Besselian elements now
Source of the elements
Ephemeris: x, y, d, μ, l1, l2 computed directly at every instant. Polynomials: the classic 4-term fit about t0 used in published tables.
Polynomial fit window
Polynomial coefficients (t in hours from t0)
Contact diagram at this site
Zoom on the umbra (central phase):
m = distance of the shadow axis from the observer in the fundamental plane. Partial eclipse while m < L1, total/annular while m < |L2|.
Obscuration vs time
Accuracy (checked offline)
The science
Geometry of an eclipse
The Sun's radius is 696,000 km, the Moon's 1,737 km. The Moon's shadow is two cones: the penumbra (partial eclipse, half-angle f1) and the umbra (total eclipse, half-angle f2):
sin f1 = (R_sun + R_moon) / g sin f2 = (R_sun - R_moon) / g
g is the Sun-Moon distance. The umbra converges: its length is about R_moon · g / (R_sun - R_moon) ≈ 374,000 km (it varies a little with the Sun's distance). The Moon is on average 384,400 km from Earth (range about 356,500 to 406,700 km), so only when the Moon is closer than about 380,000 km does the umbra tip reach the ground. Farther away you get an annular eclipse (a ring of Sun around the Moon).
Besselian elements
A fundamental plane through Earth's centre is perpendicular to the shadow axis (the line Sun-Moon). With z toward the Sun, x east and y north in that plane:
x, y : where the axis crosses the plane (Earth radii)
d : declination of the axis mu : Greenwich hour angle of the axis
l1, l2 : radii of penumbra / umbra on the plane (l2 < 0: total)
l1 = (z + k/sin f1) tan f1 l2 = (z - k/sin f2) tan f2
For an observer at (ξ, η, ζ) the radii at that height are L1 = l1 - ζ tan f1 and L2 = l2 - ζ tan f2, and the axis passes at m = √((x-ξ)² + (y-η)²). Contacts are the roots of m = L1 (C1, C4) and m = |L2| (C2, C3). Tables publish x, y, d, μ, l1, l2 as cubic polynomials in time; this lab computes them from a Sun and Moon ephemeris instead and can also fit the polynomials.
Path, width and duration
The axis (x, y) is intersected with the WGS-84 ellipsoid (a quadratic in ζ). The centre line is where it lands, and the limits are where the umbra edge (±|L2| perpendicular to the motion) lands. Totality lasts
duration ≈ 2 |L2| / v_rel (v_rel: speed of the shadow axis relative to the ground, in the plane)
The Moon's orbital speed is about 1.0 km/s, Earth's rotation carries the ground along at up to 0.46 km/s, and a low Sun stretches the footprint, so the shadow sweeps the ground at 0.5 km/s or much more. The longest possible totality is a little over 7 min 30 s.
Magnitude and obscuration
With the Sun's radius set to 1, the Moon's apparent radius is q = (L1 - L2)/(L1 + L2) and the centre separation is 2m/(L1 + L2). Magnitude = (L1 - m)/(L1 + L2) is the fraction of the Sun's diameter covered; obscuration is the fraction of the area covered (two overlapping discs). Published tables sometimes quote the diameter ratio q as "magnitude" for total eclipses; this lab shows both and labels them. Obscuration is only roughly linked to brightness: at 90 % obscuration the sky is still bright, at 99 % it looks like dusk.
Ephemeris and accuracy
Moon: Meeus (Astronomical Algorithms, ch. 47) series with extra terms fitted against JPL DE440s; Sun: Kepler orbit plus fitted perturbations, with the Earth-Moon barycentre correction and aberration. Time: UT1 taken as UTC, TT = UTC + 69.184 s. Checked offline against JPL DE440s (Park et al. 2021) the shadow-plane coordinates x, y agree to 0.6-5.5 km at the five eclipses tested (2023-2028) and the umbra radius to well under 0.1 km. The remaining approximations: the Moon's limb is a smooth circle of radius k (0.272281 Re for the umbra, 0.2725076 Re for the penumbra, as in the eclipse bulletins), no atmospheric refraction, sea-level observers, and no lunar limb-profile (Watts) corrections. Contact times are therefore good to roughly 10 s at worst, and the totality duration to about a second, but a real eclipse chaser should use published predictions (Espenak and Meeus, Five Millennium Canon of Solar Eclipses; NASA eclipse bulletins).
The corona
The corona is the Sun's outer atmosphere (millions of kelvin) seen by sunlight scattered on free electrons (K-corona) and dust (F-corona); it is about a million times fainter than the disc, which is why it is visible only in totality. Magnetic loops trap plasma into streamers: near solar maximum they appear at all latitudes, near minimum the corona is squashed into equatorial streamers with straight polar rays. The orientation of the Sun's rotation axis on the sky is computed (Meeus ch. 29). Everything else about the corona here is a qualitative model.
What is simplified
- Earth's atmosphere, refraction, cloud and the real lunar limb are not modelled.
- Sky brightness is a perceptual mapping of obscuration, not a photometric model.
- The Earth texture is built in Blender from Natural Earth coastlines (public domain) with procedural colours; city lights come from Natural Earth place data and are illustrative; the Moon surface and the star field are procedural and decorative.
- Lunar limb mountains (Baily's beads) are drawn exaggerated.
Data: Natural Earth (public domain), JPL DE440s (offline check).