Computing Solar Eclipses — Research

Foundations

workingupdated 2026-09-15besselianfundamental-planefoundations
  • Eight numbers describe the shadow. x,y,d,μ,l1,l2x, y, d, \mu, l_1, l_2 vary with time and tan⁡f1,tan⁡f2\tan f_1, \tan f_2 are held constant for the eclipse. Everything a global or local product computes follows from them and their rates 1 2.
  • Two lunar radius ratios, never one. k=0.272281k = 0.272281 for all umbral and antumbral contacts, and for penumbral contacts 0.2724880 in the Five Millennium Canon and the NASA pages derived from it, 0.2725076 in the NASA bulletins and on EclipseWise 3 4.
  • ΔT enters only through μ\mu. The elements themselves are free of it. A published μ\mu is a Greenwich hour angle carried against a TT argument, and the shift of 15.041 arcseconds of longitude per second of ΔT is applied when μ\mu becomes a longitude 1 5.
  • The solar semidiameter is a parameter, not a constant. Every published prediction uses Auwers' s0=959.63s_0 = 959.63 arcseconds at 1 au, while eclipse observations give about 959.95 arcseconds 6.
  • No open library generates elements from a JPL ephemeris as a documented, validated function. Skyfield does not project the shadow, and the Swiss Ephemeris computes the geometry live without ever exposing x,y,l1,l2x, y, l_1, l_2 7 8.

What this topic covers

The geometric core of every solar eclipse computation: Bessel's fundamental plane and its sign conventions, the eight tabulated elements that describe the Moon's shadow in that plane, the constants and time scale built into them, the polynomial form in which NASA and Meeus publish them, and the ordered equations that turn geocentric apparent places of the Sun and Moon into a table of elements. It also covers when to abandon the elements and compute topocentric circumstances directly from the ephemeris.

Notes in this topic

What this topic changes for the pipeline

The interface between the ephemeris stage and everything downstream is a table of (x,y,d,μ,l1,l2)(x, y, d, \mu, l_1, l_2) against TT plus the two cone constants and a metadata record naming the ephemeris, ΔT, k1k_1, k2k_2, s0s_0, the ellipsoid and t0t_0. Global and local circumstances consume only that record. Two design decisions follow from it: the umbral kk and the solar semidiameter are inputs rather than constants, and ΔT is carried as metadata rather than applied inside the elements. The topocentric path is built as a second, independent engine from the same ephemeris layer, and the agreement of the two is a regression test.

References

  1. 1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9B Eclipses and Transits Read in full (OCR text). Definitive almanac formulation: fundamental plane, point Z, x y z, mu from ephemeris sidereal time, sin f1 sin f2 with tabulated numerators for k = 0.272274, 0.2724807, 0.272281 and 0.2724880, c1 c2 l1 l2, sign convention, observer coordinates, ephemeris meridian 1.002738 ΔT, worked example 1961 Feb 15.
  2. 2primary Espenak, F., NASA GSFC, Besselian Elements of Solar Eclipses (Five Millennium Canon explanation page) Read. Defines the eight elements and states x, y, L1, L2 are in units of the Earth's equatorial radius. No formulas.
  3. 3primary Espenak, F., NASA GSFC, Mean Lunar Radius (reference page for the eclipse bulletins) Read. History of k: 1968-1980 NAO two values 0.2724880 and 0.272281, IAU 1982 k = 0.2725076, Espenak's use of 0.272281 for umbral contacts, 1986 Oct 03 misclassification.
  4. 4primary Espenak, F., NASA GSFC, Besselian Elements for the Total Solar Eclipse of 2024 Apr 08 Read. Polynomial coefficients, t0 = 18:00 TDT, ΔT = 70.6 s, VSOP87/ELP2000-85, k1 = 0.272488, k2 = 0.272281, tan f1 = 0.0046683, tan f2 = 0.0046450, validity 15.00 to 21.00 TDT, least-squares fit to five samples over six hours, centre-of-mass statement.
  5. 5unsourced Pearson, B., lasteclipse.com, A guide to using Besselian Elements to Calculate Eclipse Paths Read. Usage formulas: polynomials, z from the ellipsoid quadratic, L1 = l1 - zeta tan f1, mu_UT = mu_TT - ΔT x 0.004178074 deg/s, WGS-84 e^2 = 0.006694.
  6. 6peer-reviewed Quaglia, L., Irwin, J., Emmanouilidis, K., Pessi, A., Estimation of the Eclipse Solar Radius by Flash Spectrum Video Analysis, ApJS 256, 36 (2021), arXiv:2107.09416 Downloaded PDF and read the introduction and computational model sections. States 959.63 arcsec (Auwers 1891) is used in all published predictions, IAU 2015 nominal 959.23 arcsec, result 959.95 ± 0.05 arcsec; model uses DE430, LOLA SLDEM-256 and LDEM-128 in the ME frame, IAU 2006 Earth orientation, light time, deflection and planetary aberration, and no Besselian elements.
  7. 7company Rhodes, B., python-skyfield discussion 801, Finding lat/lon during an eclipse Read. Maintainer states Skyfield does not project the Moon's shadow cone and that doing so needs calculations beyond the current toolkit.
  8. 8company Swiss Ephemeris (Astrodienst), swecl.c, functions eclipse_where() and eclipse_how() Downloaded and read (6428 lines). Constants DSUN, DMOON, DEARTH, the fundamental-plane vector computation with reversed f1/f2 naming, flattening applied by scaling z, classification tests, and the direct topocentric magnitude computation.

In this section

Besselian elements and the fundamental planeThe eight elements, the fundamental-plane geometry, sign conventions, units, time scales, the constants k and s0, and the polynomial form NASA and Meeus publish.workingFrom ephemeris to elementsThe ordered equations for producing a table of Besselian elements from geocentric Sun and Moon positions, what the almanacs and open-source codes do about frames, light-time and aberration, and how direct topocentric computation differs.working