Precipitating electrons follow the dipole magnetic field down into the atmosphere. Their stopping altitude comes from the empirical Rees (1963) range–energy relation R(E₀) = 4.57×10⁻⁶·E₀1.75 g/cm² matched against the column mass of a simplified exponential atmosphere, so higher-energy electrons punch through to lower altitude:
R(E0) = 4.57e-6 * E0^1.75 [g/cm^2, E0 in keV]
z_stop = z_ref - H * ln(R(E0)/M_ref) [km]
That altitude sets the emission line — violet N₂⁺ (427.8 nm) below ~95 km where high-energy electrons stop, green O I (557.7 nm) around 100–150 km, and red O I (630.0 nm) above ~200 km, where the forbidden transition survives long enough to radiate before a collision quenches it.
Ionospheric Pedersen/Hall conductance comes from the Robinson et al. (1987) formula relating characteristic energy E₀ and energy flux Q to conductance:
Sigma_P = 40*E0 / (16+E0^2) * sqrt(Q)
Sigma_H = 0.45 * E0^0.85 * Sigma_P
The auroral electrojet current is the Hall current (Σ_H × convection E-field) integrated across the ~500 km width of the auroral belt. Magnetic latitude sets the local dipole field strength B(λ) = B₀·√(1+3sin²λ) and, together with an assumed plasma-sheet source field, the magnetic-mirror loss-cone angle that governs how much of the trapped population can precipitate at all.