## 2 Theoretical Models
### GEO Environmental Model  
The Geostationary Earth Orbit (GEO orbit) is known for severe spacecraft charging and discharging [12]. This orbit, particularly in the critical region for hot plasma injection in the magnetotail, where the rate of hot plasma injection is directed towards Earth, results in the accumulation of various positive and negative charges on the Earth-facing side of satellites. This accumulation triggers the phenomenon of high-voltage negative charging, posing potential serious safety hazards. GEO orbits not only contain a substantial amount of low-temperature plasma but are also exposed to high-flux thermal plasma generated by  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Spacecraft Components** & **Nodes** & \begin{tabular}{c} **Surface** \\ **Materials** \\ \end{tabular} & \begin{tabular}{c} **Connected** \\ **Nodes** \\ \end{tabular} & \
\begin{tabular}{c} **Equivalent** \\ **Circuit** \\ \end{tabular} \\ \hline main body (bottom) & 0 & Al & & \\ thruster (exit) & 1 & Al & 7 & 0 \(\Omega\) \\ antenna (Z) & 2 & Al & 0 & 1 \(\Omega\) \\ antenna (-Z) & 3 & Carbon fiber & 0 & 5000 \(\Omega\) \\ glass (Z) & 4 & SiO\({}_{2}\) & 0 & 50 \(\Omega\) \\ solar panel & 5 & Solar cell & 6 &
\begin{tabular}{c} 0.3 \(\times\) 10\({}^{8}\)\(\Omega\) \\ \(2.85\times\) 10\({}^{-6}\) F \\ thruster & 7 & Al & 1 & 0 \(\Omega\) \\ main body & 8 & Al & 0 & 5 \(\Omega\) \\ glass (-Z) & 9 & SiO\({}_{2}\) & 0 & 5000 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Circuit Nodes, Surface Materials, and Equivalent Circuit Parameter Settings.  
Figure 2: Spacecraft geometric model diagram.  
solar storms, geomagnetic storms, geomagnetic substorms, and other activities. The danger is especially pronounced in the extreme conditions induced by geomagnetic substorms, where particles with energies ranging from a few to several hundred keV are present [13]. In order to provide a more accurate depiction of the plasma environment in the GEO orbit, it is necessary to employ a double Maxwell distribution. This distribution effectively integrates the effects of plasma at different energy levels, contributing to a comprehensive understanding of the charging behavior of the spacecraft in the complex dynamics of GEO orbit plasma. The expression for the distribution function is as follows [14]:  
\[f=\Big{(}\frac{m}{2\pi}\Big{)}^{\frac{3}{2}}\Bigg{[}\frac{n_{1}}{(kT_{1})^{ \frac{3}{2}}}\exp\!\left(-\frac{mv^{2}}{2 kT_{1}}\right)+\frac{n_{2}}{(kT_{2})^{ \frac{3}{2}}}\exp\!\left(-\frac{mv^{2}}{2 kT_{2}}\right)\Bigg{]} \tag{7}\]  
The mass of electrons is significantly smaller than that of ions, leading to electron thermal motion velocities much higher than those of ions. Consequently, this results in electron charging currents being greater than ion charging currents. Therefore, in the absence of plume effects, spacecraft surfaces tend to carry a negative charge. Following the guidelines in the European Space Agency's Space Environment Handbook [15], negative charging risk assessments are typically conducted in extreme and harsh plasma environments, such as those found in GEO, for surface charging evaluations. Table 2 shows GEO extreme plasma environment parameters set for the simulation.