Computing Solar Eclipses — Research

Lunar and solar model

workingupdated 2026-09-15ephemerisjpl-delunar-radiuslibrationlunar-frames
  • The orbit is solved. DE440 places the Moon to about a metre in the present century, which is under 10 ms of contact time. The errors that matter live in ΔT, the limb profile and the solar radius 1 2.
  • The figure is not a sphere and kk is a compromise. The IAU k=0.2725076k = 0.2725076 is a limb mean 0.9 km larger than the LOLA mean sphere. Espenak uses k=0.272281k = 0.272281 for umbral contacts so that beaded annulars are not called total 3 2.
  • The ephemeris gives the centre of mass, the limb is set by the figure. The centre of figure sits 1.935 km from the centre of mass, mostly along the Earth-Moon line, so the sky-plane part is about 0.5 arcseconds 4 5.
  • Orientation comes from the ephemeris, not the IAU series. Use moon_pa_de440_200625.bpc with the constant principal-axis to mean-Earth rotation, and compute libration per observer, since topocentric libration differs from geocentric by up to 1 degree 6 7.

What this topic covers

Two notes on the bodies that cast the shadow. The first asks which ephemeris places the Moon and the Sun, and how far the semi-analytical theories fall behind the numerical integrations. The second asks what the Moon's solid figure does that its centre of mass does not: the radius ratio kk, the offset of the centre of figure, the principal-axis and mean-Earth frames, and the libration that orients a limb profile.

Notes in this topic

  • Lunar and solar ephemerides: the JPL DE series release by release, INPOP, EPM, ELP and VSOP87, the kernel files, the readers, and where each error lands in arcseconds, seconds and kilometres.
  • Lunar figure, radius ratio k, and libration: centre of mass versus centre of figure, the three values of kk, the DE Euler angles, the PA and ME frames, topocentric libration, and the master table of every model choice with its size.

What this topic changes for the pipeline

The pipeline's Moon is two objects, not one: a centre of mass from DE440 and a figure from LOLA oriented by DE440 librations in the mean-Earth frame. The ephemeris can therefore be fixed once, as de440s.bsp, and validation effort spent on ΔT, the limb profile and the solar radius instead. The constant kk survives only as the datum radius that profile heights are measured from and as a fallback for profile-free products. The master table of every model choice, with its size on the sky, in contact time and in path shift, is at the end of lunar figure, radius ratio k, and libration.

References

  1. 1peer-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.
  2. 2primary Williams, Boggs, Folkner (2013). DE430 Lunar Orbit, Physical Librations, and Surface Coordinates. JPL IOM 335-JW,DB,WF-20130722-016 Read in full from the PDF. 18,548 LLR ranges 1970-2012, 1.9 cm wrms, DE430 vs DE421 half a milliarcsecond, PA and ME frame definitions, the DE430 rotation Rx(-0.285") Ry(-78.580") Rz(-67.573"), 1" = 8.42 m, LOLA mean radius 1737.151 km (Neumann 2013).
  3. 3primary Espenak, Anderson (1999). Total Solar Eclipse of 2001 June 21. NASA/TP-1999-209484 Read from the PDF text (var/downloads/TP209484_2001.txt). Mean lunar radius section (k history, 1986 Oct 03 case, penumbral 0.2725076 and umbral 0.272281), lunar limb profile section (Watts datum ellipticity, centre of figure, 0.4 arcsec systematic errors, topocentric libration -3.1 to -4.6 deg, 0.364 arcsec/s), centre-of-figure shift +0.53/-0.13 arcsec, DE200/LE200 as the ephemeris, older -0.6 arcsec latitude convention.
  4. 4peer-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.
  5. 5primary Espenak, Meeus (2009). Five Millennium Catalog of Solar Eclipses: -1999 to +3000. NASA/TP-2009-214174 Read from the PDF text (var/downloads/TP2009-214174.txt). Sections 1.3 (VSOP87D, ELP-2000/82 with 37,862 terms, truncation at 0.0005 arcsec, 1/40 s), 1.4 (secular acceleration -25.858 arcsec/cy2 and the Delta T correction c), 1.5 (k = 0.2724880 penumbral, 0.272281 umbral, IAU 0.2725076 history) and the centre-of-figure paragraph (+0.50 arcsec longitude, -0.25 arcsec latitude, ignored).
  6. 6primary NAIF lunar frame kernel moon_de440_250416.tf Read. MOON_PA_DE440 and MOON_ME_DE440_ME421 definitions, TKFRAME angles (67.8526, 78.6944, 0.2785) arcsec about axes (3,2,1), 0.02886 deg = 875 m, DE440 ME vs DE421 ME at most 53.4 cm over 2000-2040.
  7. 7peer-reviewed Seidelmann (ed.) (1992). Explanatory Supplement to the Astronomical Almanac. University Science Books, chapters 7 and 8 Read from the OCR text (var/downloads/es1992_djvu.txt). sin s = k sin pi definition, IAU adoption of k = 0.2725076 in August 1982 and its rationale, limb effects of up to two seconds per contact, ET until 1981 and UT1 for surface phenomena, physical, geocentric optical and topocentric optical librations with the 1 degree bound and MICA.

In this section

Lunar and solar ephemeridesWhich ephemeris to feed an eclipse computation, what each JPL DE release changed for the Moon, how INPOP, EPM, ELP and VSOP compare, and where the errors land on the ground.workingLunar figure, radius ratio k, and librationCentre of mass versus centre of figure, the three values of k and where they came from, the PA and ME lunar frames, and how to get topocentric libration and the axis position angle for a limb profile.working