## 2 Theoretical Models  
### Surface Charging Theory Model  
#### 2.1.1 Equation for Current Balance  
Spacecraft surface charging is a complex process influenced by various factors, including the space plasma environment, surface material properties, illumination conditions, incident particle energy, etc. The equilibrium of spacecraft surface current inflow and outflow occurs when the potential reaches a steady state. During this equilibrium, the potential on the spacecraft surface reaches a stable state. While in orbit, the spacecraft surface experiences constant impacts from charged particles in the space plasma environment, leading to electronic charging current and ion charging current on the spacecraft surface. Higher-energy charged particles prompt the release of secondary electron current and backscattered electron current from the spacecraft surface materials. Sunlight exposure to the surface material excites photoelectrons. Simultaneously, the charge accumulated on the surface material migrates to neighboring material surfaces and inside the spacecraft, resulting in current leakage. This leakage current contributes to maintaining an overall charge balance by transferring excess charge to the spacecraft interior and surrounding surfaces, mitigating the surface charging effect. To assess the impact of the Hall thruster plume on spacecraft surface charging, we must consider both the returning ion currents in the thruster plume and the neutralizing electron currents from the hollow cathode [6]. These factors significantly influence the dynamic characteristics of the charging effect, introducing complexity to the spacecraft surface charging behavior. Figure 1 depicts spacecraft surface currents and the current balance equation under a balanced surface potential [7]:  
\[I_{\text{e}}+I_{\text{Xe}}=I_{\text{i}}+I_{\text{s}}+I_{\text{b}}+I_{\text{ph} }+I_{1}+I_{\text{ne}} \tag{1}\]  
where \(I_{\text{e}}\) is the incident electron current; \(I_{\text{i}}\) is the incident ion current; \(I_{\text{s}}\) is the secondary electron current; \(I_{\text{b}}\) is the backscattered electron current; \(I_{\text{ph}}\) is the photoelectron current, and the consideration involves the photoelectron current \(I_{\text{e-ph}}\) emitted by the material and the photoelectron current \(I_{\text{i-ph}}\) returning to the material surface, influenced by the electric potential in the space environment; \(I_{1}\) is the leakage current; \(I_{\text{Xe}}\) is the ion current generated by the thruster, including the preexisting ion currents, the ion currents \(I_{\text{Xe+}}\) and \(I_{\text{Xe++}}\), as well as the ion current \(I_{\text{CEX,Xe+}}\) generated by charge exchange (CEX) collisions; and \(I_{\text{ne}}\) is the neutralization electron current. The plume region's small Debye length in comparison to the characteristic length enables it to satisfy the quasi-neutral approximation condition. This condition is influenced by the surface potential, and the electron flux on the spacecraft surface is computed using the following electron collection model [8]:  
\[ \begin{cases}J_{e}=J_{e_{0}}\exp\left(\frac{e\left(\phi_{\text{s}}-\phi_{p} \right)}{T_{e}}\right),&\text{if $\phi_{\text{s}}-\phi_{p}<0$},\\ J_{e}=J_{e_{0}}\left(1+\frac{e\left(\phi_{\text{s}}-\phi_{p}\right)}{T_{e}} \right),&\text{if $\phi_{\text{s}}-\phi_{p}>0$},\end{cases}  \tag{2}\]  
where \(J_{e_{0}}=-en_{e}\sqrt{eT_{e}/2\pi m_{e}}\) is the thermal current density of the plasma within the grid, \(n_{e}\) is the electron density, \(m_{e}\) is the mass of the electron, \(\phi_{\text{s}}\) is the potential at the collecting surface, \(\phi_{p}\) is the plasma potential, \(T_{e}\) is the electron temperature, \(T_{e}\) being expressed in energy units, and \(e\) is the elementary charge. In the quasi-neutral approach, given the absence of sheath layers, the current (\(J_{e_{0}}\)) is directly calculated at the surface nodes, and the current calculation conforms to the Orbital Motion Limited law [9].