Refraction, light-time and frames

workingupdated 2026-09-15refractionaberrationlight-timetdbapparent-placeeffects-table

No major predictor applies refraction to eclipse contact times or to the path. Espenak states that "refraction corrections to the path are uncertain since they depend on the atmospheric temperature-pressure profile, which cannot be predicted in advance", the 1961 Supplement draws the rising and setting curves "neglecting refraction, that is assuming a zenith distance of 90°", and Jubier's calculator says "astronomical horizon, no atmospheric refraction" ¹ ² ³.
The neglect is justified by geometry, not by the smallness of refraction. Refraction lifts both Sun and Moon by $9.7'$ at $5^{\circ}$ and $21.7'$ at $1^{\circ}$ , but a contact is a relative-angle condition at the observer, so a common lift cancels, and a linear vertical compression of both disks preserves tangency. Only the curvature of $R(h)$ across the half-degree disk survives, seconds at first and last contact below $5^{\circ}$ and tenths of a second at second and third contact.
The almanac formulas are Bennett's for apparent altitude and Sæmundsson's for true altitude, $R = 1.02 / \tan(h + 10.3/(h + 5.11))$ arcminutes scaled by $(P/1010)(283/(273 + T))$ , with 34′ on the horizon for rise and set. Stellarium implements both with defaults of 1013 mbar and 10 °C and cuts off at $-3.54^{\circ}$ ⁴ ⁵.
The Besselian elements are built from apparent places. The 1992 Supplement: the inputs are "the apparent right ascension, declination, and distance of the Sun ... and of the Moon", and "Since the apparent positions of the Sun and Moon are used in the calculations aberration is already included, except for the inappreciable difference between the light-time from the Sun to the Moon and from the Sun to the Earth" ⁴. Using geometric positions for the Sun displaces the shadow axis by $20.5''$ , which is 38 km at the Moon's distance.
TT and TDB differ by at most about 1.7 ms, "which for the geocentric position of the Moon would correspond to an angular error of <1 mas", which is about 2 m on the shadow. The DE ephemerides are in TDB and must be entered with the JPL formula, but an eclipse product can use TT. On first use here, TT is printed as TDT or TD in NASA tables and was called ET before 1984 ⁶ ⁷.
Ranked by size on the ground: aberration consistency, 38 km if wrong; a sphere instead of an ellipsoid, 21 km; terrain, kilometres; ΔT prediction, hundreds of metres per second; geoid, up to 170 m; nutation model, under 20 m; polar motion, 10 m; TDB, 2 m; the ephemeris itself, under 1 m for DE440 against lunar laser ranging ⁸. Refraction sits at seconds of contact time only below a 5° Sun.

The question. Do eclipse predictors include atmospheric refraction in contact times and in the path near sunrise and sunset, and should they, given the size of the effect at 5° and 1° of altitude? Which refraction formulas do the Explanatory Supplement, Meeus, the Swiss Ephemeris and Stellarium use, and what does the "rise/set curves" convention assume about the horizon? How are light-time and aberration handled for the Sun and Moon, are the elements built from apparent or geometric places, does the TT versus TDB distinction matter, and what is the origin for the Sun's position? Finally, how large is every effect in this topic, in seconds of time and in metres on the ground, and which predictors include each?

Refraction: what the sources say

The Explanatory Supplement treats refraction as a refinement outside the standard computation. The 1992 edition lists among "refinements to be considered in performing precise calculations: choosing a consistent ephemeris, corrections for Earth's rate of rotation, and, for the Moon, offset of center of figure from center of mass; correcting for irregularities of the lunar limb; and effects of the Earth's atmosphere (refraction in solar eclipses and the effect on the shadow in lunar eclipses)", and then does not apply it in the algorithms that follow ⁴. The 1961 edition defines the rising and setting curves without it: "points on the rising and setting curve (neglecting refraction, that is assuming a zenith distance of 90°) may be calculated for each stated time by noting that ζ is equal to zero" ². That is the convention on every almanac and NASA map: the "eclipse begins at sunrise" curve is where the Sun's centre is on the geometric horizon, with no refraction and no semidiameter. An observer sees the upper limb rise when the centre is at $-34' - 16' = -50'$ of true altitude (the Supplement's rise and set convention is "a constant of 34′ is used for the horizontal refraction; thus for the center of the Sun or Moon to be coincident with the horizon the true altitude will be −34′" ⁴), which at the equator is about 3.3 minutes earlier than the geometric moment. On the ground the mapped curve is displaced from the visible sunrise line by $50'$ of terminator position, about 93 km, always toward the night side.

Espenak's bulletins state the policy of the NASA products: "Unless otherwise stated, all predictions are based on center of mass positions for the Moon and Sun with no corrections made for center of figure, lunar limb profile or atmospheric refraction", and for the maps, "Atmospheric refraction has not been included, as its effects play a significant role only at very low solar altitudes. In any case, refraction corrections to the path are uncertain since they depend on the atmospheric temperature-pressure profile, which cannot be predicted in advance" ¹. The local-circumstance tables say the same: "The effects of refraction have not been included in these calculations" ¹. The NASA and EclipseWise 2017 and 2024 pages carry no refraction statement at all ⁹ ¹⁰.

Jubier's calculator labels its diagram "Relative positions of the Sun and Moon (astronomical horizon, no atmospheric refraction)" ³. timeanddate does not mention refraction for eclipse contacts, and for rise and set says its times "are based on a flat horizon where the observer is at the same altitude as the horizon" ¹¹. The technical note for The Photographer's Ephemeris (Photo Ephemeris) applies refraction only to its picture: "the simulator is corrected for refraction using the US Standard Atmosphere model, but the apparent limb shape of each body remains circular" ¹². Occult's graze listings say "The path is adjusted for the effects of refraction at low moon altitudes", for lunar grazes. Nothing was found about solar-eclipse limits ¹³. The Swiss Ephemeris applies it selectively: "Algorithms for the central line is taken from Montenbruck, pp. 179ff., with the exception, that we consider refraction for the maxima of partial and noncentral eclipses", with an error budget entry "from displacement of shadow points by atmospheric refraction: a few meters" and the warning "if the sun is close to the horizon, all of these errors can grow up to a km or more" ¹⁴. It also uses refraction plus dip to decide visibility: "approximate minimum height for visibility, considering refraction and dip: 34.4556′: refraction at horizon, from Bennets formulae; 1.75′/sqrt(geohgt): dip of horizon; 0.37′/sqrt(geohgt): refraction between horizon and observer", coded as hmin_appr = -(34.4556 + (1.75 + 0.37) * sqrt(geohgt)) / 60 degrees ¹⁴.

The formulas

The 1992 Supplement's low-precision formula, "accurate to 0′.5" and adequate for "the times of rising and setting", is in terms of the apparent altitude $H$ in degrees, temperature $T$ in °C and pressure $P$ in millibars:

R = \frac{P}{1010}\,\frac{283}{273 + T}\;\frac{0.0167^{\circ}}{\tan\!\left(H + \dfrac{7.31}{H + 4.4}\right)}

"If $T$ and $P$ are unknown, assume the term in the first bracket is unity", and for $H < 15^{\circ}$ the alternative

R = \frac{P}{1010}\,\frac{283}{273 + T}\;\frac{0.1594 + 0.0196H + 0.00002H^2}{1 + 0.505H + 0.0845H^2}\ \text{degrees}

⁴. The first is Bennett's formula ( $0.0167^{\circ} = 1.002'$ ). The Swiss Ephemeris swe_refrac uses the Astronomical Almanac pair: above 15° " $D = 0.00452\,P/((273 + T)\tan h)$ " degrees with "Accuracy 'usually about 0.1′'", and below it the "Almanac for Computers" form with $N = 1/\tan(h + 7.31/(h + 4.4))$ and $D = N P' / (60 + Q(N + 39))$ , inverted by Newton iteration when converting true to apparent ¹⁴. eclipse-chasers.com uses the same two ¹⁵.

Meeus's Astronomical Algorithms gives the true-altitude form due to Sæmundsson, which Stellarium implements as its forward transformation: "refraction from Saemundsson, S&T1986 p70 / in Meeus, Astr.Alg.", coded as

R' = \frac{P}{1010}\,\frac{283}{273 + T}\left[\frac{1.02}{\tan\!\left(h + \dfrac{10.3}{h + 5.11}\right)} + 0.0019279\right]\ \text{arcmin}

with $h$ the geometric altitude in degrees, and the backward transformation "refraction from Bennett, in Meeus, Astr.Alg." with the constant 0.0013515 ⁵ ¹⁶ ¹⁷. The small added constants make each formula vanish at the zenith. Stellarium's defaults are pressure(1013.f), temperature(10.f), and it stops applying the formula below the horizon: MIN_GEO_ALTITUDE_DEG = -3.54f, MIN_APP_ALTITUDE_DEG = -3.21783f, with linear transition zones, and a polynomial inverse below $-2^{\circ}$ because "Bennett's formula is not a strict inverse of Saemundsson's" ⁵. The developer documentation adds only that "Typical horizons do not go down below -1, so strange effects (distortion) between -2 and -5 should be covered" ¹⁸.

Evaluating Sæmundsson's formula at standard conditions (computed here): $R = 29.0'$ at $h = 0^{\circ}$ , $21.7'$ at $1^{\circ}$ , $16.9'$ at $2^{\circ}$ , $9.7'$ at $5^{\circ}$ , $5.4'$ at $10^{\circ}$ , $2.7'$ at $20^{\circ}$ , $1.0'$ at $45^{\circ}$ . Bennett's gives $34.5'$ at an apparent altitude of $0^{\circ}$ , which is the origin of the 34′ convention. The gradient, which is what matters for an eclipse, is $dR/dh = 5.9'$ per degree at $1^{\circ}$ , $1.5'$ per degree at $5^{\circ}$ , $0.5'$ per degree at $10^{\circ}$ and $0.04'$ per degree at $45^{\circ}$ .

Why refraction (almost) cancels in an eclipse

A contact is the instant at which the apparent limb of the Moon touches the apparent limb of the Sun as seen from the observer. It is a condition on the angular separation and the apparent radii, both measured at the observer's eye. Refraction is a function of altitude alone. To first order it lifts the Sun and the Moon, which are within half a degree of each other, by the same $R$ , and a common displacement changes neither their separation nor their radii. To second order it compresses both disks vertically by the same factor $1 - dR/dh$ , and an affine compression applied to two circles preserves their tangency and containment. The path on the ground is the locus of observers for whom second and third contact coincide, so it is likewise unchanged. What remains is the variation of $dR/dh$ across the half-degree field, the curvature of $R(h)$ . This derivation is made here. None of the sources read states it, but it is why the Swiss Ephemeris finds only "a few meters" of shadow displacement in normal conditions and why the almanac tradition drops the term ¹⁴.

The residual can be bounded from the gradients above. At first or last contact the centres are separated by about $0.53^{\circ}$ , and if that separation is vertical the differential refraction between the two centres is $0.53^{\circ} \times dR/dh$ : $44''$ at $5^{\circ}$ and $176''$ at $1^{\circ}$ (computed here). Most of that is the affine part and cancels against the compression of the radii. The part that does not cancel is of order the change of $dR/dh$ across the field, roughly a quarter of those figures, $10''$ at $5^{\circ}$ and $40''$ at $1^{\circ}$ . At the relative Sun-Moon motion of about $0.5''$ per second that is 20 s and 80 s of first-contact timing, and it is not predictable because it depends on the temperature profile in the lowest kilometre, as Espenak says. At second and third contact the centres are within about $0.01^{\circ}$ and the differential across that gap is $0.9''$ at $5^{\circ}$ and $3.5''$ at $1^{\circ}$ , again largely affine, leaving well under a second of totality timing even at $5^{\circ}$ . The 34′ lift does matter for a different question, visibility: an eclipse in progress at geometric sunrise is already visible, and one ending at geometric sunset is still visible, which is the case the Swiss Ephemeris handles with its hmin_appr test ¹⁴.

Apparent places, aberration and light-time

The apparent place of a body is its geocentric direction corrected for light-time, aberration, precession and nutation. The Supplement is unambiguous that the Besselian elements are built from apparent places: "The basic quantities for calculations of solar and lunar eclipses are: the apparent right ascension ( $\alpha_s$ ), declination ( $\delta_s$ ), and distance ( $R_s$ ) of the Sun, and the apparent right ascension ( $\alpha_m$ ), declination ( $\delta_m$ ), and distance ( $R_m$ ) of the Moon, for every hour of TDT during the eclipse ... The apparent places of the Sun and the Moon are computed rigorously using the same ephemerides for the Sun and the Moon as are used for The Astronomical Almanac, and using the reduction methods described elsewhere in this book (see Chapter 3)" ⁴. On aberration: "Since the apparent positions of the Sun and Moon are used in the calculations aberration is already included, except for the inappreciable difference between the light-time from the Sun to the Moon and from the Sun to the Earth" ⁴.

The physics is that the shadow is formed by light that left the Sun 499 s earlier and travels through a frame in which the Earth moves at 30 km/s. In that frame the rays arrive from the aberrated direction of the Sun, which is $20.5''$ from the geometric direction. The shadow axis must therefore be built from the apparent Sun, and $20.5''$ of direction error at the Moon's distance of 384,400 km displaces the axis by 38 km on the ground (computed here). The Moon's own aberration is small, under an arcsecond, because it shares the Earth's orbital velocity, and its light-time is 1.28 s. The "inappreciable" term is the Sun's motion, $0.041''$ /s, over the 1.28 s difference in light-time between the Sun-to-Moon and Sun-to-Earth paths: $0.05''$ , or about 100 m on the shadow axis (computed here). A developer who takes the Sun's apparent place and the Moon's apparent place from the same reduction gets all of this right. One who takes "geometric" or "true" positions from an ephemeris API for the Sun, or mixes an aberrated Sun with an unaberrated Moon, is wrong by tens of kilometres.

The Supplement's second correction at this stage is the centre of figure: "Gravitational ephemerides refer to the positions of the centers of mass of the bodies concerned. Eclipses, however, are governed by the positions of the centers of figures ... The range of the corrections is less than 0″5, and for the calculations in The Astronomical Almanac, only the minimum corrections are applied. These are Δλ = +0″50 and Δβ = −0″25 for the LE200 lunar ephemeris" ⁴. Espenak does not apply it: "The lunar coordinates have been calculated with respect to the Moon's Center of Mass. They DO NOT include a correction to the Center of Figure" ¹⁹. SVS puts the offset at "about 0.5 kilometers" and absorbs it in the limb profile ²⁰. That is the sky-plane component. The full centre-of-mass to centre-of-figure vector from LOLA is 1.935 km, mostly along the Earth-Moon line ²¹. Those quantities belong to lunar figure, radius ratio k, and libration and are listed in the table below only for scale.

Espenak fits the elements rather than tabulating them: "The Besselian elements were derived from a least-squares fit to elements calculated at five uniformly spaced times over a six hour period centered at t0" ²². The Swiss Ephemeris computes Sun and Moon with SEFLG_SPEED | SEFLG_EQUATORIAL plus the caller's ephemeris flag, which by default yields apparent positions with light-time and aberration, and, as noted in the ΔT note, drops nutation only if mean sidereal time is used for the hour angle ¹⁴.

TT, TDB and the barycentric origin

The JPL ephemerides are integrated in TDB. The DE430 memo: "The coordinate time scale used for DE430 and DE431 is Barycentric Dynamical Time (TDB) as defined in terms of Barycentric Coordinate Time (TCB)", with the TDB − TT expression given in full, and "The origin of the ICRS is the solar system barycenter" ⁷. Circular 179 gives the working relation:

T_{\mathrm{eph}} \approx \mathrm{TDB} \approx \mathrm{TT} + 0.001657 \sin(628.3076\,T + 6.2401) + 0.000022 \sin(575.3385\,T + 4.2970) + \cdots\ \mathrm{s}

with $T$ in Julian centuries of TT from J2000.0, and states the consequence of ignoring it: "The total error in time in using TT as the input argument is <2 ms, which for the geocentric position of the Moon would correspond to an angular error of <1 mas" ⁶. The Moon moves $0.55''$ /s, so 1.7 ms is $0.0009''$ , or 1.7 m on the shadow axis (computed here). An eclipse product may evaluate the ephemeris at TT without measurable loss, but a developer reading a DE file directly should still enter it with TDB, because the library interfaces (SPICE, the Swiss Ephemeris) expect it.

The origin question is one of bookkeeping. The DE files give positions relative to the solar-system barycentre, with the Earth-Moon barycentre and the geocentric Moon as separate objects. The geocentric Sun needed for the elements is $\mathbf{r}_{\odot} - \mathbf{r}_{\oplus}$ , where $\mathbf{r}_{\oplus}$ is recovered from the barycentre and the Moon using the ephemeris's own Earth-Moon mass ratio. The light-time and aberration corrections are then applied to that geocentric vector, and the Moon is already geocentric. Doing this by hand invites a mismatch between the mass ratio and the ephemeris. The SPICE and Swiss Ephemeris interfaces do it internally ⁷ ¹⁴.

Every effect in this topic

The full ranked budget is in error budget and validation protocol. This table keeps the rows this topic owns. Sizes on the ground are for the umbral path or a contact position. Times are for contact instants. "Included" means documented on the pages read; a dash means not documented either way.

Effect	Typical size, time	Typical size, ground	Espenak NASA and EclipseWise	NASA SVS	Jubier	timeanddate	Occult	Swiss Ephemeris
Ellipsoid instead of sphere	tens of s at contacts	up to 21 km	Yes (implied by the elements)	Yes, WGS 84	not stated	Yes ("squashed")	Yes, WGS 84	Yes, z-scaling
Choice among IAU 1976, GRS 80, WGS 84	0	3 m in $a$ , 16 mm in $b$	any	WGS 84	not stated	not stated	WGS 84	AA 2006
Observer elevation, path limits	s to tens of s at limits	$h \cot A \sin D$ : 0.6 to 1.7 km per 1000 m at 60° to 30°	Sea level; elevation factor supplied	Yes, SRTM	Altitude input	No, sea level	Site altitude	Height input, global paths at sea level
Observer elevation, duration	under 1 s	$2h \tan f_2$ , 82 m at Everest	No	Yes	Yes	No	not stated	Yes
Geoid versus ellipsoid height	under 1 s	$N \cot A$ , up to 170 m at 30°	No	Yes, EGM96	not stated	No	not stated	Listed as error
DEM vertical error	under 1 s	under 20 m for SRTM at 30°	n/a	SRTM	not stated	n/a	not stated	n/a
ΔT prediction error	0.05 s at 1 yr, 0.6 s at 5 yr, 4.6 s at 20 yr, 52 s at 100 yr	$465\cos\phi$ m per second: 0.02, 0.2, 1.6, 18 km at 40°	Yes; 2024 was 1.4 to 2.3 s high	Yes; 2017 within 0.08 s	Yes, value not read	Yes, value not published	not stated	Yes, model switch at 1955
UT1 versus UTC	up to 0.9 s, under 1 ms in 2026	up to 0.42 km	UT1 printed	Leap seconds in "Delta UTC"	not stated	not stated	not stated	UT1
Polar motion	0	about 10 m	No	Implicit in SPICE kernel	No	No	No	Listed as error, not applied
Mean versus apparent sidereal time, if inconsistent	up to 1.1 s	up to 0.5 km	consistent	consistent	not stated	not stated	not stated	consistent by construction
Nutation and precession model (1980 vs 2000A)	0	under 20 m	any	SPICE	not stated	not stated	not stated	any
Aberration of the Sun, if omitted or mixed	tens of s	38 km	Apparent places	Apparent (DE421 via SPICE)	not stated	not stated	not stated	Apparent by default
Sun-to-Moon versus Sun-to-Earth light-time	0.2 s	about 100 m	No	not stated	not stated	not stated	not stated	No
TT versus TDB	1.7 ms	2 m	any	TDB via SPICE	not stated	not stated	not stated	TDB internally
Refraction at 2nd and 3rd contact, Sun above 5°	under 1 s	under a few metres	No	No	No	No	not stated	No
Refraction at 1st and 4th contact, Sun at 1° to 5°	of order 10 to 80 s, unpredictable	n/a	No	No	No	No	not stated	Maxima of partial eclipses only
Rise and set curves, refracted versus geometric horizon	about 3.3 min	about 93 km along the terminator	Geometric	not stated	Geometric	Flat horizon for rise and set	not stated	Visibility test with 34.4556′ plus dip
Centre of figure versus centre of mass (for scale)	under 1 s	about 0.5 to 1 km	No	Absorbed in limb profile	via limb correction	No	via limb data	No
Ephemeris error	under 0.01 s	under 1 m for DE440 against lunar laser ranging; the Swiss Ephemeris budgets 40 m for an assumed 0.01 arcseconds	DE405 on the 2017 Google-map page, VSOP87/ELP2000-85 on the 2024 element page	DE421 on the product pages, DE440 in the paper's appendix	not stated	not stated	not stated	DE431 default

Sources for the rows: ellipsoid and elevation ² ⁴ ¹ ²³ ¹¹ ¹⁴; ΔT ²⁴ ²⁵ ²⁶ ¹⁰ ²⁷; UT1 and polar motion ²⁸; sidereal time and nutation ⁶; aberration and light-time ⁴; TDB ⁶ ⁷; refraction ⁴ ⁵; centre of figure ⁴ ²⁰ ¹⁹; ephemeris error ⁸ ¹⁴ ⁹.

Sources compared

Source	Refraction in contacts	Refraction in rise/set curves	Positions used	Time argument	What it uniquely provides
Explanatory Supplement 1961 ²	Not applied	Geometric horizon, stated	Apparent places	ET	The rise/set convention in words
Explanatory Supplement 1992 ⁴	Listed as a refinement, not applied	34′ for rise/set tables, geometric for eclipse curves	Apparent places, stated; centre-of-figure +0″50, −0″25	TDT	The refraction formulas and the aberration statement
Espenak NASA bulletins ¹	Not applied, with the reason	Geometric	Centre of mass, no limb	TDT and UT	The clearest policy statement
Swiss Ephemeris ¹⁴	Only for maxima of partial eclipses; visibility test with 34.4556′ plus dip	n/a	Apparent, flags	UT in, TT internally	An explicit error budget
Stellarium ⁵	n/a (display)	n/a	n/a	n/a	The Sæmundsson and Bennett formulas as code, with cut-offs
Jubier calculator ³	Diagram without refraction	n/a	n/a	ΔT printed	Limb correction at contacts
Photo Ephemeris ¹²	Simulator only, US Standard Atmosphere	n/a	Meeus elements	USNO ΔT	A stated refraction model for the picture
Circular 179 ⁶	n/a	n/a	n/a	TT, TDB, sidereal time	The TDB series and the sizes of nutation-model effects
DE430 memo ⁷	n/a	n/a	SSB, ICRS	TDB	The time argument and origin of the ephemeris

What a developer should do

Build the elements from apparent places of the Sun and Moon from one reduction chain, and never from geometric positions. Check the chain by confirming that the Sun's apparent and geometric right ascensions differ by about $20''$ ⁴.
Enter the DE ephemeris with TDB if reading it directly, or let SPICE or the Swiss Ephemeris do it. Do not spend effort on the 1.7 ms ⁶ ⁷.
Leave refraction out of contact times and path limits, and say so in the product. Offer an optional first- and fourth-contact adjustment for Sun altitudes under 5° with an explicit "unpredictable to tens of seconds" label, using the Supplement's low-altitude formula with the user's pressure and temperature ⁴ ¹.
Draw rise and set curves for the geometric horizon, as the almanacs do, and label them. Add a visible-sunrise curve at $-50'$ of true altitude only as a separately labelled layer ² ⁴.
Use the Swiss Ephemeris visibility test, $h_{\min} = -(34.4556 + 2.12\sqrt{h_{\mathrm{obs}}})/60$ degrees, to decide whether a contact near the horizon is observable at all ¹⁴.

Read first: Section 8.12 of the 1992 Supplement for the apparent-place statement, Section 3.283 for the refraction formulas, Section 2 of Circular 179 for the time scales, and the comment block at the head of eclipse_where in swecl.c for the error budget.

What this changes

Nothing in the geometry stage. The pipeline needs an "apparent places" contract on the ephemeris interface and a stated refraction policy in the output metadata. The effects table fixes the order in which validation effort should be spent: aberration consistency and the ellipsoid first, then terrain and ΔT, then everything else.

Open questions

Obtain Meeus's Elements of Solar Eclipses 1951–2200 and check whether his elements are computed from apparent places and how he treats the Sun's aberration. The Photo Ephemeris products rest on it ²⁹.
Obtain Occult's solar-eclipse help topic to learn whether the refraction adjustment applied to graze paths is also applied to eclipse limits near the horizon ¹³.
Obtain a ray-traced test case, for example from the Swiss Ephemeris test suite or a purpose-built integration, that quantifies the second-order refraction residual at second contact for a Sun at 2° and 5°, to replace the bound derived here.
Obtain the 2013 Explanatory Supplement's eclipse chapter for any change to the apparent-place and centre-of-figure statements of 1992 ³⁰.

References

1primary Total Solar Eclipse of 2001 June 21, F. Espenak and J. Anderson, NASA/TP-1999-209484 Read from the PDF text kept in var/downloads/TP209484_2001.txt. States that predictions use centre-of-mass positions with no refraction or limb corrections, that local circumstances are at sea level unless the elevation is known, and defines the elevation factor tan(90-A) sin(D) for shifting the path limits.
2peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (HMSO, 1961) Section 6 (figure of the Earth: Hayford spheroid, the S and C functions) and Section 9B (eclipses: observer coordinates, Bessel's parametric-latitude device, rising and setting curves without refraction). Read from the archive.org OCR text kept in var/downloads/es1961_djvu.txt.
3company 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.
4peer-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.
5company Stellarium source, src/core/RefractionExtinction.cpp Read (downloaded 2026-09-15). Saemundsson forward formula and Bennett backward formula as implemented, defaults 1013 mbar and 10 C, cut-offs at -3.54 deg geometric and -3.21783 deg apparent altitude with transition zones.
6primary 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.
7primary The Planetary and Lunar Ephemerides DE430 and DE431, W. M. Folkner, J. G. Williams, D. H. Boggs, R. S. Park and P. Kuchynka, IPN Progress Report 42-196 (2014) Read from the PDF kept in var/downloads/folkner2014_de430.pdf. States that the coordinate time of DE430 and DE431 is TDB and gives the TDB - TT expression, and that the origin is the solar-system barycentre.
8peer-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.
9primary 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.
10company 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.
11company 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.
12company 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.
13company Occult tutorial (grazing occultation examples), PDF distributed with Occult Read from the local copy var/downloads/OccultTutorial.pdf. A graze prediction listing states that path coordinates are referred to WGS84 with the site altitude referenced to mean sea level and that the path is adjusted for refraction at low Moon altitudes.
14company 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.
15trade Eclipse Calculator Details, eclipse-chasers.com (B. Kramer) Read. DE200/LE200 elements, refraction from the two Astronomical Almanac formulas (0.00452 P / ((273+T) tan a) above 15 degrees and the low-altitude rational formula below), horizon dip for elevated observers, Watts or Kaguya limb data.
16trade Astronomical Algorithms, 2nd edition, J. Meeus (Willmann-Bell, 1998) Not read for this note. Its refraction chapter (Saemundsson and Bennett formulas) is quoted from the Stellarium and Swiss Ephemeris source files that cite it, and its delta-T tables are cited by swephlib.c and the Photo Ephemeris note.
17trade Astronomical refraction, Th. Saemundsson, Sky and Telescope 72, 70 (1986) Not read. The formula R = 1.02 / tan(h + 10.3/(h + 5.11)) arcmin is quoted as implemented in Stellarium, which cites this paper via Meeus.
18company Stellarium API documentation, class Refraction Read. Describes the class as following the atmospheric-optics literature and notes the handling of altitudes between -2 and -5 degrees.
19primary 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.
20primary 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.
21peer-reviewed Jones, Nichols-Fleming, Evans, Johnson, Andrews-Hanna (2025). Can the Moon's Center of Mass-Center of Figure Offset Be Explained With a Uniform Primordial Crust? Journal of Geophysical Research: Planets Abstract only, via the Semantic Scholar API. Quotes the 1.935 km lunar COM-COF offset as the constraint.
22primary 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.
23primary NASA SVS 4517: Umbra Shapes (E. Wright) Read from the Wayback Machine snapshot of 2025-12-10. Explains the limb-profile point-cloud method, states that observer elevations come from SRTM, and that the western-US elevations in 2017 shift the umbra toward the Sun's azimuth by as much as 3 km.
24primary 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.
25primary 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.
26primary 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.
27primary 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.
28primary 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.
29trade Elements of Solar Eclipses 1951-2200, J. Meeus (Willmann-Bell, 1989) Not read for this note. Cited by the Photo Ephemeris technical note as the basis of its local circumstances and paths.
30peer-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.