Computing Eclipses, Occultations and Transits

Lunar eclipses

workingupdated 2026-10-17lunar-eclipseumbrapenumbrashadow-enlargementdanjoncrater-timingsdelta-t
  • A lunar eclipse prediction is geocentric. The contacts are the same instant everywhere, so the product is a list of times, two magnitudes, gamma and a map of where the Moon is above the horizon at each contact 1 2.
  • The computation is the solar method with the Earth as the shadow caster. The Moon's two coordinates about the antisolar point and three limiting separations are the elements. A quadratic in time gives the six contacts P1, U1, U2, U3, U4 and P4 1 3.
  • The size of the Earth's shadow is a convention. Chauvenet enlarges both radii by 2 per cent, Danjon adds a layer of 1/85 of the Earth's radius, and Herald and Sinnott add 87 km above an oblate Earth. The choice changes umbral magnitudes by 0.005, contacts by 20 to 90 seconds and the type of borderline eclipses 4 5.
  • Crater timings are the test. 22,539 timings at 94 eclipses give a layer of 86.9 km with no dependence on lunar distance, season or solar cycle, and an umbra flattened only by the Earth's own figure 5.
  • None of the predictions cited here includes brightness. The Danjon scale grades it after the event, and volcanic aerosols can darken totality by half a magnitude or more 6 7.

What this topic covers

This topic covers the computation of lunar eclipse products from enumeration to validation. It sets out the classification, gamma and the Saros behaviour of lunar eclipses, Meeus's closed form, and the Explanatory Supplement's elements and contact solution. It quotes the shadow-radius formulas of Chauvenet, Danjon and Herald and Sinnott with their constants and shows which almanac and program uses each. It describes visibility maps, the role of ΔT, crater timings, the Danjon brightness scale and the effect of volcanic aerosols. It compares the main predictors and ends with the use of historical lunar eclipses as evidence for ΔT.

Notes in this topic

  • Enumeration, elements and contacts: types and catalogue codes, gamma, Saros and Inex, eclipse limits, Meeus's method, the lunar Besselian elements, contacts, magnitudes, durations, the NASA catalogue, visibility maps and ΔT.
  • Shadow geometry and enlargement: the umbral and penumbral radii, the oblate Earth, the three enlargement rules with constants, their effects on magnitude, contacts and type, and which implementation uses which.
  • Observation, brightness and validation: crater timings and their reduction, the Danjon scale, aerosols and the limits of brightness prediction, predictor comparisons, the error budget, and historical lunar eclipses as ΔT evidence.

What this topic changes for the pipeline

Lunar eclipses reuse the ephemeris, time and Besselian stages with three changes. The element set shrinks to xx, yy and the limiting separations L1L_1, L2L_2 and L3L_3. A new input, the atmospheric enlargement rule, sets those separations and must be reported with every result. The local-circumstances stage becomes a horizon computation that draws moonrise and moonset curves for each contact, and the path products disappear. Validation needs crater timings from VizieR VI/140 alongside the NASA catalogue.

References

  1. 1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9E Lunar eclipses, and Example 4.12 Semi-diameter of the Moon Read in full in the OCR text for these sections. Section 9E (from page 258), with Example 9.23: types of lunar eclipse, shadow radii with 0.998333 (0.998340 for 1968 onwards) and 1.02, limits on the Moon's latitude at opposition, x y about the antisolar point in arcseconds, the contact quadratic with n, Delta and the sign rule, the type tests, greatest eclipse, position angles, and the 1960 March 13 example (Example 9.23). Example 4.12: sin s = 0.2724880 sin pi and s = 0.0799 + 0.272453 pi for 1968 onwards. Same book as bes-es1961, which covers section 9B.
  2. 2primary Espenak, F. and Meeus, J. (2009). Five Millennium Canon of Lunar Eclipses: -1999 to +3000, NASA/TP-2009-214172, text sections 1 to 5 Read in full (text PDF, 3.2 MB, extracted with pdftotext). Source of the contact definitions, gamma and magnitude definitions, eclipse statistics (12,064 eclipses), VSOP87 D and truncated ELP-2000/82, k = 0.2724880, secular acceleration -25.858 arcsec/cy^2 and the c correction, the enlargement section 1.5 (Chauvenet and Danjon formulas, Tables 1-3 to 1-6), map accuracy and Delta T uncertainty zones, Danjon scale wording, Saros and Inex behaviour. Some tables are misaligned in extraction and were cross-checked against the web pages.
  3. 3peer-reviewed Explanatory Supplement to the Astronomical Almanac (1992), chapter 8 Eclipses of the Sun and Moon, by Alan D. Fiala and John A. Bangert: sections 8.23 (occurrence of lunar eclipses) and 8.4 (lunar eclipses) Read in full in the OCR text for these sections. Section 8.231: penumbral and umbral radii f1 = pi_m + pi_s + s_s (8.231-1), the 0.998340 reduction to latitude 45 degrees and the 1.02 enlargement (8.231-2, 8.231-3). Sections 8.2321 to 8.2331: conditions for penumbral, partial and total eclipses. Section 8.4: elements x y about the antisolar point, L1 L2 L3 (8.42-1), the 10-minute discriminant tabulation, centre-of-figure and Delta T corrections, Table 8.422.1 of contact order, greatest eclipse, magnitude (8.424), position angles (8.425), sublunar points and the horizon circle (8.426-1, 8.426-2). Same book as glob-es1992, which covers the solar sections.
  4. 4primary Espenak, F. NASA GSFC Eclipse Web Site, Enlargement of Earth's Shadows (page dated 2009 Feb 02) Read in full. Web version of Canon section 1.5 with correctly aligned Tables 1-3 to 1-6: Chauvenet formula with 0.998340, Danjon formula with 1.01 = 1 + 1/85 - 1/594, magnitude differences 0.006 and 0.026, type-change lists, and which sources use which rule.
  5. 5peer-reviewed Herald, D. and Sinnott, R. W. (2014). Analysis of lunar crater timings, 1842-2011. J. Br. Astron. Assoc. 124(5), 247-253 Read in full from the ADS scan (text extracted; Greek letters in the Symbol font are lost, so equation [4]'s angle is written theta here). Source of equations [1] to [4], the recommended rule with h = 0.0136, the NEL statistics (mean 86.9 +/- 0.2 km, mode 78, median 85), the per-eclipse heights, the ingress-egress and limb-contact biases, the shape result and the 1963 Dec 30 case.
  6. 6trade Danjon, A. (1921). Relation entre l'eclairement de la Lune eclipsee et l'activite solaire. L'Astronomie 35, 261-265 Page 262 read from the ADS scan image. The original five-step brightness scale 0 to 4 and the table of mean brightness of 70 eclipses against years since solar minimum for 1823-1867 and 1867-1920. Pages 261 and 262 read from the scan images; the title is confirmed on page 261, which also states that Danjon compiled a catalogue of about 150 eclipses.
  7. 7trade Keen, R. A. (2019). Two Centuries of Volcanic Aerosols Derived from Lunar Eclipse Records, 1805-2019. Poster, NOAA Global Monitoring Annual Conference, Boulder Read in full. The author's own statement of the method, the 147 plus 31 eclipse record, mean AOD 0.002 for 1996-2018 against 0.041 for the previous 14 years, and the 2019 January eclipse half a magnitude below prediction.

In this section

Enumeration, elements and contactsHow lunar eclipses are found and classified, and how gamma, magnitudes, contact times, durations and visibility maps are computed.workingShadow geometry and enlargementHow the Earth's umbra and penumbra are sized for lunar eclipses, and how the Chauvenet, Danjon and Herald and Sinnott rules change the results.workingObservation, brightness and validationHow crater timings test lunar eclipse predictions, why brightness is not predicted, and how historical lunar eclipses constrain ΔT.working