The Sun (radius Rs ≈ 696,000 km) and Moon (radius Rm ≈ 1,737 km), separated by d ≈ 149.6 million km, fix the umbra's half-angle and total length:
δ = asin[(R_s − R_m) / d] ≈ 0.266°
L = R_m / tan(δ) ≈ 374,370 km (umbra length from the Moon's center)
The Moon's distance D from Earth's center varies over its elliptical orbit (≈356,500 km at perigee to ≈406,700 km at apogee). The leftover cone length past that distance is L − D. Push the slider past ≈374,370 km and the cone tip never reaches the ground at all — that's an annular ("ring of fire") eclipse, computed here, not scripted:
w₀ = 2 (L − D) tan(δ) if D < L, else no umbra on the ground
Away from local solar noon the shadow axis meets the ground at incidence angle θ = 90° − γ (γ = Sun's elevation), which foreshortens the footprint along the direction of travel:
w_along = w₀ / cos(θ)
Ground-track speed is a genuine vector subtraction: the umbra's own orbital ground speed (sped up by the same 1/cos(θ) foreshortening) minus the component of Earth's rotation moving the same direction, which shrinks with latitude:
v_ground = v_shadow − v_earth, v_shadow = v₀ / cos(θ), v₀ ≈ 0.61 km/s
v_earth = v_eq·cos(latitude), v_eq ≈ 0.4651 km/s
duration = w_along / v_ground
- Moon distance — closer to perigee leaves more umbra length past the surface, widening w₀; past the L ≈ 374,370 km breakeven the umbra never touches the ground and the eclipse turns annular.
- Sun elevation γ — low Sun (dawn/dusk crossings) stretches the footprint into a long ellipse and speeds up its ground track, which is why totality is briefest near sunrise/sunset even though the shadow looks wider.
- Crossing latitude — nearer the pole, the ground itself moves slower with Earth's spin, so less gets subtracted from v_shadow and the net ground speed rises — visible directly in the vector-subtraction panel below the map.
This is a schematic, first-order model (a real path also curves with Earth's oblateness and the Moon's own along-orbit speed), but every term above is the real governing relation used to explain why some eclipse tracks are short, fast slivers and others are slow, wide sweeps.