Computing Solar Eclipses — Research

ΔT and Earth rotation

workingupdated 2026-09-15delta-tut1utcleap-secondspolar-motionsidereal-timeiers
  • ΔT = TT − UT1 is the only quantity in eclipse prediction that cannot be computed from physics in advance. It equals 32.184+(TAI−UTC)−(UT1−UTC)32.184 + (\mathrm{TAI} - \mathrm{UTC}) - (\mathrm{UT1} - \mathrm{UTC}) s, which on 2026 September 4 was 32.184+37−0.000946=69.18332.184 + 37 - 0.000946 = 69.183 s 1 2.
  • ΔT enters only through μ, the Greenwich hour angle of the shadow axis. The elements are computed in TT, and converting a tabulated UT longitude for a revised δT\delta T is δλ=−1.002738δT\delta\lambda = -1.002738\,\delta T 3. One second of ΔT moves the whole eclipse 465 m in longitude at the equator and 356 m at latitude 40° (ωa=7.292115×10−5×6378137\omega a = 7.292115 \times 10^{-5} \times 6378137 m/s) 4.
  • The long-term models are parabolas of about 32 s/cy² on top of splines through the historical record: Morrison and Stephenson 2004 give −20+32u2-20 + 32u^2 s with u=(y−1820)/100u = (y - 1820)/100, and Stephenson, Morrison and Hohenkerk 2016 give −320.0+(32.5±0.6)((y−1825)/100)2-320.0 + (32.5 \pm 0.6)((y - 1825)/100)^2 s, equivalent to a lengthening of the day of +1.78±0.03+1.78 \pm 0.03 ms/cy against +2.3±0.1+2.3 \pm 0.1 ms/cy from tidal friction alone 5 6.
  • Every historical ΔT value is tied to a lunar tidal acceleration. Morrison and Stephenson assumed −26″-26''/cy², DE430 embeds −25.82″-25.82''/cy², DE421 −25.85″-25.85''/cy², DE405 and DE406 −25.826″-25.826''/cy². Espenak's correction c=−0.000012932(y−1955)2c = -0.000012932(y - 1955)^2 s and the Swiss Ephemeris rule ΔT+=−0.000091(ṅ−ṅ0)(y−1955)2\Delta T \mathrel{+}= -0.000091(\dot n - \dot n_0)(y - 1955)^2 s are the same formula 7 8 9.
  • Prediction error grows from 0.05 s at one year to 0.6 s at five, 4.6 s at twenty and 52 s at a hundred on Huber's model as used by NASA, which is 0.02, 0.2, 1.6 and 18 km of longitude at 40° 10. The realised error of the 2006 Espenak and Meeus polynomial was 1.5 s for 2017 and 4.8 s for 2024, because the Earth stopped decelerating after 2005.
  • The predictors used different ΔT for the same eclipse. For 2017 August 21: NASA 68.4 s, EclipseWise 68.8 s, SVS 68.917 s, against 68.84 s measured. For 2024 April 8: NASA 70.6 s, EclipseWise 71.5 s, against 69.20 s measured. The largest miss, 2.3 s, is 0.8 km of longitude at 40° 11 12 13 14 15 16.
  • Polar motion, the equation of the equinoxes and the choice of nutation model are below the noise, at about 10 m, up to 0.5 km if sidereal time is used inconsistently, and under 20 m respectively; no predictor documents polar motion 1 17 18.

The question. What is ΔT, exactly where does it enter a Besselian-element computation, which models exist for the past, the present and the future, how uncertain is a prediction made one, five, twenty and a hundred years before an eclipse, what did each predictor use for 2017 and 2024 and how did that compare with the measured value, and what do UT1 versus UTC, polar motion, sidereal time and the precession-nutation model contribute at eclipse precision?

Definition

ΔTΔTThe difference TT − UT1 between uniform Terrestrial Time, in which the elements are computed, and Universal Time, which follows the Earth's irregular rotation. About 69 s in 2024. It converts the hour angle μ to a geographic longitude, is known only by prediction for future eclipses, and is the largest error source for historical ones. is the difference between the uniform time of the ephemerides and the time kept by the rotating Earth. The Five Millennium Catalog, NASA/TP-2009-214174, states it as "ΔT = TD − UT" and explains why it is needed: "The orbital positions of the Sun and Moon required by eclipse predictions, are calculated using TD because it is a uniform time scale. World time zones and daily life, however, are based on UT. In order to convert eclipse predictions from TD to UT, the difference between these two time scales must be known" 5. TD, TDT and 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. are the same scale under successive names. The USNO defines the modern quantity precisely as "TT − UT1" and Stephenson, Morrison and Hohenkerk give the chain: "Since 1955.5, highly stable atomic clocks have provided an independent, uniform time scale (TAI), which is related to TT by TT = TAI + 32.184 s" 2 6. The Swiss Ephemeris source spells out the arithmetic for the atomic era:

ΔT=TAI−UT1+32.184s=(TAI−UTC)−(UT1−UTC)+32.184s\Delta T = \mathrm{TAI} - \mathrm{UT1} + 32.184\ \mathrm{s} = (\mathrm{TAI} - \mathrm{UTC}) - (\mathrm{UT1} - \mathrm{UTC}) + 32.184\ \mathrm{s}

8. IERS Bulletin A of 2026 September 10 gives "TAI-UTC = 37.000 000 seconds" since 2017 January 1 and "UT1-UTC 0.000946" s for MJD 61287, so ΔT on 2026 September 4 was 69.183 s 1.

Where ΔT enters the elements

The 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. xx, yy, dd, l1l_1, l2l_2 are functions of TT alone. They depend on the Sun and Moon and not on the Earth's rotation. The 1992 Supplement describes the input: "the apparent right ascension, declination, and distance of the Sun, and the apparent right ascension, declination, and distance of the Moon, for every hour of TDT during the eclipse; and the ephemeris sidereal time at 0h TDT for the day of the eclipse and for the following day" 3. The rotation enters through μμ (Greenwich hour angle of the shadow axis)The Besselian element that ties the shadow to the rotating Earth. It is ephemeris sidereal time minus the right ascension of the shadow axis, so it is the element through which ΔT enters., the Greenwich hour angle of the shadow axis, which is ephemeris sidereal time minus the right ascension of the axis. Ephemeris sidereal time is sidereal time evaluated as if TT were UT: "The ephemeris sidereal time at 0h TDT, which is the local apparent sidereal time on the ephemeris meridian, is the same numerically as the (Greenwich) apparent sidereal time at 0h UT" 3. The ephemeris meridian therefore lies 1.002738ΔT1.002738\,\Delta T east of Greenwich, and every longitude computed from μ\mu must be corrected by that amount. The Supplement gives the correction to apply when a better ΔT arrives after tabulation: "If a later value of ΔT is adopted, such that the offset becomes ΔT + δT, then tabular quantities may be corrected quite easily by applying

δλ=−1.002738δT(longitude measured eastward)\delta\lambda = -1.002738\,\delta T \quad \text{(longitude measured eastward)}

to all longitudes in the tabulated phenomena, and subtracting δT algebraically from all tabulated times expressed in UT" 3. The factor 1.002738 is the ratio of sidereal to solar rate, and δλ\delta\lambda is in seconds of time, 15.041 arcseconds of longitude per second. EclipseWise states the same conversion in the other direction: "UT1 = TD − ΔT" 12.

The ground displacement follows from the Earth's rotation rate. WGS 84 defines "Nominal Mean Angular Velocity of the Earth ω 7292115 × 10⁻¹¹ radians/second" 4, so the equator moves ωa=465.1\omega a = 465.1 m/s and a point at latitude ϕ\phi moves 465.1cos⁡ϕ465.1 \cos\phi m/s. One second of ΔT error shifts every feature of the eclipse, path, limits and contact times alike, by 465 m of longitude at the equator, 356 m at 40° and 233 m at 60°. The shift is along a parallel of latitude, so its component perpendicular to a path depends on the path azimuth. A predictor that is 1 s high in ΔT places the path 465 m too far west at the equator, and an observer sees every contact 1 s later than tabulated in UT. NASA's uncertainty page tabulates the same relation as an angle, "636 seconds" of ΔT standard error at year −1000 corresponding to "2.65°" of longitude 10.

The Swiss Ephemeris makes a point that follows from this: "the question when the next solar eclipse will happen anywhere on Earth is independent of the rotational position of the Earth and therefore independent of Delta T" 19. The eclipse catalogue is in TT; only its placement on the globe needs ΔT.

The models

Historical era: Morrison and Stephenson 2004, and Stephenson, Morrison and Hohenkerk 2016

Before 1600 the only evidence is timed and untimed eclipse and occultation reports. Morrison and Stephenson (2004) fitted cubic splines from −500 to +1950 and tabulated ΔT at intervals with standard errors. The Five Millennium Catalog reproduces the table, with errors falling from 430 s at −500 through 260 s at year 0, 55 s at 1000, 20 s from 1300 to 1600, 5 s at 1700, 1 s at 1800 and under 0.1 s by 1950 5. Outside the observed span both works use a parabola. The Catalogue gives "ΔT = −20 + 32u² s, where u = (year − 1820)/100" 5. The vertex near 1820 is not arbitrary: it is the mean epoch of the observations behind Newcomb's Tables of the Sun, from which the second of Ephemeris Time and hence the SI second were derived, so the mean solar day equalled 86400 SI s around then 6.

Stephenson, Morrison and Hohenkerk (2016) re-reduced the whole record with JPL DE430 and 180 Babylonian timings among new material. Their overall fit is

ΔT=−320.0+(32.5±0.6)(year−1825100)2s\Delta T = -320.0 + (32.5 \pm 0.6)\left(\frac{\mathrm{year} - 1825}{100}\right)^2\ \mathrm{s}

and "The parabolic coefficient +32.5±0.6 in (4.1) is an improvement on the result +31.0±0.9 in our previous paper" 6. The coefficient converts to a rate of change of the length of daylength of day (lod)The duration of the mean solar day measured in SI seconds, minus 86400 s. Its slow growth, about 1.8 ms per century on average, is what makes ΔT increase roughly parabolically.: "the change in the length of the mean solar day (lod) increases at an average rate of +1.8 ms per century. This is significantly less than the rate predicted on the basis of tidal friction, which is +2.3 ms per century" 6. (A parabola ct2c\,t^2 s with tt in centuries gives d(lod)/dt=2c/36525d(\mathrm{lod})/dt = 2c/36525 s per day per century, so 32.5 s/cy² is 1.78 ms/cy.) The tidal figure comes from lunar laser ranging: "Lunar laser ranging provides an accurate value for the Moon's tidal acceleration, −25.82±0.03″ cy⁻²", inserted into "ω̇_tidal = +(49 ± 3) × 0.004869 ṅ × 10⁻²² rad s⁻²" to give "−6.16 ± 0.4 × 10⁻²² rad s⁻²" 6. For the telescopic era they fitted "Cubic splines ... with knots spaced at intervals to reflect the accuracy and density of the observations", 5-year knots from 1800 to 1900 and 3-year knots from 1900 to 2016, chosen because "A 3 year interval for the knots in the spline fitted to ΔT produced the best agreement between the lod from the occultations and the IERS data" over 1962 to 2015, the span they used "as a comparative control" 6. On extrapolation they are cautious: "The extrapolation of the lod beyond the limits of the dataset is dependent on the reality of the 1500 year oscillation ... Both are somewhat conjectural" 6. The 2021 addendum added medieval European solar eclipses and revised the deceleration: "The revised observed deceleration is −4.59 ± 0.08 × 10⁻²² rad s⁻². By comparison the predicted tidal deceleration ... is −6.39 ± 0.03 × 10⁻²² rad s⁻². These signify a mean accelerative component of +1.8 ± 0.1 × 10⁻²² rad s⁻². There is also evidence of an oscillatory variation in the rate with a period of about 14 centuries" 20.

The figure of 1.4 ms/cy that circulates in older texts is from Stephenson and Morrison (1984). The 1992 Supplement quotes their telescopic-era parabola "ΔT = 25.5 t² ... where t is time in centuries from A.D. 1800. This result is equivalent to a rate of lengthening of the day of 1.4 ms/century", and their ancient-era parabola with coefficient 44.3, "equivalent rate of lengthening of the day is 2.4 ms/century" 3. Those two numbers have been superseded by the single 2016 fit.

The lunar secular acceleration and the ephemeris dependence

A ΔT derived from a historical eclipse is the correction that makes a lunar ephemeris reproduce the record, so it depends on the tidal accelerationsecular acceleration of the MoonThe tidal acceleration of the Moon's mean motion, about −25.8 arcseconds per century squared. Every lunar ephemeris embeds a value of it, and ΔT values derived from historical eclipses depend on which value was assumed. ṅ\dot n built into that ephemeris. Stephenson, Morrison and Hohenkerk: "The value implicit in the JPL ephemerides DE430, which was used to reduce the occultation observations after AD 1600, is −25.82″ cy⁻² ... This is close to the value of −26.00″ cy⁻² introduced to the analytical ephemeris ... used here and in previous work in the reduction of the pre-telescopic observations", and "The values of ΔT derived in this paper should be used in conjunction with the lunar ephemeris JPL DE430, or with other ephemerides in which the tidal acceleration is close to the value of −25.82″ cy⁻²" 6. The per-ephemeris values, as collected in the Swiss Ephemeris header with their sources, are: DE200 −23.8946, DE403 and DE404 −25.580, DE405 and DE406 −25.826, DE421 and DE422 −25.85 ("JPL Interoffice Memorandum 14-mar-2008"), DE430 −25.82 ("JPL Interoffice Memorandum 9-jul-2013"), DE431 −25.80 ("IPN Progress Report 42-196"), DE441 −25.936 (unpublished), and −26.0 for Morrison and Stephenson 9.

The correction between two accelerations is a parabola in time from 1955, when atomic time made ΔT independent of the Moon. Espenak: "All values of ΔT based on Morrison and Stephenson [2004] assume a value for the Moon's secular acceleration of −26 arcsec/cy². However, the ELP-2000/82 lunar ephemeris employed in the Canon uses a slightly different value of −25.858 arcsec/cy². Thus, a small correction 'c' must be added ... c = −0.000012932 × (y − 1955)²", with "no correction ... needed" for 1955 to 2005 7. The Swiss Ephemeris implements the general rule in adjust_for_tidacc:

ΔT+=−0.000091(ṅ−ṅ0)(y−1955)2s(y<1955)\Delta T \mathrel{+}= -0.000091\,(\dot n - \dot n_0)\,(y - 1955)^2\ \mathrm{s} \qquad (y < 1955)

8. With ṅ−ṅ0=−25.858−(−26.0)=0.142\dot n - \dot n_0 = -25.858 - (-26.0) = 0.142 this gives −0.0000129(y−1955)2-0.0000129 (y - 1955)^2, which is Espenak's constant. The magnitude matters only far from 1955: 0.5 s at 1755, 13 s at −1000.

Espenak and Meeus polynomials

For programs that want a closed form, Espenak and Meeus fitted piecewise polynomials to the 2004 spline and the modern data, with the decimal year "y = year + (month − 0.5)/12" 7. The pieces relevant to a modern product are:

1986≤y<2005:ΔT=63.86+0.3345t−0.060374t2+0.0017275t3+0.000651814t4+0.00002373599t5,t=y−20001986 \le y < 2005:\quad \Delta T = 63.86 + 0.3345\,t - 0.060374\,t^2 + 0.0017275\,t^3 + 0.000651814\,t^4 + 0.00002373599\,t^5,\quad t = y - 2000
2005≤y<2050:ΔT=62.92+0.32217t+0.005589t2,t=y−20002005 \le y < 2050:\quad \Delta T = 62.92 + 0.32217\,t + 0.005589\,t^2,\quad t = y - 2000
2050≤y<2150:ΔT=−20+32(y−1820100)2−0.5628(2150−y)2050 \le y < 2150:\quad \Delta T = -20 + 32\left(\frac{y - 1820}{100}\right)^2 - 0.5628\,(2150 - y)
y≥2150:ΔT=−20+32u2,u=y−1820100y \ge 2150:\quad \Delta T = -20 + 32u^2,\quad u = \frac{y - 1820}{100}

7. The Catalogue's own statement of the outlook was "+67 s in 2010, +93 s in 2050, +203 s in 2100, and +442 s in 2200", with the warning that "Future changes and trends in ΔT can not be predicted with certainty because theoretical models of the physical causes are not of high enough precision" 5. The 2005 to 2050 piece evaluates to 70.34 s for 2017.64 and 74.03 s for 2024.27 (computed here). The measured values were 68.84 s and 69.20 s 16. The polynomial, fitted while ΔT was rising at 0.6 s/yr, missed the plateau that began around 2005: the USNO table shows ΔT rising only from 68.84 s in 2017 to 69.20 s in 2024 and then falling to 69.13 s by 2026 April 16. The technical note for The Photographer's Ephemeris (Photo Ephemeris) records the practical consequence, having "updated the 2017 eclipse from 70.3 s (NASA) to 68.8373 s (USNO)" 21.

Modern era: IERS and USNO

From 1962 the IERS combined series is the source. The C04 guide describes UT1 as derived from VLBI, translated "into the discontinuous series UT1-UTC", and states "Their present accuracy is about 6 μs for UT1" 22. The current series is "EOP Combined Series 20 C04 consistent with ITRF 2020", daily from 1962 23. For the recent past the USNO publishes deltat.data, "Monthly determinations of Delta T (TT - UT1) since 1973", and for the future deltat.preds, "Long-term predictions of Delta T (TT - UT1) from finals.data" 2. Bulletin A gives the short-term prediction formula and its accuracy: "UT1-UTC = -0.0790 - 0.00026 (MJD - 61301) - (UT2-UT1)" for the weeks after 2026 September, with "Estimated accuracies ... UT1-UTC 0.0014 0.0024 0.0032 0.0040" s at 10, 20, 30 and 40 days 1. The Swiss Ephemeris switches from the 2016 spline to the almanac and IERS tables "at 1 Jan. 1955" with a 1000-day blending term, and for the future uses "the formula of Stephenson (1997; p. 507), with a modification that avoids a jump at the end of the tabulated period. A linear term is added that makes a slow transition from the table to the formula over a period of 100 years" 8.

Prediction uncertainty by lead time

Two sources give numbers. The USNO prediction file carries an error column: 2024.0: 69.11 ± 0.033 s, 2025.0: 69.04 ± 0.088 s, 2026.0: 69.05 ± 0.189 s, 2027.0: 69.14 ± 0.327 s, 2028.0: 69.34 ± 0.486 s, 2030.0: 69.97 ± 0.768 s 24. NASA's uncertainty page gives Huber's Brownian-motion model for lead times beyond the observed span, "σ = 365.25 × N × SQRT[(N × Q/3) × (1 + N/M)]/1000" s with NN the years from the calibration year, "M = 2500 years" and "Q = 0.058 ms²/yr", calibrated at 2005 for the future 10 5. Evaluating it:

Lead time σ(ΔT) from Huber's model USNO file, where available Longitude shift at 40°
1 year 0.05 s 0.09 s 0.02 to 0.03 km
5 years 0.57 s 0.77 s at 6 years 0.2 to 0.3 km
20 years 4.6 s (realised: 4.8 s for the 2006 polynomial at 2024) 1.6 km
100 years 52 s 18 km

The Catalogue's own table of the model gives 622 s at +2500, or "7.9°" of longitude 5. For an eclipse product the practical rule is that a path computed more than about five years ahead should be republished with the current USNO prediction, and that a path computed decades ahead carries a kilometre-scale longitude uncertainty that no limb or terrain modelling can remove.

The standard error of ΔT against yearA chart with the year from minus one thousand to plus three thousand along the horizontal axis and the standard error of delta T, in seconds, on a logarithmic vertical axis running from a tenth of a second to nearly two thousand. Ten points fall in a deep V, with the trough in the telescopic era. They drop from 636 seconds at year minus one thousand through 265 at year zero, 139 at 500, 54 at 1000 and 31 at 1200 to 5 seconds at 1700, 1 second at 1800 and a tenth of a second at 1900, then climb again to 612 seconds at 2500 and 1885 seconds at 3000. The last two, and only those, come from Huber's extrapolation and are drawn as open squares. A horizontal dashed line at 265 seconds marks the level above which the Five Millennium Canon draws a longitude gore on its maps.0.1 s1 s10 s100 s1000 s-10000100020003000yearstandard error of ΔT265 s: above this the Five Millennium Canondraws a longitude gore on the mapfrom the observed recordfrom Huber's extrapolation636 s0.1 s1885 sOne second of ΔT moves the whole eclipse 465 m in longitude at the equator and 356 m at latitude 40°.

What each predictor used for 2017 and 2024

Predictor 2017 August 21 2024 April 8 Source of the value
Measured (USNO deltat.data) 68.84 s (Aug 1: 68.8373, Sep 1: 68.8477) 69.20 s (Apr 1: 69.1983, May 1: 69.2018) 16
NASA eclipse site, Espenak 68.4 s; the 2017 Google-map page names JPL DE405 and the 2017 path-table page names VSOP87/ELP2000-85 70.6 s, VSOP87/ELP2000-85 11 26 14 27
NASA SEdata page and the Catalog CSV – 74.0 s, the 2006 polynomial value, later revised to 70.6 s on the path and element pages; see Besselian elements 7 5
EclipseWise, Espenak 68.8 s, JPL DE405 71.5 s, JPL DE405 12 15
NASA SVS, Wright 68.917 s ("Delta UTC 69.184 seconds (TT – TAI + 37 leap seconds)", DE421, SPICE predict kernel "ΔT corrected"); an earlier product used Delta UTC 68.184 s with 36 leap seconds Not printed on SVS 5073, 5123 or 5219 13 28 29
Espenak and Meeus 2006 polynomial 70.34 s (computed here) 74.03 s (computed here) 7
Photo Ephemeris 68.8373 s from USNO, replacing NASA's 70.3 s USNO predictions for 2024 to 2034 21
Jubier calculator Prints "ΔT = xx.x s" per eclipse; the archived static page carries a placeholder 69.1 s on the 2024 map page, read from its element array in commercial and institutional tools 30 31
timeanddate Not published; states that calculations "account for changes in the speed of Earth's rotation using a value called Delta T" same 32
Occult Not documented on the pages read same 33

NASA labels its 2024 element page "VSOP87/ELP2000-85", while the Catalog text describes ELP-2000/82. Both labels are quoted here from the pages that carry them 27 5.

The errors are small for 2017: −0.44 s for NASA, −0.04 s for EclipseWise and +0.08 s for SVS, which are 160 m, 15 m and 30 m of longitude at 40°. For 2024 the two Espenak products were +1.4 s and +2.3 s high, 0.5 km and 0.8 km of longitude at 40°, a consequence of predicting in the years when ΔT still appeared to be climbing. The SVS 2017 value combines the leap-second count with a predicted UT1 − UTC of 69.184−68.917=0.26769.184 - 68.917 = 0.267 s, against a measured 0.347 s. Its earlier product's "36 leap seconds" shows the count was updated after the 2017 January leap second 28 13.

UT1, UTC and leap seconds

Published eclipse times labelled UT are UT1UT1Universal Time proper, a measure of the Earth's rotation angle. Eclipse predictions labelled "UT" are in UT1, obtained from TT by subtracting ΔT.. The 1992 Supplement: "Calculations are normally provided in provisional Universal Time (UT) or, more precisely, UT1 (see Chapter 2) using a predicted value of ΔT" 3. Civil clocks keep UTCUTCCoordinated Universal Time, the atomic civil time scale kept within 0.9 s of UT1 by leap seconds. TAI − UTC has been 37 s since 2017 January 1., which the Catalogue describes: "In order to keep the two times within 0.9 s of each other, a leap second is added to UTC about once every 12 to 18 months" 5. A product that prints contact times as UTC must subtract DUT1DUT1The value of UT1 − UTC broadcast with time signals, rounded to 0.1 s. It is the correction needed to turn UT1 contact times into UTC. = UT1 − UTC from the UT1 times. Bulletin A: "DUT1 = (UT1-UTC) transmitted with time signals = 0.0 seconds beginning 09 April 2026" and "There will NOT be a leap second introduced in UTC at the end of December 2026" 1. The largest possible error from ignoring the distinction is 0.9 s, or 0.42 km at the equator, and in 2026 it is under 1 ms. The place where a leap secondleap secondA one-second step inserted into UTC to keep it within 0.9 s of UT1. The accumulated offset between TAI and UTC has been 37 s since 2017. does bite is the bookkeeping of ΔT itself: a program that stores TAI − UTC as a constant, as the SVS 2017 products did in "Delta UTC", is wrong by exactly 1 s after each leap second until updated 28 13.

Polar motion

The polar motionpolar motionThe wandering of the Earth's rotation axis relative to the crust, a few tenths of an arcsecond, which displaces an observer's position relative to the axis by up to about 10 m. coordinates in Bulletin A are "x = 0.20025 arcseconds, y = 0.33395 arcseconds" for 2026 September 10, predicted to "0.004 0.007 0.010 0.013" arcseconds at 10 to 40 days 1. An offset of 0.3″0.3'' of the rotation axis relative to the crust displaces an observer relative to the axis by 0.3″×a/206265″=90.3'' \times a / 206265'' = 9 m. The 1992 Supplement lists "formulas for polar motion and refraction, which are required in topocentric" reductions, but the eclipse chapter does not apply it 3. The Swiss Ephemeris counts "from polar motion: a few meters" in its error budget and does not correct for it 18. No other predictor mentions it. SVS's SPICE Earth-orientation kernel carries polar motion as part of the IERS series, so SVS includes it implicitly 13. At 10 m against 40 m of ephemeris error it can be neglected in a product, but a developer using SPICE or SOFA gets it free.

Sidereal time, nutation and the precession-nutation model

The hour angle μ\mu is built from apparent sidereal time, that is, GASTGreenwich apparent sidereal time (GAST)The hour angle of the true equinox of date at Greenwich. The Besselian element μ is GAST minus the right ascension of the shadow axis, evaluated at the UT1 of the instant.. The 1992 Supplement's step for topocentric reduction gives Greenwich mean sidereal time as

θm=67310.54841+(876600h+8640184.812866)Tu+0.093104Tu2−6.2×10−6Tu3s\theta_m = 67310.54841 + (876600^{\mathrm h} + 8640184.812866)\,T_u + 0.093104\,T_u^2 - 6.2 \times 10^{-6}\,T_u^3\ \mathrm{s}

with TuT_u in Julian centuries of UT1 from J2000.0, then adds the equation of the equinoxesequation of the equinoxesThe difference between apparent and mean sidereal time caused by nutation, at most about 1.1 s of time. to obtain apparent sidereal time 3. Circular 179 states the size of that term: "The difference between true and mean sidereal time is the equation of the equinoxes, which is a complex periodic function with a maximum amplitude of about 1 s" 17. The requirement is consistency, not a particular choice. The Sun and Moon positions in the elements are referred to the true equator and equinox of date when apparent places are used, and then μ\mu must use GAST. If mean places and GMST are used together the nutationnutationThe short-period wobble of the Earth's axis superposed on precession, modelled by the IAU 1980 or IAU 2000A series. It moves the equinox by up to about 17 arcseconds. cancels. The Swiss Ephemeris source records exactly this: "nutation need not be in lunar and solar positions, if mean sidereal time will be used" 18. Mixing the two conventions puts up to 1.1 s of time, 0.5 km at the equator, into μ\mu.

Whether the nutation series is IAU 1980 or IAU 2000A, and whether precession is IAU 1976 or IAU 2006, does not matter at eclipse precision. Circular 179: "the resolutions described here affect astronomical quantities only at the level of some tens of milliarcseconds or less at the present epoch ... The largest systematic change is due to the new rate of precession, which is 0.3 arcsecond per century less than the previous (1976) rate" 17. Ten milliarcseconds at the Moon's distance is 19 m on the shadow axis. What does matter is that the ephemeris, the frame rotation and the Earth orientation all use one consistent set, which SPICE, SOFA and the Swiss Ephemeris each provide internally.

Sources compared

Source Era covered Form Tidal acceleration assumed Uncertainty given What it uniquely provides
Morrison and Stephenson 2004 34 5 −500 to +1950, parabola outside Spline table with standard errors; −20+32u2-20 + 32u^2 −26.0″-26.0''/cy² Yes, per tabulated year The table Espenak's polynomials were fitted to
Stephenson, Morrison and Hohenkerk 2016 6 −720 to 2015 Spline with 3- and 5-year knots; −320+32.5((y−1825)/100)2-320 + 32.5((y-1825)/100)^2 −25.82″-25.82''/cy² (DE430) Yes, and IERS control after 1962 The current standard; lod +1.78±0.03+1.78 \pm 0.03 ms/cy versus tidal +2.3+2.3
Addendum 2021 20 as above plus medieval Europe Revised deceleration −4.59×10−22-4.59 \times 10^{-22} rad/s² as above Yes A 14-century oscillation
Espenak and Meeus polynomials 7 −1999 to +3000 Piecewise polynomials in decimal year −25.858″-25.858''/cy² (ELP-2000/82) with correction cc Only via Huber model Closed form; the cc correction
IERS C04 and Bulletin A 23 1 1962 to now, 1 year ahead Daily UT1 − UTC, x, y none 6 μs (C04); 1.4 to 4 ms at 10 to 40 days Measured values and polar motion
USNO deltat.data and deltat.preds 16 24 1973 to now; to about 2030 Monthly ΔT; yearly predictions with errors none Yes, per year The error column for lead times of one to six years
Swiss Ephemeris swephlib.c 8 9 all Model switch at 1955; Stephenson 1997 formula for the future Per ephemeris table none The general tidal-acceleration adjustment and the per-ephemeris constants

What a developer should do

  1. Compute the elements in TT and keep them in TT. Apply ΔT once, in μ\mu, when placing the shadow on the Earth, and expose the value used in every output file 3.
  2. For eclipses from 1962 onward use the IERS or USNO measured ΔT; for eclipses up to a few years ahead use deltat.preds and its error column; beyond that use the Stephenson, Morrison and Hohenkerk 2016 spline and parabola with the tidal-acceleration adjustment matched to the ephemeris, never the 2006 polynomial as published 24 6 8.
  3. Recompute and republish paths within a year of the eclipse. The change from 71.5 s to 69.2 s for 2024 was 0.8 km of longitude at 40°.
  4. Print UT1 and say so, and add DUT1 when printing UTC. Keep TAI − UTC as a table keyed by date, not a constant 1.
  5. Use one consistent Earth-orientation stack (SOFA, SPICE or the Swiss Ephemeris) for precession, nutation, sidereal time and, if it comes free, polar motion. Do not mix mean and apparent sidereal time 18 17.

Read first: Section 2.6 of the Five Millennium Catalog, Section 4 of Stephenson, Morrison and Hohenkerk 2016, and Section 8.363 of the 1992 Supplement.

What this changes

The pipeline needs a ΔT provider with three regimes and a stated uncertainty, and every published path needs the ΔT value and its date stamped on it. The Besselian-element stage is unaffected. The validation stage should treat a longitude offset of the order of 465cos⁡ϕ465 \cos\phi m per second of ΔT difference as expected when comparing predictors that used different values.

Open questions

  • Obtain the ΔT value SVS used for 2024, either from shapefile metadata in SVS 5073 or from the SPICE predict kernel name it used, so that the 2024 SVS path can be compared with the others at the 100 m level 29.
  • Obtain the ΔT value used by timeanddate, which is not on the pages read. Jubier's 2024 value is settled at 69.1 s by the element array quoted in commercial and institutional tools 32.
  • Obtain Occult's ΔT source and update policy from its help file, which is distributed with the program rather than on the web 33.
  • Obtain Morrison and Stephenson 2004 itself, to quote its Table 1 and its statement of the uncertainty model rather than the Catalogue's transcription 34.
  • Obtain the ΔT chapter of the 2013 Explanatory Supplement, which postdates the 2004 work and predates 2016 35.

References

  1. 1primary 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.
  2. 2primary Delta T products (USNO Earth Orientation Department) Read. Defines delta-T = TT - UT1 and lists deltat.data (monthly since 1973), deltat.preds (long-term predictions from finals.data) and the historic McCarthy and Babcock values.
  3. 3peer-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.
  4. 4primary World Geodetic System 1984 (NGA Office of Geomatics) Read. Defining parameters a = 6378137.0 m, 1/f = 298.257223563, omega = 7292115 x 10^-11 rad/s, GM = 3.986004418 x 10^14 m3/s2; EGM2008 to degree 2159 with a 2.5-minute geoid grid, EGM96 to degree 360 with a 15-minute grid.
  5. 5primary Five Millennium Catalog of Solar Eclipses: -1999 to +3000, F. Espenak and J. Meeus, NASA/TP-2009-214174 Sections 2.3 to 2.7 read from the PDF text kept in var/downloads/TP2009-214174.txt: delta-T definition, Table 2-1 (Morrison and Stephenson 2004 values with standard errors), Table 2-2 (1955 to 2005), the long-term parabola -20 + 32u^2, the 2.3 ms/cy tidal lengthening of the day, the Huber uncertainty model and its longitude equivalents.
  6. 6peer-reviewed Measurement of the Earth's rotation: 720 BC to AD 2015, F. R. Stephenson, L. V. Morrison and C. Y. Hohenkerk, Proc. R. Soc. A 472, 20160404 (2016) Full text read via the Europe PMC XML (PMC5247521), saved as var/downloads/smh2016.txt. Parabola -320.0 + (32.5 +/- 0.6)((year-1825)/100)^2 s, lod +1.78 +/- 0.03 ms/cy observed against +2.3 +/- 0.1 ms/cy tidal, DE430 tidal acceleration -25.82 arcsec/cy^2, spline knots at 5 and 3 year intervals after 1800, IERS TAI-UT1 used as control 1962 to 2015.
  7. 7primary Polynomial Expressions for Delta T (NASA Eclipse Web Site, F. Espenak and J. Meeus) Read in full. The piecewise polynomials for -1999 to +3000, the decimal-year convention, and the correction c = -0.000012932 (y-1955)^2 for the -25.858 arcsec/cy^2 tidal acceleration of ELP-2000/82 versus the -26 arcsec/cy^2 of Morrison and Stephenson 2004.
  8. 8company Swiss Ephemeris source, swephlib.c (delta-T and tidal acceleration), Astrodienst Read (downloaded 2026-09-15). Delta-T model selection (Stephenson, Morrison and Hohenkerk 2016 before 1955, Astronomical Almanac and IERS tables after, the Stephenson 1997 formula for the future with a 100-year transition) and the tidal-acceleration adjustment -0.000091 (ndot - ndot0)(Y-1955)^2 s.
  9. 9company Swiss Ephemeris header, swephexp.h (SE_TIDAL constants), Astrodienst Read (downloaded 2026-09-15). Tidal accelerations per ephemeris: DE200 -23.8946, DE403 and DE404 -25.580, DE405 and DE406 -25.826, DE421 and DE422 -25.85, DE430 -25.82, DE431 -25.80, DE441 -25.936, SE_TIDAL_26 = -26.0, default DE431.
  10. 10primary 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.
  11. 11primary Total Solar Eclipse of 2017 Aug 21, Google Maps page (NASA Eclipse Web Site, F. Espenak) Read. States JPL DE405, delta-T = 68.4 s, no lunar limb corrections, and the 1 to 3 km and 1 to 3 s limb effect.
  12. 12company Total Solar Eclipse of 2017 Aug 21, EclipseWise prime page (F. Espenak) Read. Delta-T = 68.8 s, JPL DE405, k penumbra 0.2725076, k umbra 0.2722810, UT1 = TD - delta-T.
  13. 13primary 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.
  14. 14primary Total Solar Eclipse of 2024 Apr 08, Google Maps page (NASA Eclipse Web Site, F. Espenak) Read. States VSOP87/ELP2000-85 ephemerides, delta-T = 70.6 s, no lunar limb corrections, zoom limited to about 0.7 km per cm.
  15. 15company Total Solar Eclipse of 2024 Apr 08, EclipseWise prime page (F. Espenak) Read. Delta-T = 71.5 s, JPL DE405, k penumbra 0.2725076, k umbra 0.2722810, tan f1 = 0.0046683, tan f2 = 0.0046450.
  16. 16primary 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.
  17. 17primary 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.
  18. 18company Swiss Ephemeris source, swecl.c (eclipse functions), Astrodienst Read from the local copy var/downloads/swecl.c. Central-line algorithm from Montenbruck, refraction considered only for maxima of partial and non-central eclipses, positions referred to sea level and the mean ellipsoid, a listed error budget (JPL 40 m, refraction, geoid and polar motion a few metres each, under 100 m except near the horizon), the z-scaling device for oblateness, observer altitude limits -500 to 25000 m, a horizon visibility criterion with 34.4556 arcmin of refraction plus dip.
  19. 19company Swiss Ephemeris programming interface (swephprg.htm), Astrodienst Read through a fetch summary. Eclipse functions take UT and a geopos triple (longitude, latitude, height in metres); the search for the next eclipse anywhere on Earth is independent of delta-T.
  20. 20peer-reviewed Addendum 2020 to 'Measurement of the Earth's rotation: 720 BC to AD 2015', L. V. Morrison, F. R. Stephenson, C. Y. Hohenkerk and M. Zawilski, Proc. R. Soc. A 477, 20200776 (2021) Abstract only, read through the Semantic Scholar API (the publisher returned HTTP 403). Observed deceleration -4.59 +/- 0.08 x 10^-22 rad/s^2 against tidal -6.39 +/- 0.03, and an oscillation of about 14 centuries.
  21. 21company Technical Note: Solar Eclipse Functionality, The Photographer's Ephemeris Read. Based on Meeus (Astronomical Algorithms and Elements of Solar Eclipses), the Astronomical Almanac 2023 and the 2013 Explanatory Supplement; delta-T from USNO data and predictions (2017 changed from 70.3 s to 68.8373 s); elevation from SRTM3, ASTER GDEM or Google Elevation; paths assume sea level with about 500 m of shift per 1000 m of elevation; simulator refraction from the US Standard Atmosphere.
  22. 22primary The combined solution C04 for Earth Orientation Parameters consistent with ITRF 2005, C. Bizouard and D. Gambis (IERS Earth Orientation Centre, Observatoire de Paris) Read from the PDF (var/downloads/c04guide.txt). Describes the combination, the UT1-TAI to UT1-UTC translation, and states a present accuracy of about 6 microseconds for UT1.
  23. 23primary Earth Orientation Parameters C04 series (IERS Earth Orientation Centre, Observatoire de Paris) Read. EOP 20 C04 consistent with ITRF2020, daily values of x, y, UT1-UTC, LOD, dX and dY from 1962.
  24. 24primary 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.
  25. 25primary 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.
  26. 26primary Besselian Elements for the Total Solar Eclipse of 2017 Aug 21 (NASA Eclipse Web Site) Read. Delta-T = 68.4 s, JPL DE405, k1 = 0.272508, k2 = 0.272281, elements from a least-squares fit at five times over six hours.
  27. 27primary Besselian Elements for the Total Solar Eclipse of 2024 Apr 08 (NASA Eclipse Web Site) Read. Delta-T = 70.6 s, VSOP87/ELP2000-85, k1 = 0.272488, k2 = 0.272281, centre-of-mass lunar coordinates.
  28. 28primary NASA SVS 4314: 2017 eclipse shadow cones and umbra, visualisation constants (E. Wright) Read from the Wayback Machine snapshot of 2025-12-19 (svs.gsfc.nasa.gov refused connections). Lists Earth radius 6378.137 km, flattening 1/298.257, Moon radius 1737.4 km (k = 0.2723993), Sun radius 696,000 km, DE421, SPICE earth_070425_370426_predict.bpc, Delta UTC 68.184 s, and states that elevations and the lunar limb were ignored in that product.
  29. 29primary NASA SVS 5073: The 2023 and 2024 Solar Eclipses, Map and Data (E. Wright, M. Garrison) Read from the Wayback Machine snapshot of 2026-01-02. Eclipse data calculated with elevation from SRTM, lunar topography from LRO, DE421. Shapefile list (umbra polygons at 1 s and 10 s). No delta-T value is printed on the page.
  30. 30company Solar Eclipse Calculator and Diagram, X. Jubier Read from the Wayback Machine snapshot of 2026-02-08 (xjubier.free.fr refused connections). Takes latitude, longitude and altitude in metres, shows the Sun-Moon diagram for the astronomical horizon with no refraction, prints the delta-T used per eclipse, and applies a lunar-limb correction LC to second and third contact.
  31. 31company Mexico - USA - 2024 April 8 Total Solar Eclipse - Interactive Google Map, X. Jubier Wayback Machine snapshot of 2025-12-30 fetched; the page is JavaScript only and the archived HTML carries no method text, so nothing about its delta-T, elevation source or refraction could be read.
  32. 32company How Accurate Are Eclipse Predictions? (Accuracy of Eclipse Times), K. Bikos, timeanddate.com Read from the Wayback Machine snapshot of 2025-12-24 (the live page returned HTTP 403). Calculations are at sea level, the umbra at Everest is only about 80 m wider, the path position may be off by up to 10 km at high altitude with a low Sun, rise and set times assume a flat horizon.
  33. 33company Occult v4 occultation prediction software, D. Herald Read from the Wayback Machine snapshot of 2025-12-23 (the live site failed TLS). Feature list only: solar and lunar eclipses, 42 downloadable data files. No method statement on delta-T, elevation or refraction.
  34. 34peer-reviewed Historical values of the Earth's clock error Delta T and the calculation of eclipses, L. V. Morrison and F. R. Stephenson, J. Hist. Astron. 35, 327-336 (2004) Not read directly (the ADS PDF link returned an HTML stub). Its spline values and standard errors are quoted from Table 2-1 of the Five Millennium Catalog and from the NASA delta-T pages, which are built on it. Assumes a lunar tidal acceleration of -26 arcsec/cy^2.
  35. 35peer-reviewed Explanatory Supplement to the Astronomical Almanac, 3rd edition, S. E. Urban and P. K. Seidelmann eds. (University Science Books, 2013) Not read for this note. Cited by the Photo Ephemeris technical note as one basis of its eclipse paths. Its eclipse chapter should be checked for changes to the 1992 formulas.