Computing Eclipses and Occultations

Executive summary

workingupdated 2026-10-18
  • Every event is a shadow, with two cases. The observer stands in it and sees the source hidden (occultations, transits, solar eclipses), or another body stands in it and goes dark (lunar and satellite eclipses). Each event has one dominant input that is a convention or a model 1 2 3 4.
  • The classical almanac route uses Besselian elements. Direct topocentric geometry and raster limb tests are alternatives. Comparisons must match both the method and its inputs 1 5 6.
  • Solar radius, lunar limb and terrain dominate several modern path-edge comparisons. Omitted or inconsistent corrections elsewhere can still exceed 100 m. A numerical implementation check is separate from a physical uncertainty estimate 7 8 9.
  • Wright and Young 2024 specify NASA SVS's raster method. Each map pixel is placed at terrain height and tested against an 18,000-element lunar limb profile built from LRO and Kaguya (SELENE) topography. The paper is open access, and the released maps provide reference products 6.
  • The solar radius is not settled by a single convention. Recent flash-spectrum and light-curve analyses infer 959.95 to 960.01 arcseconds. Timing-based reductions also depend on the limb and detection model. Cross-program duration differences do not isolate the solar radius alone 10 11 12.
  • Catalogue enumeration has a reproduced reference comparison. The quoted Meeus series finds all 221 NASA eclipses for 1951 to 2050 with matching types. Lunar laser-ranging residuals constrain ephemerides, but they do not certify every position component to a centimetre 13 14 15.
  • Open code covers many smooth-Moon products. The surveyed repositories include single-eclipse limb and bead work, but no general derivation of limb-corrected path limits 16 17 18.

The question and the answer

The question is what a developer needs to know to compute eclipses, occultations and transits, from the list of events in a period to the second at which one begins at one place. Every one of these events is a body casting a shadow, and there are two cases, set by who stands in the shadow. In the hidden case the observer stands in it and sees the source hidden. In the darkened case another body stands in it and goes dark for everyone 1 2. The next two sections give the findings for each case. Solar eclipses are covered in most depth, at the level of NASA SVS's 2024 work: a pipeline of nine stages, given in the solar eclipse pipeline, and a short list of decisions that matter more than the rest. Extending the pipeline to other events sets out what each other event adds.

Hidden: the observer is in the shadow

Occultations, transits and solar eclipses are the hidden case, and the time of each event depends on where the observer stands. Each has one input that dominates its accuracy. For lunar occultations it is the limb and the event position angle, and most predictions now agree with video timings to 0.3 s 4. For occultations by minor bodies it is the body's orbit and the star's position, which Gaia has reduced to about 10 milliarcseconds 19. For transits of Mercury and Venus it is the solar and planetary radii, at 10 to 21 seconds of time per arcsecond of contact radius 20 21 22. For solar eclipses it is the solar radius, the lunar limb and the terrain, which the sections below cover.

Darkened: another body is in the shadow

Lunar eclipses and the eclipses of planetary satellites are the darkened case: every observer sees the same instant, apart from light-time 23 24. For lunar eclipses the dominant input is the atmospheric enlargement of the Earth's shadow, where crater timings favour a layer of about 87 km 3. For planetary satellites it is the satellite ephemeris, which mutual events test at 5 to 50 milliarcseconds 25 26.

Solar eclipses: shared geometry, different methods and inputs

Bessel's method projects the Moon's shadow onto a plane through the Earth's centre perpendicular to the shadow axis, the fundamental planefundamental planeThe plane through the Earth's centre perpendicular to the axis of the Moon's shadow. Its x axis lies in the equator pointing east, its y axis points north, and the shadow's cross-section on it is an exact circle.. Eight Besselian elementsBesselian elementsThe time-dependent quantities xx, yy, dd, μ, l1l_1, l2l_2 and the constants tanf1f_1, tanf2f_2 that describe the Moon's shadow relative to the fundamental plane, from which any eclipse circumstance can be computed.Read on Wikipedia describe the shadow there. The central line is a square root, the path limits are a root-find in one angle, the local contact times at a site are the roots of one equation in time. The 1961 Explanatory Supplement gives every formula with worked examples, the 1992 edition restates it in vector form and recommends direct root-finding over the auxiliary angle, and Stellarium's source implements it with equation numbers cited 1 27 16.

Four input choices help explain the differences between the surveyed predictors. Direct methods and raster tests also differ from the classical circular-shadow algorithm 7 6.

Constant Values in use Effect of the difference
Solar radius, angular s0s_0 or linear R⊙R_\odot 959.63″ (Auwers 1891, NASA, Espenak, Jubier, Occult); 696,000 km (SVS, USNO, Stellarium); 959.95″ (Irwin; Photo Ephemeris in its bead simulator only); 695,700 km IAU nominal (astropy, unsuitable) 0.32″ is about 600 m per limit and 1.6 to 1.8 s of central-line duration 10 28
Lunar radius ratio kk 0.2725076 (IAU 1982) for penumbral contacts; 0.272281 for umbral contacts; 0.2724880 in the Five Millennium Canon The umbral pair differs by 1.4 km of lunar radius, about 4 s of totality and 1.4 km per limit 29 30
ΔT 2024: NASA 70.6 s, EclipseWise 71.5 s, Jubier 69.1 s, measured 69.20 s 1 s is 465 cos φ metres of longitude; the 2024 misses were 0.5 to 0.8 km 31 8 32
Limb datum Mean sphere with kk; Watts charts; Kaguya LALT; LRO LOLA and SLDEM2015 on a 1737.4 km centre-of-mass sphere Omitting the profile costs 1 to 3 km per limit and 1 to 3 s per contact; LOLA brings contacts to 0.2 to 0.3 s 33 34

The cited modern ephemeris comparisons are smaller than the dominant uncorrected path-edge terms. Park and colleagues report lunar laser-ranging residuals near a centimetre for recent observations. A fitted range residual is not a bound on every three-dimensional position component. Differences between ephemerides are comparisons, not errors against truth 15 35 7.

The reference method

Wright and Young's raster method replaces the circular-shadow assumption with a per-observer test. For each map pixel at each time step, the pixel is placed on the WGS84 ellipsoid at its SRTM height corrected by the EGM96 geoid. The Moon is replaced by a 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. LL of 18,000 radii at 0.02 degree steps, built by rotating the SLDEM2015 and polar LDEM point clouds into the observer's line of sight using the topocentric librationtopocentric librationThe libration seen from a specific point on Earth rather than from the geocentre. It differs from geocentric libration by up to about one degree because of lunar parallax, so the limb profile is observer-dependent. and keeping the largest angular radius in each bin. The eclipse is total at that pixel if the profile encloses the Sun's disc, tested as ρ=a2+δ2−2aδcos(θ−ϕ)\rho = a^2 + \delta^2 - 2a\delta\cos(\theta - \phi) against 1 for every element. The path limits, central line, duration contours and obscuration contours are all extracted from the resulting raster stack, and the umbra is a polygon with as many sides as there are limb valleys on its edge, 49 at one 2017 instant 6.

The constants SVS states are DE421 (DE440 in the paper's appendix), Earth radius 6378.137 km with WGS84 flattening, a 1737.4 km lunar datum, and a Sun of 696,000 km. The 2017 run used ΔT = 68.917 s from a SPICE Earth-orientation kernel. The released shapefiles hold umbra polygons at 1 s intervals with libration and distance attributes, and a JSON of contact times for 32,174 US places. These files are public reference products for the raster method. Observed contacts and other limb-corrected predictions provide independent validation evidence 36 37 38.

What the observations say

Near-limit observations provide especially sensitive solar-radius tests. Published campaigns also include central-line observations 10 11. At Stephenville, Texas, on 2024 April 8, an observer timed 13.7 s of totality. Six public predictions for the same point ranged from 12.9 s to 65 s. The smooth-Moon products with the standard radius were 40 to 50 s too long, Jubier's limb-corrected value was 24.5 s, and Irwin's true-limb model with 959.95 arcseconds was within a second. This cross-program comparison does not isolate the radius effect from other input and solver differences 39. At Vale, Oregon, in 2017, raising the radius from 959.63″ to 960.00″ changed the predicted duration at a site 1.2 km inside the southern limit, as drawn with 959.63″, from 32.6 s to 13.3 s 7. IOTA, which for decades reported a varying solar radius from bead timings, now attributes that scatter to observational error and recommends standing at least 2 km inside Jubier's limb-corrected limit 40.

What exists to build on

The surveyed open code covers many smooth-Moon products and includes some limb work for individual eclipses. Stellarium's SolarEclipseComputer.cpp computes elements, every classic curve, KML and PNG output, with Explanatory Supplement equation numbers in the comments. NASA's program.js is the 1961 local-circumstances method in 1,200 lines of GPL JavaScript. The Swiss Ephemeris and Astronomy Engine find eclipses geometrically without elements and give the central point and local contacts. The surveyed versions of Skyfield, astropy, PyEphem, libnova, NOVAS and SOFA do not provide a complete solar-contact routine. SunPy supplies obscuration from smooth apparent discs 41. Of thirty public GitHub repositories, one builds Baily's beads from a LOLA grid for a single eclipse, and none derives a path limit from a limb profile 16 17 42 18.

The data is all public: NASA's CSV of 11,898 polynomial element sets, the SVS shapefiles, LOLA LDEM grids from 4 to 1024 pixels per degree, SLDEM2015, SRTM and Copernicus DEMs, JPL DE440 with its lunar orientation kernel, and the USNO and IERS ΔT files 43 44 45 46 31.

What to build

The solar eclipse pipeline has the detail. The decisions it rests on:

  1. Two product modes, never mixed in one file. An almanac-reproduction mode with s0=959.63″s_0 = 959.63″, the two kk values, a smooth Moon and sea level, which must match NASA's tables to 1 km and 0.1 s. An edge mode with a LOLA profile, terrain, and s0s_0 near 959.95″ with a stated uncertainty, which must match the SVS polygons and the Stephenville and Vale records.
  2. The solar radius is a parameter with an uncertainty, and each limit is evaluated at the adopted radius and its alternatives. The cited ± 0.05 arcsecond radius component gives about ± 0.1 km in that geometry 10.
  3. ΔT is metadata, applied once in the hour angle, refreshed from USNO or IERS until the eclipse, and printed on every product with its date 47 48.
  4. The Moon is two objects: a centre of mass from DE440 and a figure from LOLA oriented by the DE440 Euler angles in the mean-Earth frame. The constant kk survives only as the datum radius and as the fallback for profile-free products 49 50.
  5. Validation is a five-level test suite with pinned constants: the 1961 worked examples, NASA's tables, the Lusaka limb example, the Vale and Stephenville edge records, and the IOTA bead tables 51 52 53.

The limits of this statement

Meeus's enumeration series and Kluepfel's Saros formula have comparisons against the NASA ASCII catalogue 13 54. The end-to-end limb-and-terrain pipeline still needs validation against its named fixtures. Table reproduction and numerical consistency do not by themselves establish physical accuracy 7. Meeus's Elements of Solar Eclipses 1951-2200 and chapter 11 of the 2013 Explanatory Supplement are documented here through implementations that transcribe them. Remaining input-provenance questions concern Occult's installed help, Solar Eclipse Maestro's ephemeris and ΔT, timeanddate's ephemeris and ΔT source, and the exact SVS 2024 run 55 56 57 58 6. Open questions names the artefacts that would resolve these uncertainties.

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. 2peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961, reprint with 1973 added notes), section 10 Occultations, pp. 277-305 Read in full from the archive.org OCR text. Source of the occultation Besselian elements (T, H, Y, x', y'), the formulas for x and y and their rates, the station reduction with f, g and kn cos psi, the position angle, the a and b coefficients, the limiting parallels, the prediction limits, the k values 0.2725, 0.2724953 and 0.2725026, the reduction to delta-sigma, the limb correction arguments and section F on planetary occultations. Some equations are damaged in the OCR text.
  3. 3peer-reviewed Herald, D. and Sinnott, R. W. (2014). Analysis of lunar crater timings, 1842-2011. J. Br. Astron. Assoc. 124(5), 247-253 Read in full from the ADS scan (text extracted; Greek letters in the Symbol font are lost, so equation [4]'s angle is written theta here). Source of equations [1] to [4], the recommended rule with h = 0.0136, the NEL statistics (mean 86.9 +/- 0.2 km, mode 78, median 85), the per-eclipse heights, the ingress-egress and limb-contact biases, the shape result and the 1963 Dec 30 case.
  4. 4company Herald, D. (2016-11-20). Update to Occult and Observations Read (curl). States that a position-angle error of up to 0.2 deg in the traditional method corrupted the limb correction, and that with the LOLA lunar limb (Occult download #26) most occultation predictions are within 0.3 s of observation.
  5. 5peer-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.
  6. 6peer-reviewed Wright, E. and Young, C. A. 2024, A Raster-oriented Method for Creating Eclipse Maps, AJ 168, 163 Original publisher PDF preserved in Wayback snapshot 2024-11-19. Sections 4-6 and Appendix A checked 2026-09-30. 0.02 degree bins imply about 600 m along the Moon, not Earth edge accuracy. Printed broken-annular criterion needs common angular normalisation. PDF SHA256 5c07500302a852e21439b0cfe8b4a94c423fd92ca3315918818009d4cbf26447. CC BY 4.0.
  7. 7peer-reviewed Quaglia, Irwin, Emmanouilidis & Pessi (2021), Estimation of the Eclipse Solar Radius by Flash Spectrum Video Analysis, ApJS 256:36 Full PDF read (var/downloads/quaglia2021_flash_spectrum_ApJS.txt). Flash-spectrum video from a site a few hundred metres inside the 2017 southern limit near Vale, Oregon. S = 959.95 +/- 0.05 arcsec. Gives sensitivity of duration and limit distance to solar radius, and compares Irwin's model with Occult and Solar Eclipse Maestro.
  8. 8primary Espenak, Total Solar Eclipse of 2024 Apr 08, NASA GSFC interactive Google map page Full HTML read via curl (var/downloads/gsfc_SE2024Apr08Tgoogle.html). VSOP87/ELP2000-85 ephemerides, Delta T = 70.6 s, no limb profile, limits may shift 1 to 3 km, durations 1 to 3 s, greatest-duration point 10 to 20 km.
  9. 9primary Wright (2017), 2017 Eclipse Shadow Cones and Umbra Shape, NASA SVS 4517 Full HTML read from a saved copy. Terrain shifts the 2017 umbra south-east by as much as 3 km in the western states. Explains the polygonal umbra and the move from Watts to LRO and Kaguya profiles.
  10. 10peer-reviewed Quaglia, L., Irwin, J., Emmanouilidis, K. and Pessi, A. (2021) Estimation of the eclipse solar radius by flash spectrum video analysis. ApJS 256, 36 Full arXiv PDF read. Accepted-paper Figure 2 and conclusions checked 2026-09-30: 959.63-to-960.00 comparison gives 1.8 s central and 19.3 s Vale duration loss; fitted 959.95 +/-0.05 arcseconds gives 1.6 s central loss. Fit depends on limb and detection model. Accepted-paper PDF SHA256 dea52d9a88577023541769f62b3ddd41c601737d64ef7afdd64ab7fe31302e95.
  11. 11peer-reviewed Lamy, P., Prado, J.-Y., Floyd, O., Rocher, P., Faury, G. and Koutchmy, S. (2015) A novel technique for measuring the solar radius from eclipse light curves: results for 2010, 2012, 2013, and 2015. Solar Physics 290, 2617 Seventeen photometer light curves at 540 nm with Kaguya limb profiles: 959.99 ± 0.06 arcseconds (696,246 ± 45 km). Abstract read on the Springer page; full text paywalled.
  12. 12trade Guhl (2023), Baily's Beads Observation during the Hybrid Solar Eclipse 2023 April 20, Journal for Occultation Astronomy 2023-4, pp. 12-15 Full PDF read (var/downloads/JOA2023_4.txt). Northern limit of the total segment, Cape Range, Western Australia. Sixteen bead timings, mean correction +0.38 arcsec, result 960.01 +/- 0.12 arcsec.
  13. 13trade Meeus J. (1991) Astronomical Algorithms, first edition, Willmann-Bell, Chapter 52 Eclipses Read from the Internet Archive OCR text (chapter 52 in the 1991 edition, chapter 54 in the 1998 edition). Source of the |sin F| > 0.36 rule, the corrections to the time of maximum, P, Q, W, gamma, u, the thresholds 0.9972, 1.0260, 1.5433, 0.0047, 0.00464, the partial-magnitude formula and the stated accuracy of 0.36 min mean and 1.1 min maximum for 1951 to 2050. Graded trade as a recognised practitioner's own algorithm text.
  14. 14primary NASA GSFC, Five Millennium Catalog of Solar Eclipses, ASCII table 5MKSEcatalog.txt (2008 Oct 07) Downloaded (1.38 MB, 11,908 lines) and parsed: all 11,898 rows, type counts 4200/3956/3173/569, type-code tallies, per-century counts, gamma ranges per class, and the Kluepfel Saros formula and the Meeus chapter 54 method were verified against it.
  15. 15peer-reviewed Park, Folkner, Williams, Boggs (2021). The JPL Planetary and Lunar Ephemerides DE440 and DE441. Astronomical Journal 161, 105 Open-access HTML read through the fetch tool's extraction, not the PDF. Spans, geodetic precession on librations, LLR to 2020 March, 20 cm early and 1.3 cm recent rms, ICRF3, libration angles stored in the files, DE440 for modern data and DE441 for historical.
  16. 16company Stellarium src/core/SolarEclipseComputer.cpp Source read at pinned commit 69888f4f47af2aa4d1c9d14ec3853e874e253c73. SHA256 6ea4ddfac3c2fc921adcb75c06fd6c0062d6e7716ce566c15976c16b2020cbd5. Lines 584-587 use a small-separation angular axis approximation; global limits and observer transformations inspected. Verified 2026-09-30.
  17. 17primary JSEX program.js (JavaScript source of the Solar Eclipse Explorer) Source read in full (36.8 KB, downloaded 2026-09-15). GPL v2+. Contains the local-circumstance algorithm, the observer geocentric constants and the crude refraction handling.
  18. 18company tomasrojasc/eclipse-2026 Read live. Skyfield with DE421, LOLA LDEM_16 limb profile, moon_pa_de421 orientation, Baily's beads per bead; 84 tests; no licence stated.
  19. 19peer-reviewed Ferreira, J. F., Tanga, P., Spoto, F., Machado, P. and Herald, D. (2022). Asteroid astrometry by stellar occultations: accuracy of the existing sample from orbital fitting. A&A 658, A73 (arXiv:2201.03892) Read from the arXiv text. Source of the pre-Gaia residual level of 30 to 50 mas, the Gaia EDR3 level near 10 mas, the factor of about 5 reduction, the 10 mas FWHM of the residual core, the 3 to 4 mas along-track error level, the RUWE analysis and the statement that 61 per cent of events are single-chord.
  20. 20peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9F Transits of Mercury, pages 262 to 276 Read in full (OCR text). Node condition and the table of node limits, dates and frequencies (Mercury ascending node about 16 November 238 arcminutes, descending 8 May 108; Venus ascending 9 December 37, descending 7 June 41), Newcomb's heliocentric method and the transit fundamental plane, the light-time rule, the contact solution, least distance, position angles with V, Newcomb's q p w, the B C D reduction and Examples 9.26 to 9.32 for 1960 November 7. The node-limit formula for theta and the cone-angle equation are damaged in the OCR text and are not transcribed.
  21. 21primary IMCCE for ESO VT-2004, Extended InfoSheet B1: Predictions of the Venus transit of June 8, 2004 Read in full (curl). VSOP87, Lieske precession, Wahr nutation, Aoki sidereal time, IAU 1976 constants (solar semidiameter 15'59.63", Venus radius 6051.8 km, Earth radius 6378.140 km), flattening 1/298.257, TT - UTC 65.184 s, general and geocentric phases, the E, F, G coefficients and the Paris example. Formula (1) is an image; its form is reconstructed in the note from the Paris example.
  22. 22company Espenak, F., EclipseWise, 2019 Transit of Mercury (Observer's Handbook article) Read through a fetch summary with quoted table values. Geocentric contacts 12:35:27, 12:37:08, 15:19:48, 18:02:33, 18:04:14 UT1 with position angles, minimum separation 75.9 arcseconds, JPL DE406, local times differing by up to 2 minutes, and the remark that the thread-breaking time is delayed by some seconds, up to a minute.
  23. 23primary Espenak, F. and Meeus, J. (2009). Five Millennium Canon of Lunar Eclipses: -1999 to +3000, NASA/TP-2009-214172, text sections 1 to 5 Read in full (text PDF, 3.2 MB, extracted with pdftotext). Source of the contact definitions, gamma and magnitude definitions, eclipse statistics (12,064 eclipses), VSOP87 D and truncated ELP-2000/82, k = 0.2724880, secular acceleration -25.858 arcsec/cy^2 and the c correction, the enlargement section 1.5 (Chauvenet and Danjon formulas, Tables 1-3 to 1-6), map accuracy and Delta T uncertainty zones, Danjon scale wording, Saros and Inex behaviour. Some tables are misaligned in extraction and were cross-checked against the web pages.
  24. 24peer-reviewed Thuillot, W. (1983). Theorie des phenomenes des satellites galileens de Jupiter: une representation par les polynomes de Tchebychev. Astronomy and Astrophysics 127, 63-67 Pages 1 to 3 read from the ADS scan (in French). Source of the cone equations with an ellipsoidal Jupiter, the phase correction of 3 min (Io) and 6 min (Callisto), the light-time rules, the Chebyshev form and its 51 s maximum error.
  25. 25peer-reviewed Saquet, E., Emelyanov, N., Robert, V., Arlot, J.-E. et al. (2018). The PHEMU15 catalogue and astrometric results of the Jupiter's Galilean satellite mutual occultation and eclipse observations made in 2014-2015. MNRAS 474, 4730-4739 Read in full from the UNESP repository copy. Campaign statistics 1985 to 2015, the photometric model, the Emelyanov reduction equations, NOE-5-2010-GAL and JUP310 residuals.
  26. 26peer-reviewed Arlot, J.-E., Emelyanov, N. V., Lainey, V. et al. (2012). Astrometric results of observations of mutual occultations and eclipses of the Saturnian satellites in 2009. Astronomy and Astrophysics 544, A29 Read in full from the HAL PDF (CC BY 4.0). Source of the observation counts 14, 66 and 33 for 1980, 1995 and 2009, the photometric model E(t) = K S(X, Y), the O-C residuals against TASS 1.7, Lainey and SAT351, and the NOE-6-2011-MAIN accuracy statement.
  27. 27peer-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.
  28. 28peer-reviewed Wright, E. and Young, C. A. (2024) A raster-oriented method for creating eclipse maps. AJ 168, 163 Section 6.4 gives NASA's position: figures assume 696,000 km, nominal 695,700 km is unsuitable, eclipse values 959.99, 959.95, 959.98, 960.01 listed, Irwin's map shifted the northern limit several city blocks, 1 s of duration near a limit equals 0.03 arcseconds. Read from a saved copy of the IOP HTML (var/downloads/iop_ad6b23_wayback.html).
  29. 29primary 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.
  30. 30primary Espenak, Solar Eclipse Predictions, NASA GSFC eclipse site (2003) Full HTML read via curl. Statement of the older ephemeris basis and of k = 0.272281 instead of the IAU 0.2725076, and the responsibility statement.
  31. 31primary deltat.data: monthly determinations of TT - UT1 (USNO) Read on 2026-09-15. 2017 Aug 1: 68.8373 s; 2017 Sep 1: 68.8477 s; 2024 Apr 1: 69.1983 s; 2024 May 1: 69.2018 s; last row 2026 Apr 1: 69.1330 s.
  32. 32company Jubier: 2024 April 8 Total Solar Eclipse Interactive Google Map Fetched with curl 2026-09-15 (WebFetch refused). The page embeds its own Besselian element array with ΔT = 69.1 s and lists the JavaScript files it loads.
  33. 33primary Fred Espenak, NASA GSFC, "The Lunar Limb Profile and Eclipse Predictions" Read. Watts corrections bring predictions to better than 0.5 s, uncorrected times can be off by 2 to 3 s and more near the path limits, Kaguya and LRO data reach about 0.2 s.
  34. 34peer-reviewed Wright, E. and Young, C. A. (2024). A Raster-oriented Method for Creating Eclipse Maps. The Astronomical Journal 168, 163 Read through the IOP HTML in several targeted passes (the PDF download returned a script page). Source of the DEM-to-limb-profile algorithm, the L = 18000 bin recommendation, the 0.01 deg libration refresh threshold, the totality test rho, the 49-sided umbra, the 696000 km solar radius, DE440 and the Moon ME frame, and the Herald 1983 history.
  35. 35primary Williams, Boggs, Folkner (2013). DE430 Lunar Orbit, Physical Librations, and Surface Coordinates. JPL IOM 335-JW,DB,WF-20130722-016 Read in full from the PDF. 18,548 LLR ranges 1970-2012, 1.9 cm wrms, DE430 vs DE421 half a milliarcsecond, PA and ME frame definitions, the DE430 rotation Rx(-0.285") Ry(-78.580") Rz(-67.573"), 1" = 8.42 m, LOLA mean radius 1737.151 km (Neumann 2013).
  36. 36primary NASA SVS 4515, 2017 Path of Totality (2016) First limb- and terrain-corrected product. Constants table with WGS84, EGM96, 1737.4 km, 696,000 km (959.645 arcsec), DE421, EOP kernel, Delta T 68.917 s. Read from Wayback capture 2026-08-28.
  37. 37primary NASA SVS, 2024eclipse_shapefiles.zip (78.6 MB) Downloaded from Wayback capture 2025-02-12 and parsed: umbra_hi 6741 records at 1 s 17:56:00 to 19:48:20 UTC with 12 attributes; umbra_lo 1181 at 10 s; center, duration, ppath, ppath01, upath_hi, upath_lo; all .prj GCS_WGS_1984.
  38. 38primary NASA SVS, cities-eclipse-2024.json Downloaded from Wayback capture 2025-02-12: 32,174 objects with STATE, NAME, LAT, LON, ECLIPSE (5 or 6 UTC times); contacts to 1 s, partial phases to 10 s.
  39. 39trade Besselian Elements team, Experimentally Testing Eclipse Maps Accuracy (2024) Read via WebFetch summary. Stephenville, Texas, 2024 April 8. Observed totality 13.7 s (C2 18:39:06.6, C3 18:39:20.3 UTC) versus six predictions from 12.9 s (Irwin) to 65 s (timeanddate). Authors' own experiment, so trade grade.
  40. 40trade Dunham (2024), April 8th Total Solar Eclipse, the Ultimate Lunar Occultation, IOTA page updated 2024 May 6 Full HTML read from a saved copy (var/downloads/iota.jhuapl.edu_TSE20240408.htm.html). Solon, Maine site 3 km north of the predicted southern limit, 43 s of totality, beads over a minute each side. Compares Jubier's and Irwin's limits and recommends umbral depth of at least 2.0 km. States IOTA's revised view that its earlier solar-radius variations were observational error.
  41. 41company SunPy 8.0.0 documentation, eclipse_amount Read official API page on 2026-09-30. Returns obscuration percentage, not a 0-to-1 fraction or contact times. Constant lunar radius IAU 0.2725076 or minimum 0.272281, light-time included, JPL ephemeris recommended.
  42. 42company Swiss Ephemeris swecl.c (eclipse routines) Read from a depth-1 clone at commit 9083a12 (2026-09-14), SE_VERSION 2.10.03. Functions eclipse_where, eclipse_how, eclipse_when_loc and their constants and comments.
  43. 43primary NASA GSFC, Besselian elements for all 11,898 eclipses of the Five Millennium Canon, CSV export Downloaded (5.95 MB, 11,899 lines). Header read: catalog columns plus t0, cubic x and y, quadratic d, mu, l1, l2, tan f1, tan f2, tmin -3 to tmax +3 h, and six undocumented trailing columns PNS, UNS, NCN, nSer, nSeq, nJLE.
  44. 44primary NASA SVS 5073: The 2023 and 2024 Solar Eclipses: Map and Data Read from a Wayback Machine snapshot (2025) because svs.gsfc.nasa.gov refused connections. Lists the shapefile and KML contents and the data sources (SRTM, LRO, DE421).
  45. 45primary PDS Geosciences Node. LRO LOLA GDR label ldem_128.lbl (LRO-L-LOLA-4-GDR-V1.0, V3.0) Read. 128 pix/deg, 236.901 m/pix, A_AXIS_RADIUS 1737.4 km, OFFSET 1737400, SCALING_FACTOR 0.5, 16-bit, MEAN EARTH/POLAR AXIS OF DE421, data 2009-07-13 to 2016-11-29.
  46. 46primary NAIF generic planetary SPK summaries (aa_summaries.txt) Read. Exact start and end epochs of de430, de431, de432s, de435, de438, de440, de440s, de441 parts, de442 and de442s.
  47. 47peer-reviewed Explanatory Supplement to the Astronomical Almanac, P. K. Seidelmann ed. (University Science Books, 1992) Sections 2.553, 3.244, 3.283, 3.351, 3.352, 7.3, 8.12, 8.342, 8.353, 8.362 and 8.363 read from archived original OCR. Section 8.363 reread on 2026-09-30: the OCR gives a negative longitude correction and labels longitude eastward, followed by table interpolation. Typeset page 467 is not confirmed here. This transcription is not used for the independently derived NASA/JSEX fixed-TT east-longitude sign. Downloaded HTML/OCR capture SHA256 f43edb6c9a5c287a00ba87d8f5a27090be75ddd4e026ff009fe8449dae834e9d.
  48. 48primary deltat.preds: long-term predictions of TT - UT1 (USNO) Read on 2026-09-15. Columns MJD, year, TT-UT1, UT1-UTC, error. 2025.0: 69.04 +/- 0.088 s; 2026.0: 69.05 +/- 0.189 s; 2028.0: 69.34 +/- 0.486 s; 2030.0: 69.97 +/- 0.768 s.
  49. 49primary NAIF lunar frame kernel moon_de440_250416.tf Read. MOON_PA_DE440 and MOON_ME_DE440_ME421 definitions, TKFRAME angles (67.8526, 78.6944, 0.2785) arcsec about axes (3,2,1), 0.02886 deg = 875 m, DE440 ME vs DE421 ME at most 53.4 cm over 2000-2040.
  50. 50peer-reviewed Archinal et al. (2011). Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009. Celestial Mechanics and Dynamical Astronomy 109, 101-135 Read from the PDF (var/downloads/archinal2011_wgccre2009.txt). ME system recommended, 860 m PA/ME difference, closed formulae valid to about 150 m, DE421 the best lunar ephemeris with libration angles in the file, Moon mean radius 1737.4 +/- 1 km with equatorial and polar radii the same.
  51. 51peer-reviewed HMNAO and USNO (1961), Explanatory Supplement to the Astronomical Ephemeris, section 9 examples Local scan text read (var/downloads/es1961_djvu.txt). Worked examples 9.2 to 9.9 for the eclipse of 1961 February 15, usable as hand-checkable reference cases.
  52. 52primary Espenak & Anderson (2001), Total Solar Eclipse of 2001 June 21, NASA TP-2001-209484 Local text read (var/downloads/TP209484_2001.txt). DE200/LE200, Watts corrections of 0.4 arcsec, graze-zone accuracy +/- 0.3 arcsec, advice to stay 1 km inside the interior limit, Elev Fact terrain factor, consumer GPS +/- 100 m, worked Lusaka limb-correction example.
  53. 53trade Guhl & Tegtmeier (2018), Baily's Beads Observations during the Total Solar Eclipse 2017 August 21, Journal for Occultation Astronomy 2018-3, pp. 19-21 Original PDF and paper extraction checked 2026-09-30. Southern video is overexposed and unreliable; no universal signed timing correction is given. Northern result 959.66 arcseconds. Original PDF SHA256 2ea6655d62dce0155e18a643d4fa2963d4935bb70a8e184008c65e071b57dbe9.
  54. 54trade van Gent R. H., A Catalogue of Eclipse Cycles, list of eclipse cycles Read live 2026-09-15 via curl and text extraction. Source of the Kluepfel 1985 algorithm for the Saros number from the lunation number, the lunation-number offsets (Brown -953), the odd/even node rule, series lengths 1226 to 1550 years, and the k = m I + n S statement. Verified against all 11,898 NASA rows.
  55. 55trade Jean Meeus, "Elements of Solar Eclipses 1951-2200" (Willmann-Bell, 1989) Not read. The book's local-circumstances chapter is known here only through Jubier's acknowledgement that it supplied the algorithms of his calculator, Photo Ephemeris' method statement, and Bill Gray's review. Its formulation is the Explanatory Supplement one.
  56. 56primary USNO Astronomical Applications Department, The Explanatory Supplement to the Astronomical Almanac (page describing the 2013 third edition) Read. States the third edition is a complete revision of 1992 and links errata. Used only for the status of chapter 11. The 2013 chapter itself was not read.
  57. 57company Occult v4 (David Herald) home page Fetched with curl 2026-09-15. Feature list, C#/.NET 4.5, 42 data files, installer sizes. Eclipse-specific help is inside the Windows help file and was not read.
  58. 58company timeanddate.com: Accuracy of Eclipse Times (Bikos) Read from a Wayback Machine snapshot (2025) because the live site returned 403. States sea-level assumption, no limb profile, ΔT and solar radius caveats.