Lunar figure, radius ratio k, and libration

The question. An eclipse is a shadow cast by the Moon's solid figure, but the ephemeris tracks the Moon's centre of mass and the constant $k$ replaces the figure by a sphere. This note answers how large the centre-of-mass to centre-of-figure offset is and how eclipse predictors treat it, where the three values of $k$ came from and what LOLA says the mean radius is, how the Moon's orientation is defined in the DE files versus the IAU frames, and how a developer obtains the topocentric libration and axis position angle needed to place a limb profile.

Centre of mass versus centre of figure

The Explanatory Supplement of 1961 states the principle: "Gravitational ephemerides refer to the positions of the centres of mass of the bodies concerned. The phenomena of eclipses, however, are governed by the positions of the centres of figure of the Sun and Moon. The centre of figure of the Moon does not coincide with the centre of mass; to allow for this a correction $\Delta\beta$ of −0".5 is applied to the tabular latitude of the Moon. As from 1964, $\Delta\beta$ will be revised to −0".6" ¹⁰. The correction is converted to right ascension and declination with

evaluated once at conjunction and "treated as constant throughout each eclipse" ¹⁰. Espenak's long-range predictions from the 1980s kept that convention: "A correction of −0.6" was added to the Moon's ecliptic latitude to account for the difference between the Moon's center of mass and center of figure" ³. The 1992 Explanatory Supplement describes the same mechanism, corrections "applied in the ecliptic system" and "applied as constants for each eclipse" ⁴.

The Five Millennium Catalog, NASA/TP-2009-214174, gives the later two-component convention and then declines it: "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" ². Espenak's EclipseWise page for 2024 confirms "The lunar coordinates were calculated with respect to the Moon's Center of Mass" ¹¹.

The physical offset. The lunar geodetic grid from LOLA is built "with respect to the Moon's center of mass" to "~10 m radial and ~100 m spatial accuracy" ¹². The offset of the centre of figure from that origin is quoted in the current literature as "the 1.935-km lunar center of mass (COM)–center of figure (COF) offset" ¹. Earlier determinations, collected in a NASA technical note, give the components: Clementine altimetry put the centre of figure at "(−1.74, −0.75, 0.27) km in the (x, y, z) directions" from the centre of mass, and Apollo-era altimetry gave "−2.55 km in the direction of 25°E" with a mean radius of 1737.7 km ¹³. In the mean-Earth frame x points at the Earth, so the dominant component is along the line of sight and invisible in the sky. The y and z components, about −0.75 km and +0.27 km, project to 0.40 arcseconds and 0.15 arcseconds at the mean distance of 384,400 km, which is the size of the historical +0.5 and −0.25 arcseconds corrections. The Kaguya (SELENE) LALT determination could not be read at the time of writing (2026 September). The Science abstract is available but carries no offset value ¹⁴, and the Smith 2010 and 2017 LOLA papers were blocked at the publisher, so only their abstracts are cited here ^{12 15}.

Watts and the datum. Watts's limb charts measure heights against an adopted datum whose implicit centre is displaced from the Moon's centre of mass and whose cross-section is slightly elliptical, which Morrison and Appleby's 66,000 occultations showed reaches 0.4 arcseconds at some position angles ³. The datum story and the Morrison and Appleby correction formula are in limb profile methods.

How this resolves for a modern product. A limb profile derived from LOLA topography is already referenced to the centre of mass because the LOLA grid is ¹². The profile heights at each position angle then carry the whole figure, offset included, and no separate centre-of-figure correction exists to apply. The same is true of a Watts profile once "ellipticity and libration corrections" have been applied "to refer the profile to the Moon's center of mass" ³. The correction is only needed by a product that uses a mean sphere and no profile, and even then the Catalog's argument stands: a constant shift is a poor substitute for the actual limb at the actual libration. Pages describing how Occult, Xavier Jubier's site and NASA SVS handle the offset could not be fetched at the time of writing (2026 September). lunar-occultations.com returned a TLS error, xjubier.free.fr and svs.gsfc.nasa.gov refused connections, and the archive mirrors were unavailable. What all three share, per their published outputs, is a LOLA or Kaguya profile referenced to the centre of mass, which makes the question moot for them. The SVS specifics are in SVS products and data.

Size on the ground. A sky-plane offset of 0.5 arcseconds at 384,400 km is 0.93 km of shadow displacement and, at a Moon-Sun relative rate of 0.364 arcseconds per second ³, 1.4 s of contact time at a path edge. The full 1.0 arcseconds that the 1.935 km offset could reach if it lay in the sky plane would be 1.9 km and 2.7 s. These conversions are derived in lunar and solar ephemerides.

The lunar radius ratio k

Definition. The Explanatory Supplement defines it through the Moon's semidiameter: "the apparent semidiameter of the Moon is calculated by putting its sine equal to $k \sin\pi_m$, where $\pi_m$ is the horizontal parallax and $k$, the ratio of the Moon's radius to the equatorial radius of the Earth, is an adopted constant" ⁴. In symbols,

With $a_\oplus = 6378.137$ km the three values in use are $k = 0.2725076 \rightarrow 1738.09$ km, $k = 0.2724880 \rightarrow 1737.97$ km, and $k = 0.272281 \rightarrow 1736.65$ km. The last two differ from the IAU value by 0.125 km and 1.445 km, or 0.067 and 0.776 arcseconds of semidiameter at mean distance.

History. "From 1968 through 1980, the Nautical Almanac Office used two separate values for k in their predictions. The larger value (k=0.2724880), representing a mean over topographic features, was used for all penumbral (exterior) contacts and for annular eclipses. A smaller value (k=0.272281), representing a mean minimum radius, was reserved exclusively for umbral (interior) contact calculations of total eclipses" ³. "In August 1982, the International Astronomical Union (IAU) General Assembly adopted a value of k=0.2725076 for the mean lunar radius. This value is now used by the Nautical Almanac Office for all solar eclipse predictions [Fiala and Lukac, 1983] and is currently the best mean radius, averaging mountain peaks and low valleys along the Moon's rugged limb" ³. The 1992 Explanatory Supplement gives the motive from the almanac side: "The IAU adopted a new value of k (k = 0.2725076) in August 1982. Before that time, two different values of k were used in computing a solar eclipse ... a smaller value of k was adopted solely for calculating duration on the central line of total solar eclipses. This smaller value caused numerical inconsistencies as well as misunderstandings", and the new value was chosen "to conform to the value used in occultation predictions" ⁴. The IAU 1982 resolution text itself was not retrievable from iau.org at the time of writing (2026 September).

Why Espenak departs from the IAU. "The use of even the best 'mean' value for the Moon's radius introduces a problem in predicting the true character and duration of umbral eclipses, particularly total eclipses. A total eclipse can be defined as an eclipse in which the Sun's disk is completely occulted by the Moon. This cannot occur so long as any photospheric rays are visible through deep valleys along the Moon's limb ... the use of the IAU's mean k guarantees that some annular or annular-total eclipses will be misidentified as total. A case in point is the eclipse of 3 October 1986. Using the IAU value for k, the Astronomical Almanac identified this event as a total eclipse of 3 seconds duration when it was, in fact, a beaded annular eclipse" ³. Hence: "This publication uses the IAU's accepted value of k=0.2725076 for all penumbral (exterior) contacts. In order to avoid eclipse type misidentification and to predict central durations which are closer to the actual durations at total eclipses, we depart from standard convention by adopting the smaller value of k=0.272281 for all umbral (interior) contacts" ³. The consequence is stated: "the smaller k produces shorter umbral durations and narrower paths for total eclipses ... predictions using a smaller k result in longer umbral durations and wider paths for annular eclipses" ³. The Catalog uses the older pair instead, "the larger value (k=0.2724880) is utilized for all partial (penumbral) eclipses", with magnitudes agreeing "to within 0.0001" of the IAU value, and k = 0.272281 "for all umbral and antumbral eclipses (total, annular, and hybrid)" ². EclipseWise 2024 lists "k (penumbra): 0.2725076" and "k (umbra): 0.2722810" ¹¹. The Catalog also notes the cost of two values: "the use of two different values of k for total and annular eclipses introduces a discontinuity in the case of hybrid eclipses" ².

Size. The 1.445 km difference between the IAU and umbral radii changes the umbral path width by 2.9 km and the central duration by about 4.3 s (0.776 arcseconds at each of second and third contact, at 0.364 arcseconds per second). The 0.125 km difference between the two penumbral values is 0.25 km of penumbral width and 0.37 s of first or fourth contact.

LOLA and the modern mean radius. The IAU cartographic working group recommends a mean radius of 1737.4 ± 1 km with the equatorial and polar radii set to the same value ⁶. The LLR analysis for DE430 uses "mean radius 1737.151 m (Neumann, 2013)", where the memo's unit is a misprint for km, and notes "an arc of 1" corresponds to 8.42 m" on the surface ⁵. The LOLA mean sphere therefore corresponds to $k = 1737.151/6378.137 = 0.272361$, and the IAU 1737.4 km to $k = 0.272400$. The IAU eclipse value 0.2725076 (1738.09 km) is 0.94 km, 0.50 arcseconds, larger than the LOLA mean sphere, and the umbral value (1736.65 km) is 0.50 km, 0.27 arcseconds, smaller. This is consistent with the 1982 value being a mean of the limb as seen in profile, where at every position angle the limb is the highest terrain along the line of sight, rather than a mean of the global surface. That reading is inferred here, not stated by the sources. A "modern mean limb" from LOLA would be the average, over position angle and over the libration range, of the maximum LOLA radius along each line of sight. No published value of that quantity was found, and it is listed under open questions.

How k interacts with a limb profile. The 1992 Supplement is explicit that after 1982 "it was agreed implicitly that limb effects are no longer accounted for, but are averaged, and if an observer considers them to be important, then corrections must be calculated and applied separately. It is possible for these effects to advance or retard predicted second- or third-contact times on the central line by as much as two seconds apiece" ⁴. A limb profile supplies a radius as a function of position angle,

where $h$ is the profile height above the datum sphere at position angle $P$ for the topocentric libration $(l, b)$. The constant $k$ then matters only as the datum the heights refer to. Watts heights refer to Watts's datum and need the ellipticity and centre corrections before use ³. LOLA heights refer to a centre-of-mass sphere whose radius the profile generator chooses, so the generator's datum radius must be the $k$ used in the contact solution. Feeding LOLA heights referenced to 1737.4 km into a solver that assumes $k = 0.2725076$ inflates every radius by 0.69 km, 0.37 arcseconds, and lengthens totality by 2 s. The correct pairing is one of the interfaces the pipeline must pin down.

Lunar orientation: the DE Euler angles and the PA frame

The DE files carry the Moon's orientation as three integrated Euler angleslunar libration Euler anglesThe three angles $\varphi$, $\theta$, $\psi$, integrated with the orbit and stored in the DE files or a binary PCK, that rotate the ICRF into the Moon's principal-axis frame and thus give the Moon's orientation at any time.. "The orientation of the lunar exterior (mantle and crust, hereafter referred to by mantle) is parameterized by Euler angles, $\phi_m$, $\theta_m$, and $\psi_m$, that relate the Moon-centered, rotating lunar mantle to the inertial frame ... The Euler angles that define the rotation from the principal axis (PA) frame to the inertial ICRF2 frame are: $\phi_m$, the angle from the X-axis of the inertial frame along the XY plane to the intersection of the mantle equator; $\theta_m$, the inclination of the mantle equator from the inertial XY plane; and $\psi_m$, the longitude from the intersection of the inertial XY plane with the mantle equator along the mantle equator to the prime meridian" ¹⁶. The principal-axis frameprincipal-axis (PA) frameThe Moon-fixed frame aligned with the principal axes of the lunar moment of inertia, in which the DE Euler angles are integrated. It is rotated by a constant of about 875 m on the surface from the mean-Earth frame. is "defined by the principal axes of the undistorted mantle in which the moment of inertia matrix of the undistorted mantle is diagonal", with directions "estimated from analysis of LLR data" ¹⁶. A vector in the ICRF is taken into the PA frame by

which is the 3-1-3 sequence the report describes. DE440 added "the effect of geodetic precession on lunar librations" and stores the angles, "also known as lunar libration angles", in the DE440 and DE441 files ¹⁷. In SPICE form the DE440 angles are the binary PCK moon_pa_de440_200625.bpc, valid 1549 Dec 31 to 2650 Jan 25, with frame ID MOON_PA_DE440 defined in the frame kernel moon_de440_250416.tf ^{18 7}. The DE421 equivalent, moon_pa_de421_1900-2050.bpc with moon_080317.tf, is what Skyfield's documentation uses ^{19 9}.

The DE430 physical libration was fitted to the same LLR set as the orbit, and the DE430-to-DE421 change in orientation was at the 0.1 arcseconds level on the surface (about 1 m), so any DE from DE421 on gives the Moon's rotation to better than 10 m on the limb ⁵. Because the DE440 libration model includes geodetic precession that DE430 lacks, a product should pair its orbit and its orientation from the same release ¹⁷.

The mean-Earth/polar-axis frame and the constant rotation

Cartography does not use the PA frame. "The recommended coordinate system for the Moon is the mean Earth/polar axis (ME) system. This is in contrast to the principal axis (PA) system, sometimes called the axis of figure system. The ME system ... is recommended because nearly all cartographic products of the past and present have been aligned to it. The difference in the coordinates of a point on the surface of the Moon between these systems is approximately 860 m" ⁶. LOLA products, and therefore every LOLA-derived limb profile, are in the ME frameMoon mean-Earth/polar-axis (ME) frameThe lunar body-fixed frame whose z axis is the mean rotation pole and whose x axis points to the mean sub-Earth point, as realised by a JPL ephemeris. LRO LOLA, SLDEM2015 and Kaguya LALT grids use the ME frame of DE421; the SPICE frame MOON_ME_DE440_ME421 reproduces it from DE440..

Williams et al. define the ME frame as the system with "the mean direction toward the Earth for the X axis and the mean direction of rotation for the Z axis", and explain why it differs from PA: "An ellipsoidal Moon with only a second-degree (gravity) figure would have coinciding mean axis and principal axis systems. Third- and higher-degree coefficients of the gravity field affect the Euler angles, and cause a constant 3-axis rotation between the PA and MER frames" ⁵. The rotation has the form

"where $p_{1c}$, $p_{2c}$, and $\tau_c$ are the constant parts of the three libration parameters" and $I = 0.02692$ rad is the mean tilt of the lunar equator to the ecliptic ⁵. The numerical rotations by release are:

For DE430: "The PA X axis is 67.573" (~569 m) east and 78.580" (~662 m) south of the MER X axis. The PA Y axis is 67.573" (~569 m) east and 0.285" (~2 m) north of the MER Y axis. The PA Z axis is tilted 78.580" (~662 m) toward longitude zero and 0.285" (~2 m) toward 90° west" ⁵. The first-order form carries an error "estimated to be half" of 0.026 arcseconds, "or 0.11 m in position" ⁵. The Z rotation "is uncertain by 0.2" (1.7 m on the equator) and the Ry rotation is uncertain by <0.1"" ⁵. NAIF's DE440 frame kernel defines MOON_ME_DE440_ME421 as "a reference frame defined by a constant rotational offset from the DE440 principal axes frame, such that the frame is closely aligned with the DE421 lunar mean Earth/polar axis frame", with the total offset "approximately 0.02886 degrees; this is equivalent to approximately 875 m when expressed as a displacement along a great circle on the Moon's surface", and it reports that over 2000 to 2040 the DE440 ME frame differs from the DE421 one by at most $3.07 \times 10^{-7}$ rad, "approximately 53.4 cm" ⁷. The DE421 memo gives the surface scale, "1" corresponds to 8.4 m", and the DE421-to-DE403 ME change of "0.53" in longitude ... 5 meters" ²⁰.

The IAU closed formulae versus the DE angles. The working group kept its series for the ME pole and prime meridian "for convenience for many users" but warned that "these are valid only to the approximately 150 m level of accuracy ... For high precision work involving e.g., spacecraft operations, high-resolution mapping, and gravity field determination, it is recommended that a lunar ephemeris be used to obtain the libration angles for the Moon from which the pole position and rotation can be derived" ⁶. The report's own summary item 3 is "An algorithm is described in the text for expressing the orientation of the Moon using the JPL DE 421 lunar ephemeris, rotated to the mean Earth/polar axis system, in order to obtain the pole and rotation with high precision" ⁶. The 2015 report went further and, per the search abstract only, removed the low-precision series for the Moon and kept the high-precision realisation from the ephemeris ²¹. A limb profile builder should therefore rotate LOLA (ME) coordinates to PA with the constant matrix, orient with the DE Euler angles, and never use the trigonometric series.

Libration: geocentric, topocentric, and the axis position angle

LibrationlibrationThe slow apparent rocking of the Moon that changes which part of its surface faces an observer, expressed as a longitude and a latitude of the sub-observer point in selenographic coordinates. It ranges about plus or minus 8 degrees in each coordinate and sets which terrain lies on the limb. is the apparent rocking of the Moon that lets an Earth observer see about 59 percent of the surface. The Explanatory Supplement separates three parts. "The librations are due partly to physical rotational librations, oscillations of the actual rate of rotation of the Moon with respect to its mean rotation rate. A second, much larger part of the librations are the geocentric optical librations, which are the result of the nonuniform motion of revolution of the Moon around the Earth-Moon barycenter ... A third contribution to the librations, the topocentric optical librations, are a result of the difference between the viewpoints of an observer on the surface of the Earth and the hypothetical observer at the center of mass of the Earth. The topocentric optical librations may be as large as 1° and have important effects on the apparent contour of the limb" ⁴. The almanac tabulates geocentric values and MICA "computes the topocentric optical librations from rigorous formulas", with the almanac's approximate method agreeing to "at most, 1"" ⁴. Sign convention: "When the libration in selenographic longitude of the Earth is positive, the mean center of the disk is displaced eastward on the celestial sphere, exposing to view a region of the western limb. When the libration in selenographic latitude of the Earth is positive, the mean center of the disk is displaced toward the south, and a region of the north limb is exposed to view" ⁴.

Meeus's closed form (chapter 53). The optical libration uses the inclination $I = 1.54242^{\circ}$ of the mean lunar equator to the ecliptic. With $\lambda, \beta$ the Moon's apparent geocentric ecliptic longitude and latitude, $\Omega$ the mean longitude of the ascending node, $F$ the argument of latitude and $\Delta\psi$ the nutation in longitude,

The physical libration adds series $\rho$, $\sigma$, $\tau$ in the Delaunay arguments, whose leading terms are $\rho = -0.02752\cos M' - 0.02245 \sin F + \dots$, $\sigma = -0.02816 \sin M' + 0.02244 \cos F + \dots$, $\tau = 0.0252\,E \sin M + 0.00473 \sin(2M' - 2F) + \dots$ in degrees, and the physical part is $l'' = -\tau + (\rho\cos A + \sigma \sin A)\tan b'$, $b'' = \sigma\cos A - \rho \sin A$, so that $l = l' + l''$ and $b = b' + b''$. The position angle of the axis follows from

The optical part, the $I$ constant, the $\rho, \sigma, \tau$ coefficients and the $V, X, Y, \omega$ construction are as implemented, with chapter references, in the soniakeys Go port of Meeus, which was read for this note ⁸. The $l'', b''$ and $P$ closing formulas are transcribed from the printed second edition ²². Meeus also gives a topocentric correction for $l$, $b$ and $P$ in terms of the Moon's horizontal parallax and the observer's hour angle. The Go port notes that part is "commented out for lack of test data" ⁸, so a developer should validate any Meeus-based topocentric libration against a SPICE computation rather than trust it blind.

The SPICE and Skyfield way. With a PCK loaded, libration is a frame transformation and topocentric libration is the same transformation applied to the observer's vector. Skyfield's documentation builds the frame from the DE421 kernels and states "The Moon's libration is expressed as the latitude and longitude of the Moon location that is currently nearest the Earth", computed as p = (earth - moon).at(t); lat, lon, distance = p.frame_latlon(frame) with frame = pc.build_frame_named('MOON_ME_DE421'), and gives the worked example "Libration in latitude: −6.749" and "Libration in longitude: 1.520" degrees ⁹. Replacing earth with an observer on the ellipsoid gives the topocentric values directly. Replacing the ME frame by MOON_PA_DE440 gives the PA-frame sub-observer point. The axis position angle is the position angle of the projected +Z axis of the frame on the sky, obtained from the same rotation matrix, and the limb profile is then drawn in position angle measured from that axis.

How much topocentric libration matters for the limb. The horizontal parallax of the Moon is near 57 arcmin, so an observer at the limb of the Earth sees the Moon from a direction up to 0.95° from the geocentric one, which is the "as large as 1°" figure ⁴. On the 2001 path "the Moon's topocentric libration (physical + optical) in longitude ranges from l=−3.1° to l=−4.6°", and the bulletin concludes "a limb profile with the appropriate libration is required in any detailed analysis of contact times, central durations, etc.. But a profile with an intermediate value is useful for planning purposes" ³. One degree of libration moves the terrain seen at the limb by 30 km along the line of sight. The limb height at a given position angle is the maximum radius along that line, and over 30 km lunar topography varies by hundreds of metres to a few kilometres, so profile heights can change by up to roughly 1 arcsecond, which is 2.7 s of contact time or 1.9 km at the path edge. This estimate is derived here. The same 1 arcsecond is the size of the whole Baily's-bead structure a limb-corrected product exists to predict, so the profile must be topocentric. Across a single eclipse the topocentric libration changes by about 1° from one end of the path to the other and by about 0.02° in the four minutes of totality at one site, so one profile per site suffices, and one profile per path segment of a few hundred kilometres is adequate for path-limit work.

Model choices and their size

The full ranked budget is in error budget and validation protocol. This table keeps the rows this topic owns, which are the lunar model choices. The conversions use one arcsecond of Sun or Moon direction = 1.86 km on the ground, and 0.364 arcseconds per second of relative Moon-Sun motion = 2.7 s of contact time per arcsecond ³. A radius change $\Delta r$ at the Moon changes the umbral width by $2\Delta r$ and the central duration by $2\Delta r$ divided by 1.86 km per arcsecond divided by 0.364 arcseconds per second. Rows marked "derived" are arithmetic performed here on the cited numbers.

Sources compared

What a developer should do

1. Compute the Moon's centre of mass from DE440 and never apply a centre-of-figure shift. Use a limb profile referenced to the centre of mass instead, which LOLA products are by construction ^{12 2}.
2. Carry two radii explicitly: the datum radius of the limb profile (state it in km, for example 1737.4 km) and, for products without a profile, k = 0.2725076 for penumbral contacts and k = 0.272281 for umbral contacts as Espenak does, with the reason documented ^{3 11}.
3. Orient the Moon with the DE440 Euler angles from moon_pa_de440_200625.bpc and the constant PA to ME rotation from moon_de440_250416.tf. Do not use the IAU trigonometric series ^{18 7 6}.
4. Compute libration topocentrically, as the selenographic latitude and longitude of the observer-to-Moon vector in the ME frame, per site and per time. Take the axis position angle from the same rotation ^{9 4}.
5. Use Meeus chapter 53 only as a cross-check of the geocentric values, and validate its topocentric correction against SPICE before use ^{8 22}.
6. Read first: Williams, Boggs and Folkner 2013 sections 5 and 6, the 2001 bulletin's "Mean lunar radius" and "Lunar limb profile" sections, and the 1992 Supplement's eclipse and physical-ephemeris chapters ^{5 3 4}.

What this changes

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 ME frame. The constant $k$ survives only as (a) the datum radius the profile heights are measured from and (b) a fallback for profile-free products, where the Espenak two-value convention is the defensible choice. Libration must be a per-site topocentric quantity computed from the frame kernels, which makes the limb-profile module a consumer of the same SPICE kernel set as the ephemeris module. The centre-of-figure correction of the almanacs is retired.

Open questions

• Obtain Smith et al. 2010 (GRL 37, L18204) and Smith et al. 2017 (Icarus 283, 70) full texts and record the LOLA centre-of-figure offset components in km in the ME frame.
• Obtain Araki et al. 2009, Science 323, 897, in full and record the Kaguya (SELENE) LALT mean radius and centre-of-figure offset for comparison with LOLA.
• Compute, from the LOLA LDEM, the mean limb radius as the average over position angle and over the libration range of the maximum radius along the line of sight, and compare it with 1738.09 km (k = 0.2725076) and 1736.65 km (k = 0.272281). No published value was found.
• Obtain Fiala and Lukac 1983 and the IAU 1982 General Assembly resolutions to record the exact wording under which k = 0.2725076 was adopted.
• Obtain the Occult 4 documentation, Jubier's limb-profile explanatory page and the SVS 2017 and 2024 method pages, whose hosts refused connections at the time of writing (2026 September). Record which datum radius and which frame, principal-axis or mean-Earth, each uses for its Kaguya (SELENE) or LOLA profile.
• Read Archinal et al. 2018 (CeMDA 130, 22) in full to confirm that the low-precision lunar series was removed and which DE the 2015 report names.
• Extract Meeus's topocentric libration formulas (chapter 53, equations for $\Delta l$, $\Delta b$, $\Delta P$) and test them against a SPICE topocentric computation for one eclipse.