Computing Solar Eclipses — Research

Global circumstances

workingupdated 2026-09-15global-circumstancespathlimitsbesselian
  • One algorithm, many implementations. Every world-map product is derived from the eight Besselian elements by the method Bessel set out and Chauvenet, the 1961 Explanatory Supplement and its 1992 and 2013 revisions formalised. Stellarium is the closest thing to a complete open reference implementation 1 2.
  • The oblate Earth is handled by Bessel's substitution. Two auxiliary radii and a rotated declination turn the ellipsoid into a unit sphere, so the central line, limits and outlines all become root finds in one angle or one height 1 3.
  • Constants matter more than the algebra. The lunar radius ratio kk, the solar radius, the ellipsoid and ΔT move the path edges by kilometres. A ΔT shift is 15.041 arcseconds of longitude per second 4 5.
  • Smooth-Moon limits are wrong by 1 to 3 km. NASA, Jubier and SVS all say so. Only the SVS shapefiles publish a limb-profiled, terrain-corrected umbra polygon, and no open-source implementation of a limb-corrected path edge was found 6 7.
  • Magnitude and obscuration contours are rasterised, not solved. The Supplements obtain them by inverse interpolation on the curves of maximum eclipse. SVS and Eclipse-Engine evaluate maximum obscuration per grid cell and contour the field 1 8 5.

What this topic covers

Everything that goes on a world map, computed from a table of Besselian elements: the central line, the northern and southern limits of the umbra and of the penumbra, the outline of the shadow at an instant, duration, path width, greatest eclipse and greatest duration, the ground speed of the umbra, the rise and set curves, the curves of maximum eclipse, the magnitude and obscuration contours, and the observer-height and terrain treatment. It also covers the published products that carry these curves, which of them are limb and terrain corrected, and the open-source code that computes them.

Notes in this topic

  • Path and limits: the central line, northern and southern limits, outline curves, duration, path width, greatest eclipse, ground speed, umbra shape.
  • Global maps and contours: rise and set curves, curves of maximum eclipse, equal magnitude and equal obscuration contours, contact-time contours.
  • Data products and code: NASA path tables, SVS shapefiles, Jubier KML, NOAA releases, Stellarium, Swiss Ephemeris, GitHub implementations, and a pipeline to GeoJSON.

What this topic changes for the pipeline

The global stage is one routine that maps a time, a position angle and a cone to a surface point through Bessel's substitution. The central line, limits, outlines, rise and set curves and maximum-eclipse curves are all constraints on that routine, and width and duration are by-products of it. The partial-eclipse map is a separate product, because magnitude and obscuration contours need a per-cell maximisation in time and a contouring step rather than a per-time root find. The smooth-Moon path and the limb-profiled polygon are two distinct products and must never share a file without a label. Validation is contractual: smooth-Moon output must match the NASA path table to 1 km and 0.1 s, and limb-corrected output is compared against the SVS one-second umbra polygons.

References

  1. 1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), chapter 9 Eclipses and Transits, section B Solar eclipses fundamental equations and section C predicted data The primary algebra for Besselian elements, the observer in the fundamental frame, the auxiliary elements a' b' c', central line, limits, outline, maximum-eclipse, rise/set and greatest-eclipse curves, with worked examples for 1961 Feb 15. Read the OCR full text on archive.org (djvu text); OCR errors were resolved against the 1992 edition and the Stellarium code.
  2. 2company Stellarium, src/core/SolarEclipseComputer.cpp (master) Read the source (var/downloads/stellarium_SolarEclipseComputer.cpp). Besselian elements at run time with 6378.1366 km, 696,000 km, k=0.2725076 and s=0.272281; zetaFromQ from ES 2013 eq. 11.81; Newton solve of the limit polynomial; central line, duration, Mikhailov path width, outlines, rise/set by ellipse-circle Newton solve, maximum at rise/set; PNG and KML output. Cites ES 1961, 1992, 2013 and IERS 2003.
  3. 3peer-reviewed Explanatory Supplement to the Astronomical Almanac (1992), chapter 8 Eclipses of the Sun and Moon, by Alan D. Fiala and John A. Bangert Sections 8.353 to 8.3565 and 8.361: the conditional equation with tan^2 f, the flattening iteration in gamma, the Q-scan and 1e-5 tolerance for limits, Mikhailov's path-width formula (8.3553-5), discriminants for contacts, eclipse-map conventions. Read the OCR full text; equation 8.3553-5 is scan-damaged and was reconstructed from Stellarium's transcription.
  4. 4primary Espenak, F. and Anderson, J., Total Solar Eclipse of 2001 June 21, NASA/TP-1999-209484 Read the PDF (extracted with pdftotext, var/downloads/TP209484_2001.txt). Gives the polynomial evaluation, Table 2-8 definitions, k values, DE200/LE200, the off-axis duration formula d = D(1-(2a/W)2)1/2, the graze-zone algorithm, Elev Fact, the limb time-correction formula, and narrative umbra speeds.
  5. 5company RHerAle/Eclipse-Engine, js/besselian.js and README (eclipseradar.com, AGPL-3.0) Read the README summary and the source file (var/downloads/rherale_besselian.js). WGS84 flattening, Delta T shift at 1.002738*15 arcsec/s, central line at 6-s steps, limits perpendicular to ground-relative motion with 5 iterations and the 5 km low-latitude remark, outlines at 181 angles to convergence, obscuration grid 640x320x121 and contour refinement to 0.5 km, antimeridian and pole handling. Validation only against the project's own Python chain.
  6. 6primary Total Solar Eclipse of 2024 Apr 08, interactive Google map (NASA GSFC eclipse web site) Read. States VSOP87/ELP2000-85, Delta T = 70.6 s, that predictions do not include lunar limb effects, that limb corrections shift limits by ~1-3 km, durations by ~1-3 s and greatest duration by ~10-20 km, and that corrected predictions are posted 12-18 months ahead.
  7. 7primary NASA SVS 4518, 2017 Total Solar Eclipse Map and Shapefiles, Ernie Wright Read from the Wayback Machine snapshot of 2025-12-20. Lists the nine 2017 shapefiles and the 1-second set (6000 umbra shapes 17:12-18:52 UTC at ~100 m precision, path at 250 m, centre polyline, durations at 30 s), WGS84 lat-lon projection, LRO/Kaguya limb, DE421.
  8. 8primary NASA SVS 5123, The 2024 Total Solar Eclipse (map and shapefiles), Ernie Wright and Michala Garrison Read from the Wayback Machine snapshot of 2025-12-09 (svs.gsfc.nasa.gov refused connections). Lists 2024eclipse_shapefiles.zip contents (center, duration 30 s, ppath 5%, ppath01 1%, umbra_hi 1 s, umbra_lo 10 s, upath_hi, upath_lo), and the SRTM, LRO, DE421 inputs.

In this section

Path and limitsThe formulas for the central line, the umbral and penumbral limits, the shadow outline, duration, path width, greatest eclipse and umbra speed, with the oblate-Earth treatment and the limb and terrain corrections that the main implementations apply.workingGlobal maps and contoursHow the curves of the partial-eclipse world map are computed: rise and set curves, maximum eclipse at sunrise and sunset, curves of maximum eclipse at stated times, outline (contact-time) curves, and contours of equal magnitude and equal obscuration.workingData products and codeThe published path products (NASA tables, SVS shapefiles, Jubier KML, NOAA releases, GreatAmericanEclipse, Irwin), which of them carry limb and terrain corrections, the open-source code that computes paths, and a worked pipeline from Besselian polynomials to GeoJSON.working