1. Shadow geometry
The Sun (radius RS) and the Earth (RE) share two pairs of common tangents. The inner pair bounds the umbra, where the whole Sun is hidden, with half-angle sinα = (RS − RE)/d; the outer pair bounds the penumbra, sinβ = (RS + RE)/d. At the Moon's distance L the radii are
Ru = (RE − L sinα)/cosα and Rp = (RE + L sinβ)/cosβ. With the mean L = 384,400 km and a 1.2 % enlargement Ru ≈ 4,680 km (0.70°, about 2.7 lunar radii), Rp ≈ 8,250 km. The umbra tip lies about 1.4 million km from Earth.
The enlargement is an old empirical rule: the Earth's opaque atmosphere makes the shadow slightly larger than the solid planet's. Chauvenet used 1/50, Danjon 1/85. Here it only moves contact times; the colour model below is computed from first principles and, for a clean atmosphere, puts the edge of the dark core (where the luminance has fallen to 1 % of a full Moon) about 2 % out, consistent with the older rule.
2. The Moon's path, magnitudes and contact times
The Moon's centre moves along a straight line through the shadow plane at a distance γRE from the axis, at relative speed v = (ωMoon(L) − ωSun) L ≈ 0.95 km/s (Kepler's second law makes the Moon slower far from Earth). Distance of centres: D(t) = √(γ2RE2 + v2t2). Contacts occur when D equals Rp + RM (P1, P4), Ru + RM (U1, U4) and Ru − RM (U2, U3), so for example t = ±√(D2 − γ2RE2)/v. Umbral magnitude at greatest eclipse: (Ru + RM − γRE)/(2RM); totality needs γ < (Ru − RM)/RE ≈ 0.45. Longest totality (≈ 107 min) is for γ = 0 with the Moon near apogee.
Simplifications: straight path, constant L and v during the eclipse, no parallax, the Sun's apparent motion taken at its mean rate (scaled by 1/AU2). A real catalogue (Espenak and Meeus, Five Millennium Canon of Lunar Eclipses) uses the full ephemeris, so expect errors of a few percent in durations.
3. Why the Moon turns red
Light reaching the umbra has all passed through the Earth's atmosphere near the limb. A ray whose lowest point (tangent height) is h is bent toward the shadow axis by δ(h) ≈ (n0−1) e−h/H √(2π(R+h)/H), about 1.1° for h = 0 (twice the 34′ horizon refraction). Over the 384,000 km to the Moon this moves the ray by Lδ ≈ 7,500 km, so rays that skim the surface cross the axis and rays at 10–15 km land inside the umbra. The Moon plane position of the ray from the solar disc centre is a = b(1 + L/d) − Lδ(h), with b = R+h.
On the way the ray loses blue light. Rayleigh scattering has τR(λ) ≈ 0.00857 λ−4(1 + 0.0113λ−2 + …) per vertical atmosphere (λ in μm) and the grazing path has a slant air mass of about 70 at h = 0 (X(h) = e−h/H√(2π(R+h)/H)), so the transmitted fraction is T(λ) = exp(−τR(λ)X). At h = 10 km, X ≈ 20 and red (650 nm) is transmitted about 25 times better than blue (450 nm). Ozone absorbs in the broad Chappuis band (peak near 600 nm), removing orange and letting blue pass at higher altitude: that is the bluish-turquoise fringe at the umbral edge. Stratospheric aerosol (volcanic) adds extinction at about the 20 km level, exactly where the light that lights the umbra passes, so after big eruptions (El Chichón 1982, Pinatubo 1991) eclipses are very dark (Keen 1983).
4. How the colour on screen is computed
For every point of the shadow plane the page integrates, exactly over the 2-D solar disc, all grazing rays that land there: E(ρ,λ) = Σ T(h,λ) b Δb Φ(a,ρ) / N, where Φ is the azimuth angle for which the disc image covers the point and N normalises un-shadowed light to 1. A 24-wavelength spectrum (400–720 nm) is multiplied by a 5772 K Planck curve and a reddish lunar albedo slope, converted to CIE XYZ with the Wyman, Sloan and Shirley (2013) fit to the colour matching functions, then to sRGB and white-balanced to a full Moon. Your screen shows it through an exposure setting and a filmic tone map. The Danjon number is an indicative linear map of the brightness drop (L = 4 − (Δm − 8.7)/1.6, clamped to 0–4); the real scale is a visual estimate and only meaningful for totality.
Honest limits: scattered (not refracted) light, multiple scattering, airglow, aerosol layers other than the one modelled, a non-uniform cloud limb, the actual weather around the terminator and the 3-D structure of the limb are not modelled, so the model tends to give a deeper, redder Moon than photographs. The relief shading of the Moon texture is cosmetic (a real full Moon is flat-lit). Earth continents are stylised. Cloud cover is an azimuthal average. In the “From the Moon” view the atmosphere is stretched by the stated factor (the real ring is only 0.015° thick) and brightness follows the same ray model: a ray at tangent height h is visible from a Moon-plane point P only if the Sun point it came from lies inside the solar disc.
5. Sources
Danjon (1921) brightness scale; Keen (1983) Science 222, volcanic aerosols and lunar eclipse brightness; Pallé et al. (2009) Nature 459, 814, the Earth's transmission spectrum from lunar eclipse observations; García Muñoz and Pallé (2011) JQSRT 112, lunar eclipse theory; Bodhaine et al. (1999) Rayleigh optical depth; Wyman, Sloan and Shirley (2013) CIE fit; Espenak and Meeus, Five Millennium Canon of Lunar Eclipses (NASA TP-2006-214172). Not a forecasting tool.