Path of Totality: Umbra Footprint & Ground-Track Speed

Shadow-Cone Geometry

Moon distance from Earth 361,000 km

Sun elevation at crossing (γ) 55°

Crossing latitude 10°

Animate ground track (time-scaled)

Live Readouts

Across-track width

— km

Along-track width

— km

Ground-track speed

— km/s

Totality duration

— s

Earth & footprint drawn to true relative scale; the Moon's distance along the shadow beam is compressed for legibility.

How it works

The Sun (radius R_s ≈ 696,000 km) and Moon (radius R_m ≈ 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) ≈ 374,300 km  (umbra length from the Moon's center)

If the Moon sits a distance D from Earth's center (356,500–406,700 km over its elliptical orbit), the leftover cone length at the surface is L − D. The umbra's across-track diameter there is:

w₀ = 2 (L − D) tan(δ)

That circle only stays a circle if the shadow axis meets the ground perpendicularly. Away from local solar noon the axis strikes at an incidence angle θ = 90° − γ from the surface normal (γ = Sun's elevation), and simple foreshortening stretches the footprint along the direction of travel:

w_along = w₀ / cos(θ)

The same 1/cos(θ) foreshortening speeds up the point where the cone axis meets the ground, on top of the umbra's intrinsic orbital ground speed v₀ (~0.61 km/s). Earth's own rotation, moving the same direction the shadow travels, subtracts from that:

v_ground = v₀ / cos(θ) − v_eq cos(latitude),   v_eq ≈ 0.465 km/s
duration = w_along / v_ground

Moon distance — closer to perigee leaves more umbra length past the surface, widening w₀; near 374,300 km the leftover (and the path) shrinks to nothing.
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 — higher latitude means the ground under the shadow is moving slower with Earth's own spin, changing how much gets subtracted from the ground-track speed.

This is a schematic, first-order model (a real path also curves with Earth's oblateness and the Moon's 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.

Moon / umbra beam Umbra footprint Ground track

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