Corrections that move contact times, Baily's beads, and public code

The question. The smooth-Moon Besselian or topocentric reduction is exact for its inputs. Which physical effects left out of those inputs move the contact times at an observer, and by how many seconds each? How are the limb-profile correction and the resulting sequence of Baily's beads computed at one site? What do the publicly readable implementations actually do about each effect?

Corrections, one at a time

Lunar limb profile

Every reduction in contact-times-and-magnitude.md uses a mean lunar radius $k$. The real limb at the contact position angle is higher or lower than that mean. NASA's bulletin text defines the correction: "For any given position angle, there will be a high mountain (annular eclipses) or a low valley (total eclipses) in the vicinity that ultimately determines the true instant of contact. The difference, in time, between the Sun's position when tangent to the contact point on the mean limb and tangent to the highest mountain (annular) or lowest valley (total) at actual contact is the desired correction to the predicted contact time" ⁹. The bulletins plot "curves of corrections to the times of second and third contact for most position angles". The observer reads them at the $P$ of each contact. The 1998 Maracaibo example is "C2 = -2.5 seconds" and "C3 = -1.0 seconds" ⁹.

Magnitudes: "The correction of eclipse predictions using Watts limb data results in agreement between predicted and observed contact times and durations to better than 0.5 seconds. Without the corrections, the times and durations may be in error by as much as 2 to 3 seconds (and more near the path limits where the geometry is far more critical)", and with Kaguya and LRO "these data can be used to improve the precision of eclipse contact times and durations to the ~0.2 second level of accuracy" ^{1 16}. The Photographer's Ephemeris, which does not apply the correction, states the range as "typically affects the timing of C2 and C3 by a few seconds, but up to ~15 s in extreme cases" ². Jubier's calculator says the correction "can produce a few seconds time difference on the start and end of totality or annularity" ⁶. A repository that reports both corrected and uncorrected values for 2027 gives 1 to 3 s in duration and 1 to 2 km in path position ¹⁷. The 2013 November 3 timing paper notes a sampling effect inside the correction itself: a limb profile at 0.2 degree spacing "could produce errors up to 0.25 seconds" in a contact time, against 0.03 s at 0.02 degrees ⁴.

The local computation, once the topocentric libration and the profile $h(\psi)$ in arcseconds relative to the datum are in hand, is this. At the contact position angle $\psi_c$ the solar limb approaches the mean lunar limb radially at the rate $\dot r(\psi) = n_\mathrm{rel}\cos(\psi - \psi_n)$. Here $n_\mathrm{rel}$ is the relative angular speed of the two centres in arcseconds per second and $\psi_n$ the position angle of the relative motion. The time correction at any limb point is then $\Delta t(\psi) = h(\psi)/\dot r(\psi)$ with the sign that brings the true limb earlier for a valley at C2 and later at C3. The corrected contact is the extreme of $t_\mathrm{mean}(\psi) + \Delta t(\psi)$ over a neighbourhood of $\psi_c$: the last point of the photosphere to disappear at C2 sits in the deepest valley near $\psi_c$, not exactly at $\psi_c$. That is the analytical form of the graphical overlay procedure and is the author's restatement, not a quotation. The sources describe the procedure in words and pictures ^{9 10}. The datum matters. Watts heights refer to a slightly elliptical reference "with center displacement from the Moon's center of mass" ⁹. Solar Eclipse Maestro plots LRO heights against the IAU mean radius $k = 0.2725076$ (1738.091 km) and draws the reduced radius $k_2$ used for the uncorrected contacts as a separate line ¹⁰. The profile datasets themselves are the subject of Datasets: Watts, Kaguya, LOLA.

Solar radius

The measured values and their provenance are in Solar radius values and their provenance. This section keeps the sizes the local stage owns.

$L_1$ and $L_2$ contain the solar radius through $f_1$ and $f_2$. NASA's elements use 696,000 km, "calculated from arctan of IAU 1976 solar radius (959.63 arcsec at 1 au)" in Stellarium's paraphrase of the convention ¹⁸. The Besselian Elements team's test on the 2017 path: "On a location on the centreline, the duration of totality drops from 130s to 128s" and "for a location very near the southern edge of the eclipse path, the duration of totality goes from 34s to 13s" when the radius goes from 959.63 to about 960 arcseconds ³. That southern-edge case is the Vale, Oregon site of the 2017 eclipse, given more precisely as 32.6 s falling to 13.3 s in observed against predicted, which is the home page for the observed cases. Their 2013 November 3 flash-spectrum timing found "the measured time of second contact is around 1 second later than predicted" and "third contact is around 1 second earlier", with a compatible radius "in the range 959.90 to 960.00 arcsec" ⁴. At Stephenville, Texas, on 2024 April 8, near the edge, the observed totality was 13.7 s, John Irwin's prediction gave 12.9 s, and other published sources "often" differed by "several tens of seconds" ¹⁹. Pasachoff, Jubier and Wright add that the IAU 2015 conversion factor of 959.22 arcseconds "would be even longer, up to two seconds longer, than with the IAU 1976 value, which gives a duration that is already far too long", and propose 959.98 plus or minus 0.02 arcseconds ²⁰. The Photo Ephemeris bead simulator defaults to 959.95 arcseconds and found 960.01 arcseconds the best match to a 2023 annular video ¹². Every one of these numbers moves the interior contacts by about one second per 0.3 arcseconds on the central line and by much more near the edge.

The lunar radius $k$

The full history of the two values is in Besselian elements. This section keeps what the choice costs at one observer.

Espenak's history: $k = 0.2724880$ (USNO 1968 to 1980) for penumbral contacts and annular eclipses, $k = 0.272281$ "reserved exclusively for umbral (i.e., interior) contact calculations of total eclipses", and $k = 0.2725076$ adopted by the IAU in 1982 and "currently used by USNO for all solar eclipse predictions" (the Astronomical Almanac; the USNO online computer uses 1737.4 km, as the direct topocentric method records). The smaller value "produces shorter central durations and narrower paths for total eclipses" and the 2017 Illinois example ranges from 2m40.3s to 2m44.3s ⁵. The Five Millennium Canon, NASA/TP-2006-214141, and the NASA element pages use 0.272281 for all umbral and antumbral contacts and 0.2724880 for penumbral contacts. The 1998 bulletin uses 0.272281 and 0.2725076, as do EclipseWise and the later bulletins ^{21 22}. The reduced $k$ is an empirical stand-in for the limb valleys. Once a real limb profile is applied, the mean IAU radius is the right datum, and Solar Eclipse Maestro displays both for that reason ¹⁰.

$\Delta T$

Times computed in Terrestrial Time (TT)Terrestrial 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 converted to UT by subtracting $\Delta T$, and the observer's longitude is shifted to the ephemeris longitude by $1.002738\,\Delta T$ ⁸. An error $\delta T$ in $\Delta T$ therefore shifts every reported UT by $\delta T$ directly and moves the site by $1.002738\,\delta T$ in longitude, which changes the contacts by a second-order amount through $\xi$ and $\eta$. The 1961 Supplement handles the second part with its differential corrections: set "$\delta\lambda = 1.002738\,\delta T$, $\delta\phi = \delta H = 0$" and apply the $\partial t/\partial\lambda$ coefficients ⁸. The Five Millennium Canon states that the ephemeris truncation error in eclipse phase times is "of the order of 1/40 s, which is considerably smaller than the uncertainties in predicted values of $\Delta T$" ²¹. Jubier: "It is not possible to predict the exact value of $\Delta T$ in advance, although the extrapolated value should be good to better than 0.5 seconds" ⁶. The Photographer's Ephemeris updates the $\Delta T$ inside NASA's element files case by case from USNO deltat.data and deltat.preds ². Stellarium's export notes that "Local circumstances for eclipses during thousands of years in the past and future are not reliable due to uncertainty in $\Delta T$" ¹⁴.

Observer height

Height enters through $\rho\sin\phi'$ and $\rho\cos\phi'$. The 1961 Supplement gives the derivative of an exterior contact time with respect to height as

in hours per metre with $a$ the equatorial radius in metres, and the corresponding coefficients $r_m$ for maximum and $r_s$ for the semi-duration ⁸. NASA's tables state that "the elevation does not play a significant role in the predictions unless the location is near the umbral path limits and the Sun's altitude is relatively small (<10°)" ⁷. Geometrically the effect is a displacement of the observer by $h$ along the vertical, whose component across the shadow axis is $h\sin z$ with $z$ the zenith distance of the Sun. The 2017 poster describes the ground-level consequence: "accounting for Earth terrain tends to shift the shape toward the Sun's azimuth by approximately $h\cot(a)$" ²⁰. For a central line observer the duration change is second order. For an observer near the edge, where the umbral polygon side is only tens of metres away, a kilometre of height at a 30 degree Sun is a 1.7 km shift. That can make the difference between a short totality and none, the case the Stephenville comparison illustrates ¹⁹. Bill Kramer's calculator adds horizon dip above 100 m to the altitude ²³. Solar Eclipse Maestro asks for the site to about 200 m ²⁴.

Refraction

Refraction raises both discs and, near the horizon, flattens them differently because the differential refraction across the two discs differs with their altitudes. NASA's local circumstances exclude it: "The effects of refraction have not been included in these calculations" ⁷. Jubier's calculator "does not account for atmospheric refraction, which makes a difference if the eclipse occurs close to sunrise or sunset (a good example of this was the midnight Sun eclipse in Antarctica on November 23, 2003)" ⁶. Solar Eclipse Maestro lists refraction as supported ²⁴. The Photographer's Ephemeris corrects the altitude "using the US Standard Atmosphere model, but the apparent limb shape of each body remains circular" ². Bill Kramer's calculator applies refraction "using averaged temperatures and barometric pressures" with altitude-dependent approximations ²³. Swiss Ephemeris and USNO refract only the reported altitude ^{15 25}. The JSEX authors left a note on what a correct treatment requires: a different virtual observer position for each contact ¹³. No source read for this note quantifies the contact-time effect of refraction in seconds. That is a negative finding: the searches on refraction and eclipse contact timing returned only qualitative statements and the sunrise-eclipse examples. A derived bound, for the exterior contacts at low Sun altitudes and for the interior contacts, is in Refraction, light-time and frames.

Centre of figure versus centre of mass

The Five Millennium Canon: "The center of figure of the Moon does not coincide exactly with its center of mass. To compensate for this property in their eclipse predictions, many of the national institutes employ an empirical correction to the center of mass position of the Moon. This correction is typically +0.50 arcsec in longitude and -0.25 arcsec in latitude. Unfortunately, the large variation in lunar libration from one eclipse to the next minimizes the effectiveness of the empirical correction. The authors have chosen to ignore this convention and have performed all calculations using the Moon's center of mass position" ²¹. The NASA bulletins say the same ²⁶. Bill Kramer's calculator applies it only implicitly, "when the lunar limb corrections are incorporated the actual observed center of figure is determined and used" ²³. Since the Sun and Moon separate at about 0.5 arcseconds per second of time during an eclipse, a 0.5 arcsecond offset along the direction of motion is about 1 s in a contact time. That rate and conversion are the author's arithmetic from the synodic motion, not a quotation. A limb profile referenced to the centre of mass, as the LOLA-based profiles are, absorbs the offset, so the empirical correction and a modern limb correction must not both be applied.

Ephemeris

The Canon's truncated ELP series introduces "about 0.0006 s of time in right ascension, and about 0.006 arcsec in declination", hence 1/40 s in phase times ²¹. NASA's older bulletins used DE200/LE200 ^{22 26}. sunpy warns that astropy's built-in Moon is "appreciably inaccurate" without a JPL ephemeris ²⁷. Between modern JPL ephemerides the difference is below 0.1 s in contact times. The 2024 Stephenville comparison nevertheless shows published predictions spread by tens of seconds at an edge site. The authors attribute the spread to the radius and limb treatment rather than the ephemeris ¹⁹.

Summary of magnitudes

The full ranked budget is in error budget and validation protocol. This table keeps the rows the local-circumstances stage owns.

Baily's beads at the observer

The bead sequence at a site is the limb correction computed at every position angle rather than only at $\psi_c$. Herald's 1983 procedure, as NASA describes it: "For a given position angle, the solar limb overlay is moved radially from the mean lunar limb contact point until it is tangent to the lowest lunar profile feature in the vicinity ... This operation may be used for predicting the formation and location of Baily's beads" ⁹. Around second contact each valley near $\psi_c$ admits photospheric light until the solar limb, which is receding behind the Moon at rate $\dot r(\psi)$, drops below the valley floor. The disappearance time of the bead in valley $i$ is $t_i = t_\mathrm{mean}(\psi_i) + h(\psi_i)/\dot r(\psi_i)$ and the beads vanish in order of increasing $t_i$, the last one being the corrected C2. Around third contact the sequence runs in reverse. The photosphere's own limb-darkening profile and the chosen solar radius shift every $t_i$ together. That common shift is what makes bead timings a measurement of the solar radius, as Effect of the radius on eclipse products sets out ^{4 12}.

The 2017 DPS poster gives the global picture of the same computation: "The limb of the Moon produces a polygonal umbra shape on the Earth's surface. Each side of the polygon corresponds to a single valley on the lunar limb, either the last valley admitting photospheric light prior to second contact, or the first valley admitting sunlight just after third contact. Here, with 18,000 points equally spaced in position angle around the limb, the umbra is determined by just 49 points ... Observers at a particular edge, where two umbra polygon sides meet, see a first or last Baily's bead in the corresponding valley on the lunar limb" ²⁰. Zeiler's double diamond ring page uses the same polygon: "each of these chords represent the influence of a single Baily's Bead", and an observer at a vertex between two chords sees two last beads at once ²⁸.

Implementations:

• Solar Eclipse Maestro (Jubier, macOS). The Baily's Beads Study window "displays a graph of the LRO lunar limb profile and the Sun in the background as seen from the observer's current location at either second or third contact and maximum eclipse", with a time slider. "The mauve limb is LRO. The brown dotted line charts the reduced mean lunar radius $k_2$, usually used to compute the uncorrected second and third contacts ... The blue dotted line is the mean radius used for the data of the LRO and Kaguya probes. The vertical axis unit is arc-seconds and the unit of the position angle of axis on the horizontal axis is degrees." Uncorrected contacts are labelled C2 and C3, corrected ones C2' and C3', the solar limb is drawn at each, and a click returns the position angle of axis and the height relative to $k = 0.2725076$ (1738.091 km). The profile for the current topocentric libration can be exported as text ¹⁰. The program lists "refraction, lunar limb profile effects, and $\Delta T$ correction supported", "UTC event times for subsecond accuracy" and "lunar limb corrections and Baily's beads simulation" ²⁴. Its Google Maps front end gives durations "both with and without limb profiles incorporated" ²⁰.
• Occult (Herald, Windows, C#). The RASNZ software list states that Occult can "predict solar and lunar eclipses for any date (includes: plot maps and compute local circumstances for solar eclipses)" and "compute Baily's bead predictions (Windows version only)" ¹¹. Herald reduced the Kaguya altimetry into the limb profiles that Zeiler and Jubier used for the 2017 double diamond ring predictions ²⁸. The Occult program page itself lists eclipses among its phenomena without method detail ²⁹. Occult's help text was not obtained. The Windows help is inside the installer.
• Photo Ephemeris simulates beads with Kaguya (SELENE) and Herald profiles and a default solar radius of 959.95 arcseconds, and publishes frame-by-frame comparisons with eclipse video, for example beads captured "3.2 seconds before C2 and 2.7 seconds before C3" at Kirtland, New Mexico ¹². Its contact times, by contrast, remain smooth-Moon ².
• Bill Kramer's calculator optionally applies "Watts Lunar Limb Profile data or Kaguya/Herald laser altimeter data" and claims "for the majority of eclipses the deviation is less than a second in totality duration time" ²³.
• Jubier's web calculator (2007) applies a limb correction "LC" to C2 and C3 when online, and otherwise refers the user to a per-eclipse correction diagram read at the contact's $P$ ⁶.

Public code that computes local circumstances

Negative findings from the GitHub searches run for this note: no Go implementation of local circumstances was found, and the single Rust project found has not implemented them ³³. No general library applies a limb correction from a LOLA or Kaguya (SELENE) profile. Two single-eclipse repositories and one npm package do, and the inventory is in GitHub repositories. The limb-aware general-purpose programs are Occult, Solar Eclipse Maestro, Photo Ephemeris and Kramer's calculator, all closed, and the open codes stop at the smooth Moon.

Sources compared

What a developer should do

Treat the smooth-Moon contact times as the datum and apply corrections in this order, each as a separately reported term. First the solar radius, chosen explicitly: 959.63 arcseconds for NASA compatibility, 959.95 to 960.0 arcseconds for observational agreement, the two modes decided in Solar radius values and their provenance. Then $k = 0.2725076$ as the datum for a real limb profile. Then the limb correction at C2 and C3 from the profile at the topocentric libration. Then $\Delta T$ from USNO's current file rather than the value frozen in an element file. Then height. Do not apply the +0.50 / −0.25 arcsecond centre-of-figure convention together with a centre-of-mass limb profile. Report refraction as unmodelled unless the Sun is below 10 degrees, and then flag the result. For beads, evaluate the limb correction at every 0.02 degrees of position angle, following the 2013 analysis' finding that 0.2 degree sampling costs 0.25 s ⁴. Read first: NASA's limb help page, the RP 1383 limb section with the Maracaibo example, the Besselian Elements solar-radius post, and the Solar Eclipse Maestro beads window description.

What this changes

It fixes the interface between the local-circumstances stage and the limb-profile stage. For each interior contact the local stage must export the time, the position angle $P$ of the contact, the position angle and speed of the relative motion, and the topocentric libration. Those are exactly the inputs the limb correction and the bead sequence consume. It also settles that the solar radius must be a run-time parameter of the element generator, not a constant.

Open questions

• Obtain a quantitative statement, or compute one, of the effect of standard refraction on C1 to C4 at Sun altitudes of 1, 3 and 10 degrees. No source read gives seconds.
• Obtain Occult's help text on its Baily's beads and "limb-corrected predictions" features, which lives inside the Windows installer, and record the profile datum and solar radius it uses ¹¹.
• Obtain Herald (1983), the graphical procedure NASA cites, to confirm the tangent-overlay definition of the correction and its sign convention ⁹.
• The "LC" column of Jubier's web calculator is answered by the client code read in commercial and institutional tools, which shows a server call that chooses Watts charts when the libration in latitude exceeds 1.6 degrees in absolute value and Kaguya (SELENE) otherwise. What remains to obtain is the server-side computation and the ephemeris behind it ⁶.
• Confirm timeanddate's accuracy page against the page itself rather than a secondary report, and record the $\Delta T$ source and the solar radius it names ³⁵.