) (V) & 1050 & 1330 & 26.6\% \\ \(|V_{accl}\) (V) & 299 & 266 & 11.0\% \\ \(\textit{n}_{fp}\) (mg/s) & 0.70 & 0.71 & 1.4\% \\ \(|B_{cusp}|\) (G) & Not available & 2417 & / \\ \(|B_{i}|\) (G) & Not available & 36 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the characteristics and performances of XIPS-13 cm ion thruster (nominal operating condition) [24] and the results of the developed model. For what concerns the comparison with the NSTAR ion thruster, the error is small for every parameter except in the calculation of the accelerator grid voltage \(V_{acc}\), as shown in Table 2. This is related to the fact that this parameter is obtained from the difference between the total beam voltage \(V_{T}\) and the net beam voltage \(V_{b}\), which are both affected by errors. The acceleration grid voltage \(|V_{acc}|\), moreover, is quite a lot lower than the total beam voltage \(V_{T}\) and the net beam voltage \(V_{b}\), so even if the error, in percentage terms, is very small for these two parameters, the error for the acceleration grid voltage \(|V_{acc}|\), in percentage terms, is quite high. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **XIPS-25 cm (Mission Level 3)** [25] & **Model** & **Error** \\ \hline \(T\) (mN) & 61.8 & 61.8 & 0.0\% \\ \(P_{in}\) (kW) & 1.60 & 1.56 & 2.5\% \\ \(I_{sp}\) (s) & 3242 & 2962 & 8.6\% \\ \(\eta_{T}\) & 0.62 & 0.58 & 6.5\% \\ \(V_{b}\) (V) & 1215 & 1275 & 4.9\% \\ \(I_{b}\) (A) & 1.11 & 1.21 & 9.0\% \\ \(V_{T}\) (V) & 1495 & 1275 & 14.7\% \\ \(|V_{acc}|\) (V) & 280 & 255 & 8.9\% \\ \(\dot{m}_{p}\) (mg/s) & 1.95 & 2.15 & 10.3\% \\ \(|B_{cusp}|\) (G) & Not available & 2027 & / \\ \(|B_{l}|\) (G) & Not available & 12 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between the characteristics and performances of XIPS-25 cm ion thruster (mission level 3) [25] and the results of the developed model. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **ETS-8 [33]** & **Model** & **Error** \\ \hline \(T\) (mN) & 23.0 & 23.0 & 0.0\% \\ \(P_{in}\) (kW) & 0.58 & 0.58 & 0.0\% \\ \(I_{sp}\) (s) & 2640 & 2784 & 5.4\% \\ \(\eta_{T}\) & 0.58 & 0.54 & 6.9\% \\ \(V_{b}\) (V) & 1003 & 1310 & 30.6\% \\ \(I_{b}\) (A) & 0.47 & 0.41 & 12.7\% \\ \(V_{T}\) (V) & 1500 & 1637 & 9.1\% \\ \(|V_{acc}|\) (V) & 473 & 327 & 30.8\% \\ \(\dot{m}_{p}\) (mg/s) & 0.89 & 0.84 & 5.6\% \\ \(|B_{cusp}|\) (G) & Not available & 2894 & / \\ \(|B_{l}|\) (G) & Not available & 110 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison between the characteristics and performances of a working condition of ETS-8 ion thruster [33] and the results of the developed model.