Error budget and validation protocol

The question. For a modern total eclipse, how many seconds of contact time and how many metres of path edge does each input contribute at a mid-path site and at the edge, who has published those numbers, and what tests with what tolerances would show that a new implementation is right?

The budget

This table is the home of the sourced raw rows. The ranked version, which orders the same terms by size, is in error budget. The table combines published statements with arithmetic from published constants. Rows marked "arithmetic" are this note's own conversions and are explained below the table. "Mid-path" means a site near the central line of a 2017-class eclipse with a shadow speed near 1 km/s. "Edge" means within a few hundred metres of a limit. The edge column is in metres of limit position or, where the source gives it, seconds of duration.

Arithmetic used. One arcsecond at the Moon's mean distance of 384,400 km is 1.864 km, so $0.32''$ is 0.60 km. One second of $\Delta T$ is $15.041''$ of longitude at the sidereal rate, which is 465 m at the equator and 356 m at $40^{\circ}$ latitude. The Five Millennium Canon, NASA/TP-2006-214141, states the same rule as 240 s per degree ¹⁵. The $k$ difference is $0.2725076 - 0.272281$ Earth radii of 6378.137 km ^{24 21}. The terrain shift is $h \cot a$ ³.

A smooth-Moon, standard-radius, sea-level prediction omits several effects that reach kilometres in the cited cases. A reduced umbral $k$ already approximates valley depths, so its contribution overlaps limb omission. Quaglia describes a 100 m fuzzy band for its fitted radius and geometry, and a same-radius implementation spread equivalent to 60 to 130 m. These are different quantities. Neither establishes a universal coverage probability or combined accuracy bound ¹.

Who has published budgets

• Quaglia et al. 2021, section 1, states the accuracy of DE440, of the ITRS-GCRS orientation, of UT1 and of LOLA topography and concludes that "the solar radius is the most uncertain parameter of all these quantities" ¹.
• Espenak's map pages state the limb term for limits, durations and greatest-duration location, and the Canon states the ephemeris and $\Delta T$ terms with tables of $\sigma$ against year ^{2 15 7}.
• The 2001 bulletin states the Watts-era limb term, the graze-zone accuracy, the terrain factor and the GPS term ¹¹.
• The 1992 Explanatory Supplement states the limb term for central-line contacts and refers to Herald 1983 for "the total effect and its components" ^{21 25}.
• Wright and Young state the terrain rule and the limb sampling requirement ³.
• Dunham states the observational margin ²⁶.
• The compared SVS, Quaglia and Espenak sources do not provide a complete calibrated covariance model. Herald's graphical method and scan are documented in limb profile methods. Its original component budget still requires a numerical transcription ²⁷.

Validation of ΔT models against historical eclipses

Stephenson, Morrison and Hohenkerk analysed 180 timed Babylonian eclipse observations from $-720$ to $-9$, 111 timed Chinese observations from 434 to 1280, 11 Greek and 54 Arab timings. They added a set of untimed total and annular eclipses from $-708$ to 1567 and about half a million telescopic lunar occultations from 1623 onward ²⁸. The scatter of the timed records is 16 minutes for Babylonian observations after $-560$, 20 and 16 minutes for Chinese solar and lunar timings, about 14 minutes for Greek, and 5 and 13 minutes for Arab solar and lunar timings ²⁸. Untimed records constrain differently: "a report that an eclipse was total or near-total at a known place fixes the rotational frame of the Earth to within the projected width of the band of totality parallel to the Earth's equator. This width is usually only a few minutes of time" ²⁸. The long-term fit is

corresponding to a length-of-day increase of $+1.78 \pm 0.03$ ms per century, against $+2.3 \pm 0.1$ ms per century expected from tidal friction alone ²⁸. The 2021 addendum extends the spline fits to 2025 and tabulates uncertainties in HMNAO Table S15.2020. The spline and table details are reported secondhand through the ytliu0 implementation and later citations, and remain unconfirmed against the original table ^{29 30}.

Espenak's uncertainty page gives the Morrison and Stephenson 2004 rule $\sigma = 0.8\,t^2$ s with $t = (\text{year} - 1820)/100$, valid from $-1000$ to 1200. Decade fluctuations give about 20 s from 1300 to 1600. The telescopic era gives 5 s at 1700, 1 s at 1800 and 0.1 s at 1900. Huber's Brownian-motion model covers the years outside the record ⁷. The tabulated values are:

Source: ⁷. The Canon draws reference gores on every map whose $\sigma$ exceeds 265 s, which is every year before 1 and after 2300, and for $-1996$ shows the gore at $\pm 15.5^{\circ}$ of longitude for $\sigma = 3712$ s ¹⁵.

Two checks show what these sigmas mean in practice. Espenak's table of Stephenson and Houlden 1986 against Stephenson 1997 values differs by 644 s at $-500$, $-752$ s at 0, $-1294$ s at 300 and 210 s at 1300 ⁸, differences larger than the $\sigma$ rule at the same dates. Hayakawa et al. derived from Ming-dynasty records the bounds $-408 \le \Delta T \le 601$ s at 1361, $277 \le \Delta T \le 890$ s at 1514, $-328 \le \Delta T \le 332$ s at 1542 and $-1762 \le \Delta T \le 1091$ s at 1575. They state that these tighten the variations relative to the Morrison 2021 spline and require adjustments around 1361 and 1542 ⁹. The lesson for a developer is that a past-eclipse product must carry the $\sigma$ and should draw it, as the Canon does.

For the present, the realised errorsrealised ΔT errorThe difference between the ΔT value adopted when a prediction was published and the value later measured for the eclipse date. It shifts the whole path east or west by about 15 arcseconds of longitude per second of error. are small. The measured values are 68.8373 s on 2017 August 1 and 69.1983 s on 2024 April 1 ⁶. The 2017 predictions were within 0.4 s. The 2024 predictions made from the 2010s were 1.4 to 2.3 s high, because Earth's rotation sped up after 2016 and the extrapolations assumed continued slowing ^{2 18}. The Besselian Elements team frames the general rule: ephemerides, topography and the radius are fixed inputs, EOPEarth orientation parameters (EOP)The measured quantities UT1 − UTC, polar motion (x, y), length of day and celestial pole offsets, published by the IERS, that relate the Earth-fixed frame to the celestial frame. They cannot be known in advance and are the only prediction inputs that must be re-fetched before an eclipse. are the only inputs that must be re-fetched, and $\Delta T = 32.184\text{ s} + \text{leap seconds} - \text{dUT1}$ ³¹.

Accuracy statements by the predictors

Collected in the predictor-disagreements note and repeated here in one line each. Espenak: limits 1 to 3 km and durations 1 to 3 s from the limb, greatest-duration point 10 to 20 km, and full personal responsibility ^{2 24}. SVS: 100 m umbra shapes, 250 m path ³². Jubier: limb correction a few seconds, $\Delta T$ better than 0.5 s, refraction omitted ²⁰. Besselian Elements: $959.95'' \pm 0.05''$, 1-sigma limit lines ^{33 34}. USNO: no limb, no centre-of-figure correction ²³. Photo Ephemeris: smooth Moon, a few seconds and up to 15 s ⁵. Stellarium: 0.1 s iteration tolerance and no other statement ^{35 36}. Occult and timeanddate: none found ^{37 12}.

Validation protocol for a new implementation

The protocol has five levels. Each level names a reference casereference caseA published set of local circumstances (site, contact times, duration) from an authoritative implementation, used to test a new implementation to a stated tolerance., quantity and proposed agreement tolerance. These are numerical regression goals, rather than universal physical-accuracy bounds. Match source-specific inputs before applying a tolerance. TTTerrestrial Time (TT)The uniform time scale of the ephemerides and of the Besselian elements, equal to TAI + 32.184 s. Older eclipse tables call it TDT, TD or Ephemeris Time (ET). Elements are computed in TT and converted to UT1 with ΔT before any Earth rotation is applied. appears as TDT or TD in NASA and EclipseWise tables and as ET before 1984. Times labelled UT are UT1 as published, and observed times are UTC. Convert absolute timestamps to one scale before comparing contacts. Obtain DUT1 = UT1 − UTC from the precise IERS Earth-orientation record for the event. Use UT1 = UTC + DUT1 and UTC = UT1 − DUT1. Store the value, epoch and source. The permitted difference below 0.9 s cannot be ignored for subsecond tolerances ³⁸. A nearly constant DUT1 cancels in a short C3-minus-C2 duration.

Level 1: Besselian elements and global circumstances

• Reproduce the 1961 Explanatory Supplement worked examples 9.2 to 9.9 for the eclipse of 1961 February 15: test for occurrence, Besselian and auxiliary elements, a point on the central line and its duration, and outline curves ¹⁰. These are hand calculations from tabulated elements, so agreement is to the last printed digit once the same elements are input.
• Reproduce EclipseWise's 2017 August 21 greatest eclipse at 18:26:40.3 TD, path width 114.7 km, central duration 02m40.12s, with DE405, $\Delta T = 68.8$ s, $k = 0.2725076$ and $0.2722810$ ¹⁷. Tolerance: 0.1 s in time, 0.1 km in width.
• Reproduce the NASA eclipse site's 2024 greatest duration of 04m28.2s with the "VSOP87/ELP2000-85" the page names and $\Delta T = 70.6$ s, or the 2017 value of 2m40.2s with DE405 and 68.4 s ^{2 16}. Tolerance: 0.2 s, since the ephemerides differ.

Level 2: Local circumstances, smooth Moon

• The 2001 June 21 Lusaka case from the NASA bulletin with DE200/LE200: C2 13:09:19.3 UT at $P_2 = 118^{\circ}$, C3 13:12:32.8 UT at $P_3 = 247^{\circ}$ ¹¹. $P$ is the position angleposition angle PThe angle of a contact point on the solar limb, measured eastward from celestial north. The local formulas use P. The 1961 Explanatory Supplement calls it Q, distinct from this site's Q for global shadow-edge geometry. of the contact measured eastward from north, which the 1961 Supplement writes $Q$. Tolerance: 0.5 s, allowing for the ephemeris difference between DE200 and a modern DE.
• USNO's 2024 Solar Eclipse Computer, run for any city with height entered, as the check of a smooth-Moon solution using 696,000 km and 1737.4 km with no centre-of-figure correction ²³. Tolerance: 0.5 s after matching $\Delta T$.
• Stellarium's AstroCalc contact times for the same sites, with the knowledge that it uses $959.63''$ and iterates to 0.1 s ³⁵. Tolerance: 0.5 s.

Level 3: Limb corrections

• The Lusaka example: the bulletin's chart corrections of $+4.0$ s at C2 and $-1.2$ s at C3 give 13:09:23.3 and 13:12:31.6 UT, and the bulletin states these are within 0.2 s of a rigorous calculation with the actual limb profile ¹¹. A LOLA-based implementation should land within 1 s of the corrected values, the difference being Watts against LOLA.
• Jubier's LC column at any site, read from the interactive map, as a second opinion ²⁰. Tolerance: 1 s.

Level 4: Edge sites against observation

• Vale, Oregon, 2017: $117^{\circ} 13' 09.8''$ W, $43^{\circ} 57' 10.9''$ N, 711 m. With $s_0 = 959.63''$ the reference contacts are 17:25:34.3 and 17:26:06.9 UTC, 32.6 s. Occult gives 2 s or more longer and Solar Eclipse Maestro 4 s or more. The limit distance should fall from 1200 m to under 400 m when $s_0$ is raised to $960.00''$, and the duration-versus-radius curve should bend at $960.15''$ ¹. Tolerance: 2 s of duration and 100 m of limit distance.
• Stephenville, Texas, 2024: observed C2 18:39:06.6, C3 18:39:20.3 UTC, 13.7 s, error about 2 s ¹². With a true limb, terrain and $959.95''$ the tolerance is 3 s. With $959.63''$ and a true limb the expected result is about 24 s. That run serves as a negative control.
• Cape Range, 2023 April 20, Site 1 of Quaglia et al.: totality should reach zero at $s_0 = 960.28''$ and the observed contacts support $959.90'' < s_0 < 960.02''$ ³⁹.

Level 5: Bead events

• Guhl and Tegtmeier's Table 1 for 2017USN1 at Thermopolis, Wyoming, 16 events from 17:39:59.9 to 17:40:42.9 UTC with axis angles ¹³, and Guhl's Table 1 for 2023AUN1 with 16 events from 03:28:52.7 to 03:30:05.7 UTC ⁴⁰. A bead simulator with LOLA should reproduce each event's axis angle and time to within 1 s when run with the radius correction each paper derived, $+0.03''$ and $+0.38''$ respectively. Per-bead scatter of $\pm 0.1''$ is the floor.
• Photo Ephemeris lists eight sites across 2017 (Madras, Warrensburg), 2019 (Cerro Tololo) and 2023 (Bisti Badlands, Mentmore, Lamesa, Grand Vista Overlook, Kirtland, Kailis, Ned's Camp) for which it compared its simulation with recordings, but publishes no residuals ⁴¹. The site list is a starting point for obtaining videos.

Data sets to obtain

USNO deltat.data and IERS bulletins for $\Delta T$ and polar motion ^{6 31}. The LOLA LDEM and SLDEM2015 grids ³. The Journal for Occultation Astronomy issues 2018-3 and 2023-4 for bead tables ^{13 40}. Quaglia et al.'s flash-spectrum video, which the paper says is available on request ¹. Dunham's 2023 and 2024 bead videos, linked from IOTA's graze pages ^{42 26}.

Sources compared

What a developer should do

1. Implement the five-level protocol as an automated test suite, with the constants of each reference case pinned in the test and the tolerance stated beside it.
2. Correct known significant biases before estimating uncertainty. The uncertainty of the correction still propagates. This includes Type B uncertainty evaluated from calibration, bounds or other evidence rather than repeated measurements. Full omission shifts and inter-program spread are not automatically independent standard deviations. Combine compatible standard uncertainties in quadrature only after establishing independence. Otherwise propagate covariance or report recomputed scenario ranges ⁴³. Validate a limit band against reference observations before assigning a coverage probability ¹.
3. Store $\Delta T$ with its source and date and warn when the eclipse is more than one year past the last measured value, or more than 300 years from the present, where the Canon's own maps switch to gores.
4. Read first: Quaglia et al. 2021 section 1 and section 4, Espenak's uncertainty page, Stephenson et al. 2016 section 2, and the 2001 bulletin's limb section.

What this changes

The pipeline design gains an explicit uncertainty output and a test corpus. The solar radius moves from a constant to a configuration item with a stated uncertainty. Products for dates before 1600 or after 2300 must carry a longitude gore for the $\Delta T$ standard error. That threshold is this research's own recommendation and is stricter than the Five Millennium Canon, which draws gores only where $\sigma$ exceeds 265 s, before +0001 and after 2300. Between 1 and 1600 a product should at least print $\sigma$, which exceeds 20 s before 1600 ^{15 7}. Nothing else in the computational chain changes.

Open questions

• Transcribe the remaining error-budget components from the available Herald 1983 scan and compare them with the modern derived terms. The graphical construction is in limb profile methods ²⁷.
• Obtain Morrison et al. 2021 and HMNAO Table S15.2020 to tabulate the current $\sigma(\Delta T)$ by year and replace the 2004 rule ²⁹.
• Obtain the DE440 versus DE421 lunar position difference at the 2024 epoch, in metres, to close the ephemeris row with a number rather than "negligible".
• Obtain the per-site residuals behind Photo Ephemeris's eight-site verification, or the recordings themselves, to add bead-level cases from 2019 and 2023 October.
• Obtain Lamy et al. 2015 in full to record the limb data version and the per-site timing precision of the photometers, which would set the noise floor for a light-curve test.