How an eclipse is computed

Two cones

Hold a coin in front of a lamp and it casts a dark inner shadow that narrows to a point, and a lighter outer shadow that widens. The Moon does the same with the Sun. The inner cone is the umbra. Where its tip reaches the ground, observers see a total eclipse. If the tip falls short of the ground, its extension beyond the tip, the antumbraantumbraThe extension of the Moon's shadow cone beyond its vertex, within which the Moon appears smaller than the Sun. Where the antumbra reaches the ground the eclipse is annular., sweeps the ground instead and observers see a ring of Sun around the Moon, an annular eclipse. The outer cone is the penumbra, and anyone inside it sees a partial eclipse.

The whole computation is the geometry of those cones against a rotating, slightly flattened Earth. The orbits of the Earth and Moon are known to centimetres from laser ranging, so the cones are known almost perfectly ¹. What is not known perfectly is the size of the Sun, the shape of the Moon's edge, and how far the Earth has turned.

Bessel's plane

Friedrich Bessel's contribution in 1829 was to stop working on the Earth's surface and work instead on a plane through the Earth's centre, perpendicular to the line from the Sun through the Moon, the shadow axis. On that plane the shadow of a round Moon is a circle, and it moves almost in a straight line. Eight numbers describe it: the position of the axis on the plane ($x$, $y$), the direction of the axis on the sky ($d$, $\mu$), the radii of the penumbra and umbra on the plane ($l_1$, $l_2$), and the half-angles of the two cones ($\tan f_1$, $\tan f_2$). Published tables give the first six as short polynomials in time, valid for about six hours, and the two cone angles as constants ^{2 3}.

The elements are computed from the apparent positions of the Sun and Moon and from four constants: the Sun's radius, the Moon's radius as a fraction of the Earth's ($k$), the Earth's equatorial radius and flattening. Two values of $k$ are in use, one for the penumbra and a smaller one for the umbra, because the smaller value reproduces observed durations of totality better ⁴. Everything downstream inherits these choices.

From the plane to a map

To find where the shadow falls at a given instant, draw the line from the shadow's centre on the plane parallel to the axis and see where it meets the Earth's ellipsoid. That point is on the central linecentral lineThe locus of points where the shadow axis meets the Earth's surface. It is found at each instant from $= x$, $_1 = y/_1$ and $_1 =$, and exists only while the axis intersects the Earth.. Repeat for the edges of the umbral circle and you get the northern and southern path limitpath limitThe northern or southern boundary of the umbral (or antumbral) path on the ground, where the duration of totality (or annularity) falls to zero. Smooth-limb limits assume a spherical Moon and ellipsoidal Earth. True-limb limits include the lunar limb profile and terrain. lines. Repeat for the penumbral circle and you get the outline of the region seeing a partial eclipse. The mathematics is a square root for the centre and a one-dimensional root-find for the limits, both given in full in the global circumstances notes ⁵.

The duration of totality at a point on the central line is the umbral diameter divided by the speed of the shadow relative to the ground. The shadow moves at about a kilometre a second, so a 100 km umbra gives about two minutes ⁶.

From the plane to one place

For a single observer the question is reversed: project the observer onto the fundamental plane, and ask when the distance from the observer's projected position to the shadow centre equals the shadow radius. That gives first and fourth contact for the penumbra and second and third contact for the umbra. The eclipse magnitudeeclipse magnitudeThe fraction of the solar diameter covered by the Moon at maximum eclipse. In the partial phase it is $(L_1' - m)/(L_1' + L_2')$. Inside the umbra or antumbra NASA reports instead the ratio of the apparent lunar to solar diameters, $(L_1' - L_2')/(L_1' + L_2')$. is the fraction of the Sun's diameter covered at maximum; the obscurationobscurationThe fraction of the Sun's apparent disc area covered by the Moon, computed from the overlap of two discs and always 1 during totality. is the fraction of its area, which is a different number found from the overlap of two circles. NASA's own JavaScript calculator does exactly this in 1,200 lines, and the local circumstances notes walk through it ^{7 8}.

The Earth turns

The elements are computed in Terrestrial Time, a uniform clock. Longitudes on the Earth depend on Universal Time, which follows the Earth's irregular rotation. The difference, ΔTΔTThe difference TT − UT1 between uniform Terrestrial Time, in which the elements are computed, and Universal Time, which follows the Earth's irregular rotation. About 69 s in 2024. It converts the hour angle μ to a geographic longitude, is known only by prediction for future eclipses, and is the largest error source for historical ones., is about 69 seconds today and drifts unpredictably by a fraction of a second a year. One second of error moves the whole eclipse about 350 m east or west at mid-latitudes. Predictions made years ahead for 2024 were 1.4 to 2.3 s off, and a canon of eclipses two thousand years ago carries an uncertainty of over four minutes, which is about a hundred kilometres ^{9 10}.

The Moon is not round

At the scale of an eclipse edge the Moon's silhouette departs from a circle by up to about 3 arcseconds, which is 6 km of lunar mountain and valley. A valley on the trailing edge lets sunlight through a few seconds longer; a mountain cuts it off early. Those are Baily's beadsBaily's beadsPoints of photospheric light shining through valleys on the lunar limb in the seconds around second and third contact. Their times at a site follow from the limb profile evaluated at each position angle.. Laser altimeters on the SELENE and Lunar Reconnaissance Orbiter missions have mapped the Moon to a few metres, so the lunar limb profilelunar limb profileThe height of the Moon's silhouette edge above a reference sphere, tabulated as a function of position angle around the disk for a given libration. It is what turns a smooth-disk eclipse prediction into one that knows where the valleys are. seen from any place at any instant can be built from a digital elevation model. Applying it moves second and third contact by 2 to 3 s anywhere in the path and by tens of seconds near the edge, and turns the umbra from an ellipse into a polygon with one side per valley ^{11 12}.

The Sun has no edge

Totality means the last of the bright photosphere has gone. Where exactly the photosphere ends is a question of physics, not geometry. The value almost every prediction uses, 959.63 arcseconds at one astronomical unit, dates from 1891. Timing the beads at the edge of the path in eclipses since 2010 gives a Sun about a third of an arcsecond larger. That third of an arcsecond moves each edge of the path inward by about 600 m and, for someone standing near the edge, can take ten seconds off a short totality, and, combined with the lunar limb, a minute. It is the largest unresolved number in the subject ^{13 14}.

Where the products come from

Fred Espenak's tables on the NASA eclipse site and EclipseWise are computed from the elements with a round Moon and observers at sea level, and Espenak states that the limb moves the limits by 1 to 3 km. Xavier Jubier's interactive maps add limb corrections at a clicked point. NASA's Scientific Visualization Studio, in Ernie Wright's work for 2017, 2023 and 2024, puts every map pixel at its true height and tests it against the true limb, and is the reference standard against which the others are measured. Dave Herald's Occult is the occultation community's tool for beads. The rest of the world's apps embed one of these or NASA's element tables ^{15 12 16}.

The reports turn this into a pipeline a developer can build, with the constants, the data files and the tests.