Solar eclipses: finding them, classifying them and measuring them, from the Besselian elements of the Moon's shadow.

A solar eclipse can only happen at a new moon near a node of the Moon's orbit, when the shadow the Moon casts away from the Sun falls on the Earth. The shadow is two cones with a common axis, the line through the centres of the Sun and the Moon: the penumbra, from which some of the Sun is seen, and the umbra, from which none of it is. The umbra narrows to a vertex about as far from the Moon as the Earth is; beyond it, the antumbra, the Moon is seen inside the Sun's disk.

The Besselian elements

The shadow is described on the fundamental plane, through the centre of the Earth and perpendicular to the shadow axis, with the Earth's equatorial radius as the unit of length. At any instant the Besselian elements give:

• x and y, where the axis crosses the plane, with y towards the north and x towards the east.

• d and μ, the declination and Greenwich hour angle of the axis, which turn the plane into places on the Earth.

• l1 and l2, the radii of the penumbra and the umbra on the plane, l2 negative where the umbra reaches beyond it.

• tan f1 and tan f2, the slopes of the two cones, which give their radii at an observer's distance above the plane.

elements/1 computes them from the Sun's and the Moon's apparent geocentric positions and distances, the method of the Explanatory Supplement: the axis is the direction from the Moon to the Sun, the Moon's position is turned into the frame of the plane, and the cones follow from the radii of the Sun and the Moon. The Moon's radius is k1 = 0.272488 Earth radii for the penumbra and k2 = 0.272281, a mean below the peaks of its limb, for the umbra, as NASA's catalogues use them, so that a total phase is not overstated.

NASA publishes μ as an ephemeris hour angle, computed as if Terrestrial Time were Universal Time, and an observer's longitude is corrected by ΔT to use it. elements/1 returns the Greenwich hour angle at the moment itself, which needs no correction, so the two differ by ΔT's worth of the Earth's rotation, about 0.29° in 2024.

Greatest eclipse, gamma and the type

Greatest eclipse is the moment the shadow axis passes closest to the centre of the Earth, the least √(x² + y²), found by golden section search around each new moon. Gamma is that distance, positive when the axis passes north of the centre.

On the plane the Earth is an ellipse, of semi-axes 1 and √(1 − e² cos² d) for the Earth's flattening. Where the axis meets it, the eclipse is central, and its kind depends on the umbra where the axis meets the surface, at a height ζ above the plane: total where l2 − ζ tan f2 is negative, annular where it is positive. The point of greatest eclipse is nearer the Sun than the limb, where the central line begins and ends, so a total eclipse whose umbra stops short of the surface at either end is hybrid. Where the axis misses the Earth but the umbra or antumbra still reaches it, the eclipse is total or annular without a central line; where only the penumbra does, it is partial.

Magnitude

For a central eclipse the magnitude is the ratio of the Moon's apparent diameter to the Sun's at the point of greatest eclipse, (L1 − L2) / (L1 + L2), where L1 and L2 are the cones' radii there. Otherwise it is the fraction of the Sun's diameter covered at the point of the Earth's limb nearest the axis, (l1 − Δ) / (l1 + l2), where Δ is that point's distance from the axis; NASA's catalogues use it for a total or annular eclipse without a central line too.

How an eclipse is seen from a place, its local circumstances, is computed from the same elements by Astro.Solar.Eclipse.Local.

The Sun's and the Moon's apparent geocentric positions and distances are those of Astro.sun_position_at/1 and Astro.moon_position_at/1: from the JPL ephemeris where the loaded kernel covers the moment, and from the analytical series elsewhere.