Every exposed spacecraft surface floats to whatever potential V makes its net current zero. Three currents compete for each surface patch:
Thermal electron flux: J_e = e·n_e·√(kT_e / 2πm_e) · exp(eV/kT_e), V≤0
Thermal ion flux: J_i = e·n_i·√(kT_i / 2πm_i) (≈ constant, m_i ≫ m_e)
Photoemission (sunlit): J_ph = J_ph0 · exp(-eV/kT_ph), V>0 ; else J_ph0
C · dV/dt = J_i + J_ph − J_e (surface capacitance C)
Electrons are ~40–1800× lighter than ions, so their thermal flux dominates and every unlit surface drifts negative — this is exactly what happens to the shadowed side of a real GEO satellite during a substorm injection of hot (1–20 keV) plasma. Sunlit surfaces are rescued by photoemission: UV knocks electrons out, and that outward current keeps them near a small positive potential, self-limited because photoelectrons with kTph ≈ 1–2 eV fall back once V rises.
- kTe / ne — hotter, denser plasma (a substorm) pushes shadowed surfaces to a deeper negative equilibrium.
- Dielectric coating — can't bleed charge sideways to the grounded bus, so it reaches a more extreme potential than a conductive surface before anything neutralizes it.
- Differential ΔV = Vsun − Vshade — the real driver of arcing. A single uniformly-charged spacecraft floating at −5 kV relative to open space is harmless; a −1 kV gap between two adjacent surfaces punches through the intervening insulator.
- When |ΔV| crosses the ~1 kV arc threshold, a discharge fires: it locally re-equalizes the two potentials and can inject a damaging current transient into nearby electronics — the leading cause of GEO satellite anomalies attributed to space weather.
Fixed in this 2D build: the photoemission saturation current used here is 2.0×10⁻⁵ A/m² (the ~20 µA/m² value standard in spacecraft-charging literature, e.g. NASCAP/Purvis-era papers). Numerically re-integrating this same current-balance ODE with the original 3D page's constant of 3.2×10⁻⁹ A/m² showed photoemission staying roughly 200–2000× smaller than the thermal electron flux at every temperature on the slider — so the "sunlit surface rescued by photoemission" behavior described above and drawn in the legend never actually happened there; both surfaces just charged negative together and ΔV rarely reached the arc threshold. With the literature value the sunlit face visibly resists charging (and goes mildly positive at higher kTe) while the shadowed face still drives deeply negative, which is what produces a real, frequent ΔV arc.
This model integrates the charging ODE in real time with a scaled capacitance so the multi-second GEO charging transient is visible within a few seconds on screen — the physics (current balance, exponential retardation, photoemission self-limiting) is the same used in NASA's NASCAP spacecraft-charging codes, just sped up for demonstration.