The Tipped Dipole Bench

Both models below are exact, closed-form reproductions of the two pieces of arithmetic in the article: where a tilted dipole's field lines land, and how hard an H₃⁺ ion glows at a given temperature. Nothing is randomised and nothing is looked up from a table; move a slider and the page recomputes the same formulas the analysis script evaluates.

Model 1. Where the field lines land

For a centred dipole, a field line crossing the magnetic equator at $L$ planetary radii meets the surface at magnetic colatitude $\theta_p = \arcsin\sqrt{1/L}$ from the magnetic pole. Tip the dipole by angle $t$ from the rotation axis and read the point off at magnetic azimuth $\varphi$ around the ring, and its geographic latitude is

Model 2. Why the lamp went dim

H₃⁺ only glows in its $\nu_2$ band from levels that are populated thermally. The page sums every rotational level to $J=22$ across the ground state and two vibrational states above it, exactly as the analysis script does, to build the partition function $Q(T)$ and the fraction of ions able to emit.

What the two models say together

Model 1 says the aurora sits where it does because of where the magnet points, nothing else. Model 2 says the aurora is as faint as it is because of how cold the gas is, nothing else. Neither model needs the other to be true, and the article's whole argument is that both turned out to matter at once: the geometry explains why nobody looked in the tropics, and the thermometry explains why, even where people had looked, there was nothing bright enough to see.