Computing Solar Eclipses — Research

Executive summary

workingupdated 2026-09-30
  • 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 2 3.
  • 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 4 5 6.
  • 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 3.
  • 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 7 8 9.
  • 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 10 11 12.
  • 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 13 14 15.

The question and the answer

The question is what a developer needs to know to compute every solar eclipse product, from the list of eclipses in a period to the second at which totality begins at one place, at the depth of NASA SVS's 2024 work. The answer is a pipeline of nine stages, given in pipeline design, and a short list of decisions that matter more than the rest. The decisions follow.

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 tan⁡f1f_1, tan⁡f2f_2 that describe the Moon's shadow relative to the fundamental plane, from which any eclipse circumstance can be computed. 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 16 13.

Four input choices help explain the differences between the surveyed predictors. Direct methods and raster tests also differ from the classical circular-shadow algorithm 4 3.

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 7 17
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 18 19
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 20 5 21
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 22 23

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 12 24 4.

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 3.

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 25 26 27.

What the observations say

Near-limit observations provide especially sensitive solar-radius tests. Published campaigns also include central-line observations 7 8. 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 28. 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 4. 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 29.

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 30. 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 13 14 31 15.

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 32 33 34 35 20.

What to build

The pipeline design 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 7.
  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 36 37.
  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 38 39.
  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 40 41 42.

The limits of this statement

Meeus's enumeration series and Kluepfel's Saros formula have comparisons against the NASA ASCII catalogue 10 43. 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 4. 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 44 45 46 47 3. 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 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.
  3. 3peer-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.
  4. 4peer-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.
  5. 5primary 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.
  6. 6primary 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.
  7. 7peer-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.
  8. 8peer-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.
  9. 9trade 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.
  10. 10trade 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.
  11. 11primary 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.
  12. 12peer-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.
  13. 13company 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.
  14. 14primary 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.
  15. 15company 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.
  16. 16peer-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.
  17. 17peer-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).
  18. 18primary 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.
  19. 19primary 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.
  20. 20primary 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.
  21. 21company 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.
  22. 22primary 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.
  23. 23peer-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.
  24. 24primary 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).
  25. 25primary 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.
  26. 26primary 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.
  27. 27primary 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.
  28. 28trade 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.
  29. 29trade 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.
  30. 30company 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.
  31. 31company 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.
  32. 32primary 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.
  33. 33primary 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).
  34. 34primary 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.
  35. 35primary 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.
  36. 36peer-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.
  37. 37primary 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.
  38. 38primary 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.
  39. 39peer-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.
  40. 40peer-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.
  41. 41primary 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.
  42. 42trade 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.
  43. 43trade 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.
  44. 44survey 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.
  45. 45primary 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.
  46. 46company 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.
  47. 47company 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.