Computing Solar Eclipses — Research

Executive summary

workingupdated 2026-09-16
  • Every published eclipse product derives from one algorithm, Bessel's fundamental-plane method as set out in the 1961 Explanatory Supplement and restated in 1992. The algebra is settled and quoted in full in the raw notes. What separates predictors is their constants and data, not their mathematics 1 2.
  • Three inputs decide the edge of the path: the solar radius, the lunar limb profile, and the observer's terrain height. Each is worth 0.6 to 3 km at a path limit and seconds to tens of seconds of totality there. Everything else is below 100 m once ΔT is refreshed and the umbral kk or a limb profile is used; the ephemeris is under 1 m 3 4 5.
  • The state of the art is published and reproducible in principle. Wright and Young 2024 describe the NASA SVS raster method: every map pixel is lifted to terrain height and tested against an 18,000-element lunar limb profile built from LRO and SELENE topography. The paper is open access. The code is not 6.
  • The solar radius is the unresolved constant. Almost every predictor uses Auwers' 1891 value of 959.63 arcseconds. Four eclipse campaigns since 2010 measure 959.95 to 960.01 arcseconds. The difference moves each limit by about 600 m and, at one 2024 edge site, was the last 11 s of a 65 s smooth-Moon prediction that observation cut to 13.7 s 7 8.
  • Enumeration and the ephemeris are solved problems. Meeus's approximate method, implemented from its quoted constants, finds all 221 eclipses of 1951 to 2050 with no type errors, and JPL DE440 places the Moon to about a metre. Effort belongs in ΔT, the limb and the radius 9 10 11.
  • No open-source code computes a limb-corrected path edge. Stellarium is the complete open Besselian implementation with a smooth Moon, and NASA's GPL JavaScript is the reference for local circumstances. A limb-and-terrain pipeline has to be written 12 13.

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.

One algorithm, many constants

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 2 12.

The predictors differ because they choose differently on four numbers.

Constant Values in use Effect of the difference
Solar radius s0s_0 959.63″ (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 14
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 15 16
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 17 4 18
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 19 20

The ephemeris is not on the list. Every JPL ephemeris since DE421 puts the Moon within a few metres of truth in the present century, which is under 10 ms of contact time 11 21.

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 the only public ground truth for a limb-corrected product 22 23 24.

What the observations say

Every quantitative test of a prediction has been made at a path limit, because the sensitivity there is about ten times that of the central line 7. 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″ was within a second 8. 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 3. 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 25.

What exists to build on

Open code covers the smooth-Moon problem completely and the limb problem not at all. 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. Skyfield, astropy, PyEphem, libnova, NOVAS and SOFA ship no solar-eclipse routine. 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 12 13 26 27.

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 28 29 30 31 17.

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 every limit is drawn as a band. With ± 0.05″ the band is about ± 0.1 km 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 32 33.
  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 34 35.
  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 36 37 38.

The limits of this statement

Only two results here are reproduced independently: Meeus's enumeration series and Kluepfel's Saros formula, each implemented from the quoted constants and checked against the NASA ASCII catalogue 9 39. Every Besselian, limb and terrain number is quoted from its source rather than recomputed. Meeus's Elements of Solar Eclipses 1951-2200 and chapter 11 of the 2013 Explanatory Supplement are cited only through implementations that transcribe them. Occult's help file, Solar Eclipse Maestro's statement of its ephemeris and ΔT, timeanddate's ephemeris and ΔT source, and the SVS 2024 ΔT are all unavailable. These are the first entries in open questions.

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 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.
  3. 3peer-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.
  4. 4primary 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.
  5. 5primary 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.
  6. 6peer-reviewed Wright, E. and Young, C. A. 2024, A Raster-oriented Method for Creating Eclipse Maps, AJ 168, 163 The paper. Read in full from the Wayback capture of the IOP PDF dated 2024-11-19 (iopscience.iop.org/article/10.3847/1538-3881/ad6b23/pdf), extracted with pdftotext. CC BY 4.0. Sections 4 and 5 give the limb-profile construction and the limb test; Appendix A gives the SPICE calls.
  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 Defines the eclipse solar radius, reports 959.95 ± 0.05 arcseconds from the 2017 flash spectrum at the southern limit, gives duration and limit-distance sensitivity to the radius and compares Occult, Solar Eclipse Maestro and Irwin's model. Full arXiv PDF read.
  8. 8trade 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.
  9. 9trade 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.
  10. 10primary 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.
  11. 11peer-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.
  12. 12company Stellarium src/core/SolarEclipseComputer.cpp Read from a sparse clone at commit 69888f4 (2026-09-14). 2,183 lines. Besselian elements, k and s constants, Explanatory Supplement references, KML and PNG map export.
  13. 13primary 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.
  14. 14peer-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).
  15. 15primary 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.
  16. 16primary 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.
  17. 17primary 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.
  18. 18company 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.
  19. 19primary 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.
  20. 20peer-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.
  21. 21primary 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).
  22. 22primary 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.
  23. 23primary 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.
  24. 24primary 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.
  25. 25trade 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.
  26. 26company 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.
  27. 27company 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.
  28. 28primary 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.
  29. 29primary 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).
  30. 30primary 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.
  31. 31primary 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.
  32. 32peer-reviewed Explanatory Supplement to the Astronomical Almanac, P. K. Seidelmann ed. (University Science Books, 1992) Sections 2.553 (Stephenson and Morrison 1984 parabolas), 3.244 (terrestrial coordinates, MERIT 1983 ellipsoid), 3.283 (low-precision refraction, 34 arcmin horizontal refraction), 3.351 and 3.352 (IAU 1976 ellipsoid, GMST), 7.3 (apparent places include aberration), 8.12, 8.342, 8.353, 8.362 and 8.363 (eclipses: apparent places, shadow radius at height, height above the geoid, refraction as a refinement, the delta-T longitude correction). Read from the archive.org OCR text in var/downloads/es1992_djvu.txt.
  33. 33primary 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.
  34. 34primary 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.
  35. 35peer-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.
  36. 36peer-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.
  37. 37primary 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.
  38. 38trade Guhl & Tegtmeier (2018), Baily's Beads Observations during the Total Solar Eclipse 2017 August 21, Journal for Occultation Astronomy 2018-3, pp. 19-21 Full PDF read (var/downloads/JOA2018_3.txt). Two IOTA/ES stations, Thermopolis WY 700 m inside the northern limit and Cape Girardeau MO region. Bead-by-bead radius residuals reduced with Occult 4.5.3.0 using LOLA and Kaguya.
  39. 39trade 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.