Computing Solar Eclipses — Research

Error budget

workingupdated 2026-09-30
  • The sources compared here quantify different error components. Quaglia and colleagues analyse ephemeris, Earth orientation, topography and solar radius. Espenak describes limb and ΔT effects. Wright and Young specify terrain treatment. Their shifts and bounds are not all independent standard uncertainties 1 2 3.
  • Omitted terrain and limb can move limits by kilometres. Solar-radius choices move the cited modern limits by hundreds of metres. Other omitted corrections and inconsistent conventions can also exceed 200 m 4 5 6.
  • Mixed reduction levels and incorrect time conventions can cause much larger errors. The quoted inconsistent Sun/Moon reduction example reaches 38 km. A 69 s Earth-rotation mistake is about 32 km at the equator 7 8.
  • The cited solar-radius component is about 100 m for ± 0.05 arcseconds. It is separate from the reported 60 to 130 m inter-implementation spread. Neither is a universal prediction-accuracy floor 1.
  • Huber's future ΔT model grows with lead time. Its values are about 0.05 s at one year, 4.6 s at twenty and 52 s at a hundred. The separate historical rule σ=0.8t2\sigma=0.8t^2 applies to 1000 BCE through 1200 CE, with tt in centuries from 1820 9 10.

The ranked budget

"Mid-path" is a site near the central line of a 2017-class eclipse with a shadow speed near 1 km/s. "Edge" is within a few hundred metres of a limit. Conversions used: 1″ at the Moon's mean distance is 1.86 km on the ground; 1 s of ΔT is 15.041″ of longitude, 465 m at the equator and 356 m at 40°; the relative Moon-Sun motion of 0.364″/s gives 2.7 s of contact time per arcsecond 11 12.

Rank Term Mid-path contact time Edge position or duration Included by
1 Geometric Sun with apparent Moon 55 s 38 km Everyone avoids it; test for it 7
2 TT used as UT Contact-time effect depends on geometry; 69 s of rotation is not a uniform contact shift 32 km at the equator Everyone avoids it; test for it 13
3 Sphere instead of ellipsoid tens of s up to 21 km Everyone 14
4 Terrain omitted seconds where the Sun's azimuth lies along the track hcot⁡ah\cot a; up to 3 km in the 2017 western states SVS only, for the path; single-site altitude in Jubier, Occult, Swiss Ephemeris 4 15
5 Lunar limb omitted 1 to 3 s, up to 15 s in extreme geometry 1 to 3 km per limit; 33 s of duration at Stephenville 2024 SVS, Jubier, Occult, Irwin, Photo Ephemeris simulator 5 16
6 Solar radius convention 1.8 s of central duration for 959.63 to 960.00 arcseconds in the cited case About 600 m per limit for 959.63 to 959.95 arcseconds. The Vale comparison for 959.63 to 960.00 arcseconds reduces duration by 19.3 s. Stephenville is a cross-program comparison, not an isolated radius effect Irwin uses 959.95″, Photo Ephemeris in its bead simulator only; everyone else 959.63″ or 696,000 km 6 17
7 Lunar radius kk, umbral, if the IAU value is used about 4 s of duration 1.4 km per limit Espenak, Stellarium, Jubier use 0.272281; USNO and Astronomy Engine use a mean radius; the Swiss Ephemeris uses 1738.15 km globally but scales the lunar radius by 0.99916, which is 0.272281, for C2 and C3 18 19
8 ΔT prediction error, realised in 2024 1.4 to 2.3 s 500 to 800 m east-west at 40° All; the miss depends on when the prediction was frozen 20 21
9 Centre of figure against centre of mass, smooth-Moon products up to 1.4 s 0.5 to 1 km Absorbed by any DEM-based profile; not applied by USNO 22 23
10 Watts-era limb data 0.5 s 600 to 750 m systematic at some position angles Superseded by Kaguya and LOLA 24 25
11 Geoid against ellipsoid height under 1 s Ncot⁡aN\cot a, up to 170 m at a 30° Sun SVS (EGM96); nobody else documents it 26
12 Implementation spread on identical inputs 2 to 4 s at Vale 2017 0.03″ to 0.07″ of radius, 60 to 130 m Occult, Solar Eclipse Maestro, Irwin 1
13 Solar radius residual, ± 0.05″ 0.1 s 100 m band Radius component for the cited geometry 6
14 Limb sampling coarser than 18,000 elements 0.25 s for 0.2° bins part of the implementation spread SVS at 0.02° 27 28
15 Refraction, Sun above 5° No validated general contact-time bound in the cited sources Common angular limb remapping preserves contact; physical ray geometry is a separate model Nominal geometric contacts omit refraction 15 6
16 Refraction near the horizon No validated contact-time bound in this corpus Changes displayed altitude and visibility; anomalous conditions need separate treatment State the atmospheric and horizon model 7 15
17 Sidereal time inconsistency, mean against apparent up to 1.1 s up to 0.5 km Consistent in every predictor documented 29
18 UT1 against UTC up to 0.9 s up to 0.42 km UT1 printed by Espenak, SVS, Swiss Ephemeris 30
19 Moon light-time omitted No bound follows from geocentric lunar speed alone A reception calculation retards the barycentric target. Observer aberration is separate and can partly cancel common orbital motion Use one consistent reduction chain 31 32
20 Ephemeris, DE421 to DE440 Depends on the projected position difference and radial rate This pair's displacement is not quantified by the cited Park paper. Its range residuals do not certify a three-dimensional position bound Pin the releases, epoch and vector comparison 33
21 Ephemeris, DE403 to DE421, at 2020 0.02 s 16 m Irrelevant 34
22 ELP-2000/82 truncated as in the Canon 0.02 s 10 m Irrelevant for paths 35
23 Polar motion 0 about 10 m SPICE kernels include it; nobody documents it 30
24 Nutation model, 1980 against 2000A 0 under 20 m Irrelevant 29
25 TT against TDB 1.7 ms 2 m Irrelevant 29
26 Observer position, consumer GPS none on time ± 100 m The observer's problem 25
27 Observer clock, visual timing "a couple of seconds" same The observer's problem 16
The ranked error budget, as position error at an eclipse limitA horizontal bar chart on a logarithmic axis running from one metre to ten kilometres. Twenty terms from the error budget are ranked from largest to smallest. Two implementation bugs head the list at tens of kilometres: a geometric Sun paired with an apparent Moon at 38 km and TT used where UT belongs at 32 km, with a sphere used instead of the ellipsoid at 21 km behind them. Three shaded bars follow at kilometre scale and are the terms the corpus calls dominant: terrain omitted at up to 3 km, the lunar limb omitted at 1 to 3 km, and the choice of solar radius at 600 m. Below them the lunar radius constant, the centre-of-figure offset and the delta T prediction error lie between 400 m and 1.4 km. Watts-era, sidereal-time and UT1/UTC effects also exceed 200 m. The ephemeris terms at the foot are comparisons between releases, not accuracy bounds against truth. Bars mix model omissions, differences and uncertainty components; they cannot all be summed in quadrature.1 m10 m100 m1 km10 km38 km1Geometric Sun with apparent Moon32 km2TT used as UTup to 21 km3Sphere instead of ellipsoidup to 3 km4▸ Terrain omitted1 to 3 km5▸ Lunar limb omitted1.4 km7Lunar radius k, umbral, IAU value0.5 to 1 km9Centre of figure against centre of mass500 to 800 m8ΔT prediction error, realised in 2024600 to 750 m10Watts-era limb data600 m6▸ Solar radius 959.63″ against 959.95″up to 500 m17Sidereal time, mean against apparentup to 420 m18UT1 against UTCup to 170 m11Geoid against ellipsoid height60 to 130 m12Implementation spread, identical inputs100 m band13Solar radius residual, ± 0.05″under 20 m24Nutation, 1980 against 2000A16 m21Ephemeris, DE403 to DE421, at 202010 m22ELP-2000/82 truncated as in the Canonabout 10 m23Polar motion2 m25TT against TDBedge position error, metres, log scalerankterm, as the error budget ranks it▸ the three terms the corpus calls dominant. A tick inside a bar marks the low end of a stated range.

ΔT by lead time and epoch

The a priori uncertainty of ΔT, from Huber's model as used by NASA for the future and from Morrison and Stephenson's σ=0.8t2\sigma = 0.8t^2 rule with tt in centuries from 1820 for 1000 BCE through 1200 CE. The other historical rows use NASA's epoch-specific estimates 9 10.

Lead time or year σ(ΔT)\sigma(\Delta T) Longitude at 40°
1 year ahead 0.05 s 18 m
5 years ahead 0.6 s 0.2 km
20 years ahead 4.6 s 1.6 km
100 years ahead 52 s 18 km
Year 1900 0.1 s 36 m
Year 1700 5 s 1.8 km
Year 1000 54 s 19 km
Year 0 265 s 94 km
Year −1000 636 s 226 km
Year 3000 1885 s 670 km

Successive historical analyses differ by more than these sigmas: the 1986 and 1997 Stephenson curves differ by 1294 s at AD 300. A past-eclipse product must draw the gore, as the Canon does for every year before 1 and after 2300 36 37.

The two modes

The budget separates cleanly into what a product can control and what it chooses.

Almanac mode. Smooth Moon with k2=0.272281k_2 = 0.272281 and k1=0.2724880k_1 = 0.2724880 to match the NASA GSFC tables or 0.27250760.2725076 to match EclipseWise and the bulletins, s0=959.63″s_0 = 959.63″, sea level, the ΔT printed with the table. The result matches NASA or EclipseWise to 1 km at the limits and 0.1 s at a site, and carries their modelling choices. Omitted limb and terrain and a different radius can move a limit by kilometres. The table does not establish a combined 2 km standard uncertainty for those omissions. This is the mode for catalogues, for reproducing published tables, and for regression tests.

Edge mode. LOLA profile at 18,000 elements, terrain with geoid, s0=959.95″±0.05″s_0 = 959.95″ ± 0.05″, ΔT refreshed to the latest USNO or IERS value. The cited comparisons give a radius component near 100 m and implementation spread of 60 to 130 m, with 2 to 4 s duration differences at the Vale site. These are case-specific scales, not a calibrated combined uncertainty. Dunham's practical margin of 2 km of umbral depthumbral depthThe perpendicular distance from an observing site to the nearer limit of the umbral path, as reported by Jubier's map. IOTA recommends at least 2 km when the limit was computed with the canonical radius. covers the instrumental sensitivity to faint beads that no computation removes 38.

The two modes differ by design and must not share a file. A map that draws an almanac-mode limit next to an edge-mode duration misleads by 600 m.

How to report it

Correct known biases before estimating uncertainty. Keep unresolved model choices and implementation discrepancies separate from quantified input uncertainties. Express each input's standard uncertainty in the measurement model, rather than treating an omitted correction's full size as its standard deviation. Quadrature requires justified independence. Correlated inputs require covariance terms 39.

For y=f(x1,…,xN)y=f(x_1,\ldots,x_N), JCGM 100:2008 equation (13) gives

uc2(y)=∑i=1N(∂f∂xi)2u2(xi)+2∑i=1N−1∑j=i+1N∂f∂xi∂f∂xju(xi,xj).u_c^2(y)=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial x_i}\right)^2u^2(x_i) +2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j}u(x_i,x_j).

Here u(xi)u(x_i) is the standard uncertainty of input xix_i, and u(xi,xj)u(x_i,x_j) is their covariance. The derivatives convert each input's units to the output's units. Apply this separately to each contact time, duration or path coordinate. If distributions and correlations are not established, report scenario ranges rather than assigning a coverage probability 39.

Draw limits for the adopted solar radius and its stated alternatives. Treat the cited 100 m radius effect as one component for that geometry. Show umbral depth near a limit, and state the detection convention. Propagate ΔT uncertainty for historical and distant-future products. Store the configuration with every result 6 1.

References

  1. 1peer-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.
  2. 2primary 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.
  3. 3peer-reviewed Wright & Young (2024), A Raster-oriented Method for Creating Eclipse Maps, AJ 168:163 Full text read from a saved copy (var/downloads/wright2024_aj_clean.txt). Documents the SVS method: DE421, SLDEM2015 and LDEM, limb profile of 18,000 elements, h cot a terrain shift, polygonal umbra, and the 2024 solar-radius controversy.
  4. 4primary 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.
  5. 5primary 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.
  6. 6peer-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.
  7. 7peer-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.
  8. 8company Skyfield documentation: Positions Read. Definitions of barycentric, astrometric (light-time) and apparent (aberration and deflection) positions, ICRS and GCRS usage, ICRS axes within 0.02 arcsec of J2000.
  9. 9primary Uncertainty in Delta T (NASA Eclipse Web Site) Read in full. Huber Brownian-motion model for sigma outside the observed span (Q = 0.058 ms^2/yr, M = 2500 yr), the 0.8 t^2 parabola for 1000 BCE to 1200 CE, and the longitude equivalents of the errors.
  10. 10primary Espenak, Uncertainty in Delta T, NASA GSFC eclipse site (2007), adapted from the Five Millennium Canon Full HTML read via curl. Morrison & Stephenson 2004 sigma = 0.8 t^2, tables of sigma and longitude uncertainty from -4000 to +5000, Huber 2000 model.
  11. 11primary Espenak & Meeus (2009), Five Millennium Catalog of Solar Eclipses, NASA TP-2009-214174, and the Canon text (2006) section 1.6 Map Accuracy Local text read (var/downloads/5MCSE-Text11.txt and TP2009-214174.txt). Lunar ephemeris better than an arcsecond within centuries, 240 s of Delta T equals 1 degree of longitude, reference gores when sigma exceeds 265 s.
  12. 12primary Espenak, Anderson (1999). Total Solar Eclipse of 2001 June 21. NASA/TP-1999-209484 Read from the PDF text (var/downloads/TP209484_2001.txt). Mean lunar radius section (k history, 1986 Oct 03 case, penumbral 0.2725076 and umbral 0.272281), lunar limb profile section (Watts datum ellipticity, centre of figure, 0.4 arcsec systematic errors, topocentric libration -3.1 to -4.6 deg, 0.364 arcsec/s), centre-of-figure shift +0.53/-0.13 arcsec, DE200/LE200 as the ephemeris, older -0.6 arcsec latitude convention.
  13. 13company Espenak, EclipseWise: Total Solar Eclipse of 2024 Apr 08, prime page Read. Predictions from JPL DE405, k (penumbra) 0.2725076 and k (umbra) 0.2722810, Delta T 71.5 s, lunar coordinates with respect to the centre of mass, UT1 = TD - Delta T.
  14. 14peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (HMSO, 1961) Section 6 (figure of the Earth: Hayford spheroid, the S and C functions) and Section 9B (eclipses: observer coordinates, Bessel's parametric-latitude device, rising and setting curves without refraction). Read from the archive.org OCR text kept in var/downloads/es1961_djvu.txt.
  15. 15primary Total Solar Eclipse of 2001 June 21, F. Espenak and J. Anderson, NASA/TP-1999-209484 Read from the PDF text kept in var/downloads/TP209484_2001.txt. States that predictions use centre-of-mass positions with no refraction or limb corrections, that local circumstances are at sea level unless the elevation is known, and defines the elevation factor tan(90-A) sin(D) for shifting the path limits.
  16. 16trade 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.
  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 Espenak, Circumstances calculator for the Total Solar Eclipse of 2024 Apr 08, EclipseWise Read via WebFetch summary. DE406, Delta T = 71.3 s, coordinates relative to the Moon's centre of mass. City rows are generated by the calculator and were not extracted.
  22. 22peer-reviewed Jones, Nichols-Fleming, Evans, Johnson, Andrews-Hanna (2025). Can the Moon's Center of Mass-Center of Figure Offset Be Explained With a Uniform Primordial Crust? Journal of Geophysical Research: Planets Abstract only, via the Semantic Scholar API. Quotes the 1.935 km lunar COM-COF offset as the constraint.
  23. 23primary USNO, 2024 April 8 Total Solar Eclipse, Astronomical Applications Department Read via WebFetch. IAU radii Sun 696,000 km and Moon 1737.4 km, no limb profile, no centre-of-figure correction, contacts found by iterating topocentric positions.
  24. 24peer-reviewed Morrison, L. V. and Appleby, G. M. (1981). Analysis of lunar occultations III. Systematic corrections to Watts' limb-profiles for the Moon. MNRAS 196, 1013-1020 Read in full from the ADS scan (OCR text). Source of the harmonic correction formula, the 1737.97 km datum radius, the +0.04 arcsec radius term, the -0.18 arcsec latitude shift, the -0.09 arcsec ellipticity, the +0.50 arcsec sin Q centre-of-figure term, and the 0.4 arcsec peak error.
  25. 25primary 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.
  26. 26primary NASA SVS 4515: 2017 Total Solar Eclipse in the U.S., umbra animation with terrain and limb (E. Wright) Read from the Wayback Machine snapshot of 2026-01-14. Lists Earth radius 6378.137 km, Ellipsoid WGS84, Geoid EGM96, DE421, SPICE earth orientation kernel, Delta UTC 69.184 s and delta-T 68.917 s, DEM SRTM (SIR-C), lunar DEMs LOLA and SLDEM2015.
  27. 27peer-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.
  28. 28trade Luca Quaglia, Konstantinos Emmanouilidis and John Irwin, "Timing of the internal contacts of the 2013 Nov 03 total solar eclipse by flash spectrum analysis" (Besselian Elements) Read. Observed C2 about 1 s later and C3 about 1 s earlier than predicted with 959.63 arcsec, duration about 2 s shorter; limb-profile sampling of 0.2 degrees can move a contact by 0.25 s; compatible radius 959.90 to 960.00 arcsec.
  29. 29primary The IAU Resolutions on Astronomical Reference Systems, Time Scales, and Earth Rotation Models, G. H. Kaplan, USNO Circular 179 (2005) Read from the PDF (var/downloads/circ179.txt). Equation 2.6 for TDB - TT (0.001657 s leading term), the statement that using TT for TDB errs by under 2 ms and under 1 mas for the Moon, the equation of the equinoxes with amplitude about 1 s, and the statement that the IAU 2000 resolutions change quantities only at the level of tens of milliarcseconds.
  30. 30primary IERS Bulletin A, Vol. XXXIX No. 037 (10 September 2026) Read. UT1-UTC = 0.000946 s on MJD 61287, TAI-UTC = 37 s since 2017 Jan 1, no leap second in December 2026, DUT1 = 0.0 s from 2026 Apr 9, polar motion x = 0.20025 arcsec and y = 0.33395 arcsec, prediction accuracies for UT1-UTC of 1.4, 2.4, 3.2 and 4.0 ms at 10, 20, 30 and 40 days.
  31. 31primary NAIF CSPICE documentation: spkezr_c Read. Input et in seconds past J2000 TDB; aberration options NONE (geometric), LT (light time, planetary aberration), LT+S (plus stellar aberration), CN, CN+S (converged), XLT and XCN transmission cases.
  32. 32primary Folkner, Williams, Boggs, Park, Kuchynka (2014). The Planetary and Lunar Ephemerides DE430 and DE431. JPL IPN Progress Report 42-196 Read in full from the PDF (var/downloads/folkner2014_de430.txt). Frame ICRF2, TDB definition and integrated TT-TDB, lunar core-mantle damping and the 1550-2650 span, Euler angle definitions, LLR data table, mass table (Sun/Jupiter 1047.348625, Sun/Saturn 3497.901768), au = 149597870.700 km.
  33. 33peer-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.
  34. 34primary Williams, Boggs, Folkner (2008). DE421 Lunar Orbit, Physical Librations, and Surface Coordinates. JPL IOM 335-JW,DB,WF-20080314-001 Read from the PDF. DE421 vs DE403 differences (6 m in 2008 to 16 m in 2020, 3 to 9 mas), DE421 PA to ME rotation Rx(-0.30") Ry(-78.56") Rz(-67.92"), 1" = 8.4 m on the surface.
  35. 35primary Espenak, Meeus (2009). Five Millennium Catalog of Solar Eclipses: -1999 to +3000. NASA/TP-2009-214174 Read from the PDF text (var/downloads/TP2009-214174.txt). Sections 1.3 (VSOP87D, ELP-2000/82 with 37,862 terms, truncation at 0.0005 arcsec, 1/40 s), 1.4 (secular acceleration -25.858 arcsec/cy2 and the Delta T correction c), 1.5 (k = 0.2724880 penumbral, 0.272281 umbral, IAU 0.2725076 history) and the centre-of-figure paragraph (+0.50 arcsec longitude, -0.25 arcsec latitude, ignored).
  36. 36primary Espenak, Historical Values of Delta T, NASA GSFC eclipse site (2012) Full HTML read via curl. Table comparing Stephenson & Houlden 1986 with Stephenson 1997 values, differences up to 1294 s at AD 300.
  37. 37primary Espenak F., Meeus J. (2006) Five Millennium Canon of Solar Eclipses: -1999 to +3000, NASA/TP-2006-214141 Read in full as PDF text (2,625 lines). Sections 1.3 to 1.6 give ephemerides, secular acceleration, k values and the map-accuracy statement with the reference-gore example for -1996 Oct 04. Note the Text10 link on the NASA publication page returns 404, Text11 is the live file.
  38. 38trade 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.
  39. 39primary JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement (2010 corrected edition) Sections 3.2, 5.1 and 5.2 read on 2026-09-30. Equation (13) supports propagation with covariance, and the uncorrelated special case requires justified assumptions. Does not validate an eclipse-specific physical error budget.