Computing Eclipses and Occultations

How eclipses, occultations and transits are computed

workingupdated 2026-10-18
  • Every event is a body casting a shadow. The light can come straight from a source, the Sun or a star, or be reflected by a lit body such as a planet. A blocking body casts a shadow in that light.
  • Who stands in the shadow decides what happens. If the observer stands in it, the observer sees the source hidden: every occultation, every transit and every solar eclipse is this case. If another body stands in it, that body goes dark for everyone: lunar eclipses and the eclipses of planetary satellites.
  • The hidden source is a point or a disc. A star is a point, so the shadow is a cylinder and the star vanishes at once. The Sun, a planet or a satellite is a disc, so the shadow is a cone with an umbra, a penumbra and an antumbra, which give total, partial and annular events 1 2.
  • This is the Explanatory Supplement's own split. Its eclipse is a shadow falling on a body, and its occultation is a block in the line of sight. It notes that a solar eclipse is really an occultation 3.

Every event is a shadow

A blocking body casts a shadow in whatever light falls on it. The light may come straight from a source that shines itself, the Sun or a star, or from a body that the Sun lights, such as a planet or a satellite. There are two cases, and they differ in who stands in the shadow:

  • the observer, who then sees the source hidden behind the blocking body; or
  • another body, which then loses its sunlight and goes dark.

The two sections below take each in turn. In every diagram each body has two labels: the part it plays, and what it is in that example.

Hidden: the observer is in the shadow

When the observer stands in a blocking body's shadow, the observer sees the source of that light hidden behind the body. This is an occultationoccultationThe passage of a nearer body in front of a farther one that looks smaller, so that the observer cannot see the farther one. The Moon occulting a star, and Jupiter hiding one of its satellites, are occultations.Read on Wikipedia. Someone standing outside the shadow sees nothing happen, so the time depends on where the observer stands, and a prediction is a path or a map 2.

Hidden: the observer stands in the Moon's shadow in starlightLeft, a star, labelled light source, sending parallel lines of light to the right. Middle, the Moon, labelled blocking body. The lines that meet the Moon stop there, and a pale band behind the Moon marks where the star cannot be seen. Lines above and below the Moon carry on. Right, an observer on the Earth, labelled observer, inside the band.Schematic, not to scaleLight sourcea starBlocking bodythe MoonObserveron the Earththe Moon's shadow in starlight

A star is so far away that its light arrives in parallel lines, so the Moon's shadow in starlight is a cylinder exactly as wide as the Moon. It darkens nothing, but it is how these events are computed: an occultation is predicted as the path of that shadow across the Earth 2 4.

The hidden source can be a disc rather than a point. Then the shadow is a cone, and the event can be total, partial or annular. A solar eclipse is this case, with the Sun as the source and the Moon as the blocking body: the observer in the Moon's umbra sees the whole Sun hidden. That is why the Supplement calls a solar eclipse really an occultation 3.

Hidden: the observer stands in the Moon's umbra and the Sun is hiddenLeft, the Sun, labelled light source. Middle, the Moon, labelled blocking body. Behind the Moon a pale penumbra spreads out and a darker umbra narrows towards the Earth on the right, where both fall on the Earth's surface. An observer, labelled observer, stands inside the umbra on the Earth. A framed panel on the right shows what the observer sees: a total eclipse, the Moon's dark disc covering the Sun with a faint corona fading outwards around it.Schematic, not to scaleWhat the observer seestotalityLight sourcethe SunBlocking bodythe MoonObserveron the Earthin the Moon's umbraumbra

A transittransitThe passage of a nearer body across the face of a farther one that looks larger, such as Mercury or Venus crossing the Sun or a satellite crossing its planet. Geometrically it is an occultation in which the nearer body covers only part of the farther one.Read on Wikipedia is the same with a body far too small to cover the source. Venus's umbra closes long before it reaches the Earth, so the observer stands in its antumbra and sees a dot cross the Sun 5.

The observer need not stand on the Earth. On 2026 April 6 the Artemis II crew watched the Moon hide the Sun from the Orion spacecraft, close behind the Moon. From there the Moon looked huge and hardly moved, so the Sun stayed hidden for 54 minutes 6 7. On Mars, the Perseverance rover filmed Phobos crossing the Sun on 2022 April 2. Phobos is too small to cover the Sun, so the rover stood in its antumbra, and the crossing took a little over 40 seconds 8.

Hidden: the observer stands in Venus's antumbra and sees a dot on the SunLeft, the Sun, labelled light source. Middle, Venus, labelled blocking body, small. Its dark umbra closes to a point a short way behind it, and beyond that point the antumbra opens out towards the Earth on the right. An observer on the Earth, labelled observer, stands in the antumbra. A framed panel on the right shows what the observer sees: a small dark dot on the Sun's disc.Schematic, not to scaleantumbraWhat the observer seesVenus, a dot on the SunLight sourcethe SunBlocking bodyVenusObserveron the Earthin Venus's antumbra

The hidden source can also be a body that the Sun lights rather than one that shines. Then its phase shapes the outline that the blocking body hides. Chauvenet treats a planet hidden by the Moon this way 9.

Hidden: Jupiter blocks the observer's view of IoTop left, the Sun, labelled light source, lighting Io along a line. Left, Io, labelled lit body. Middle, Jupiter, labelled blocking body, with a pale band stretching left from it: seen from the Earth, anything in that band is behind Jupiter. Io sits inside the band. Right, an observer on the Earth, labelled observer, whose dotted line of sight towards Io ends at Jupiter.Schematic, not to scaleseen from the Earth, Io is behind Jupiter hereLight sourcethe SunLit bodyIoBlocking bodyJupiterObserveron the Earth
Hidden: an asteroid blocks the starlight reaching part of the EarthLeft, a star, labelled light source, sending parallel lines of light. Middle, a small asteroid, labelled blocking body. Behind it a narrow band, its shadow in starlight, reaches the Earth on the right, which is drawn as a disc with the band crossing it. An observer inside the band, labelled observer, loses the star. Observers outside the band see nothing happen.Schematic, not to scalethe Earthoutside the band: nothing happensLight sourcea starBlocking bodyan asteroidObserveron the Earth, in the shadow path
  • The Moon hides the Sun, a solar eclipse 1.
  • Mercury or Venus crosses the Sun, a transit 5.
  • The Moon hides a star or a planet 2.
  • An asteroid or a trans-Neptunian object hides a star. Only observers inside a narrow band on the Earth see it 4.
  • A planet hides its own satellite, and a satellite crosses its planet 10.
  • One satellite hides another, a mutual occultation 10.
  • One planet hides another, which last happened in 1818 11.
The Moon's shadow crossing the Earth, and what one observer seesLeft: the Earth's disc with the Moon's shadow on it, a wide pale penumbra around a small dark umbra, centred on an observer marked with a dot. Right: the observer's sky, with the Sun covered by the Moon. When played, the shadow slides across the Earth from left to right while, on the right, the Moon slides across the Sun, covering it fully only while the umbra is on the observer.The Moon's shadow crossing the EarthObserveron the EarthWhat the observer seesstill frame: the Sun covered, totalityLight sourcethe Sun, behindBlocking bodythe Moon, the dark discSchematic, not to scale

Darkened: another body is in the shadow

When another body stands in the blocking body's shadow, the sunlight reaching it is cut off and it goes dark. This is an eclipseeclipseThe passage of a body into the shadow of another, so that the sunlight reaching it is cut off. A lunar eclipse is the Moon in the Earth's shadow. A solar eclipse is also an occultation of the Sun by the Moon, seen from inside the Moon's shadow.Read on Wikipedia in the strict sense. Everyone who can see the darkened body sees it at the same instant, apart from the time light takes to reach them, so a prediction is a list of times rather than a map of paths. Where the observer stands only decides whether the body is above their horizon 12.

Darkened: the Earth blocks the sunlight reaching the MoonLeft, the Sun, labelled light source. Middle, the Earth, labelled blocking body, with sunlight reaching it and its shadow stretching to the right. Right, the Moon inside that shadow, labelled darkened body. An observer on the Earth's night side, labelled observer, looks along a dotted line at the darkened Moon.Schematic, not to scaleLight sourcethe SunBlocking bodythe EarthDarkened bodythe Moonin the Earth's shadowObserveron the Earth's night side
Darkened: Jupiter blocks the sunlight reaching IoLeft, the Sun, labelled light source. Middle, Jupiter, labelled blocking body, with its shadow stretching to the right. Inside the shadow, Io, labelled darkened body. Below, an observer on the Earth, labelled observer, looks up along a dotted line at the darkened Io.Schematic, not to scaleLight sourcethe SunBlocking bodyJupiterDarkened bodyIo, in Jupiter's shadowObserveron the Earth
Darkened: Io blocks the sunlight reaching Jupiter's cloud topsLeft, the Sun, labelled light source. Middle, Io, labelled blocking body, casting a long narrow shadow to the right. Right, Jupiter, labelled darkened body, with a small dark spot where Io's shadow meets its cloud tops. Below, an observer on the Earth, labelled observer, looks along a dotted line at the spot.Schematic, not to scaleLight sourcethe SunBlocking bodyIoDarkened bodyJupiter's cloud tops, a patchObserveron the Earth
  • A lunar eclipse is the Moon in the Earth's shadow 12.
  • An eclipse of a satellite is, for example, Io in Jupiter's shadow 10.
  • A shadow transit is a satellite's shadow falling on its planet's cloud tops, a patch of the planet in eclipse 10.
  • A mutual eclipse is one satellite in another satellite's shadow 10.
The Moon passing through the Earth's shadowThe Earth's shadow at the Moon's distance, seen face on: a dark inner disc, the umbra, inside a paler ring, the penumbra. The Moon sits in the middle of the umbra, tinted copper. When played, the Moon moves from left to right through the penumbra and the umbra and out again, its copper tint deepening inside the umbra and fading outside it.the Earth's umbrathe Earth's penumbraDarkened bodythe MoonLight sourcethe Sun, behind the EarthBlocking bodythe Earth, casting this shadowObserveranyone on the Earth's night sideStill frame: totality. The Moon is in the umbra and lit only by sunlight bent through the Earth's atmosphere.Schematic, not to scale

A planet and its satellites: both cases

A planet with satellites shows both cases in one orbit. When Io enters Jupiter's shadow, Io is darkened. When the observer stands in Jupiter's shadow in Io's reflected light, Io is hidden behind Jupiter. When Io's shadow falls on Jupiter, a patch of cloud tops is darkened. When the observer stands in Io's shadow in Jupiter's light, Io crosses in front of Jupiter 10.

One orbit of Io: four kinds of eventA view from above Jupiter. Sunlight comes from the left, so Jupiter's shadow is a band to its right. The Earth lies off to the lower left, so Jupiter hides a band behind it to the upper right as seen from the Earth. Io's orbit is a dashed circle with four labelled places: on the Sun side its shadow falls on Jupiter; on the Earth side it crosses in front of Jupiter; behind Jupiter as seen from the Earth it is occulted; in the shadow it is eclipsed. Io sits in the shadow at the still frame. When played, Io goes round the orbit. A framed panel on the right shows what an observer on the Earth sees: Jupiter's disc with Io beside it, crossing in front, hidden behind or dark in the shadow, and Io's shadow as a dot on Jupiter, with the event's name lit.Jupiter's shadowhidden from the EarthJupiterthe planetLight sourcethe Sun, this wayObserveron the Earth, this wayshadow transittransitoccultationeclipseIoWhat the observer seesfrom the Earthtransitshadow transitoccultationeclipseIo and Jupiter swap roles from one event to the next.Schematic, not to scale; angles exaggerated

The shape of a shadow

The source's size decides the shadow's shape. The Sun is a disc, so a body casts a cone with three parts 1:

  • the umbraumbraThe cone of the Moon's shadow within which the Sun is completely hidden. Where the umbra reaches the ground the eclipse is total.Read on Wikipedia, where the whole Sun is hidden and the event is total;
  • the antumbraantumbraThe extension of the Moon's shadow cone beyond its vertex, within which the Moon appears smaller than the Sun. Where the antumbra reaches the ground the eclipse is annular.Read on Wikipedia, beyond the point where the umbra closes, where the body looks smaller than the Sun and the event is annular;
  • the penumbrapenumbraThe outer part of the Moon's shadow within which the Sun is only partly hidden. Any place inside the penumbra sees at least a partial eclipse.Read on Wikipedia, where part of the Sun is hidden and the event is partial.

A transit is the annular case with a body far too small to cover the Sun: the observer is in the planet's antumbra and sees a dot on the Sun 5. A star is a point, so its shadow is a cylinder with no penumbra. The star does not fade but vanishes at once, apart from faint diffraction fringes 2.

Cone and cylinder: the two shapes of a shadowTop: a large source disc on the left and a small body in the middle. The umbra narrows behind the body to a point and opens out again beyond it as the antumbra. The wider penumbra spreads around both. Three observers sit in the umbra, the penumbra and the antumbra. A framed panel below shows what each one sees of the source disc: wholly covered with a faint corona around it, partly covered, and covered except for a ring. Bottom: parallel lines from a star at the far left pass a body and leave a straight band behind it, the cylinder, with an observer in it for whom the star is hidden.Light sourcethe Sun, a discBlocking bodythe MoonumbraantumbrapenumbraWhat the observer seesin the umbra: totalin the penumbra: partialin the antumbra: annularLight sourcea star: a point at infinity, so its light arrives in parallel linesBlocking bodythe Mooncylinder: no penumbraObserverthe star is hiddenSchematic, not to scale

The computation follows the same steps for every event. Choose a plane perpendicular to the shadow, the fundamental planefundamental planeThe plane through the Earth's centre perpendicular to the axis of the Moon's shadow. Its x axis lies in the equator pointing east, its y axis points north, and the shadow's cross-section on it is an exact circle.. Find the body's shadow on it as a circle. Find the observer, or the darkened body, on it as a point. An event begins and ends when the point crosses the circle 1 2.

Which events fall where

The source column says whether the light that casts the shadow comes from a point or a disc.

Event Case Source Where it is covered
Solar eclipse Hidden disc Solar eclipses and transits
Transit of Mercury or Venus Hidden disc Transits
Lunar occultation of a star Hidden point Lunar occultations
Lunar occultation of a planet Hidden disc Lunar occultations
Occultation by an asteroid or TNO Hidden point Occultations by planets and minor bodies
Occultation of a satellite by its planet Hidden disc Jupiter's satellites
Transit of a satellite Hidden disc Jupiter's satellites
Mutual occultation Hidden disc Planetary satellites
Occultation of one planet by another Hidden disc Occultations by planets and minor bodies
Lunar eclipse Darkened disc Lunar eclipses
Eclipse of a satellite Darkened disc Jupiter's satellites
Shadow transit Darkened disc Jupiter's satellites
Mutual eclipse Darkened disc Planetary satellites

How the Explanatory Supplement names them

The 1992 Explanatory Supplement uses four names 3:

  • an eclipse, when one body passes into the shadow of another;
  • an occultation, when a larger body passes in front of a smaller one;
  • a transit, when a smaller body passes in front of a larger one;
  • a shadow transit, when the shadow of a smaller body passes in front of a larger one.

Its chapter on eclipses draws the line between shadow and line of sight. It adds that long observation from the Earth gave the events their names before the distinction was understood 3. In the terms of this page, its eclipse and its shadow transit are the darkened case. Its occultation and its transit are the hidden case, told apart only by which body looks larger. A solar eclipse is an occultation of the Sun, seen from inside the Moon's shadow.

The name of this site, "Computing Eclipses and Occultations", uses the two broad terms. An eclipse is the darkened case, and an occultation is the hidden case. A transit is an occultation in which the nearer body looks smaller than the body behind it, so the site name does not list it apart. The Supplement's own section heading names all three, "Eclipses, Occultations, and Transits", because it keeps the traditional names 3.

Where to go next

  • The worked example follows one solar eclipse from the positions of the Sun and the Moon to the second totality begins at one place.
  • The glossary defines every term the site explains inline.
  • The research notes are grouped by subject: shared inputs, solar eclipses and transits, lunar eclipses, occultations and planetary satellites.
  • The reports set out the computation pipeline, its error budget and the open questions.

References

  1. 1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9B Eclipses and Transits Read in full (OCR text). Definitive almanac formulation: fundamental plane, point Z, x y z, mu from ephemeris sidereal time, sin f1 sin f2 with tabulated numerators for k = 0.272274, 0.2724807, 0.272281 and 0.2724880, c1 c2 l1 l2, sign convention, observer coordinates, ephemeris meridian 1.002738 ΔT, worked example 1961 Feb 15.
  2. 2peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961, reprint with 1973 added notes), section 10 Occultations, pp. 277-305 Read in full from the archive.org OCR text. Source of the occultation Besselian elements (T, H, Y, x', y'), the formulas for x and y and their rates, the station reduction with f, g and kn cos psi, the position angle, the a and b coefficients, the limiting parallels, the prediction limits, the k values 0.2725, 0.2724953 and 0.2725026, the reduction to delta-sigma, the limb correction arguments and section F on planetary occultations. Some equations are damaged in the OCR text.
  3. 3peer-reviewed Seidelmann, P. K. (ed.) (1992). Explanatory Supplement to the Astronomical Almanac: section 1.54 Eclipses, Occultations, and Transits; section 8.1 Introduction to chapter 8, Eclipses of the Sun and Moon (Fiala, A. D. and Bangert, J. A.); and the glossary entry "occultation" Read in the archive.org full text. Source of the definitions of eclipse, occultation, transit and shadow transit; the statement that by strict application a solar eclipse is really an occultation; the chapter 8 description of a transit as a planet's shadow falling on the Earth; and the glossary rule that an occultation cutting off a reflecting body's illumination is also called an eclipse.
  4. 4peer-reviewed Gomes-Júnior, A. R., Morgado, B. E., Benedetti-Rossi, G. et al. (2022). SORA: Stellar Occultation Reduction and Analysis. MNRAS 511, 1167-1181 (arXiv:2201.01799) Read in full from the arXiv LaTeX source. Source of the closest-approach equations (Assafin et al. 2010 geometry), the shadow velocity, the magnitude-drop equation, the Fresnel scale f = sqrt(lambda D / 2), the star radius at the body, the exposure length r_t = V_S t_exp, the timing uncertainty sigma_t = sigma_LC t_exp / Delta f, the tangent-plane equations 6 to 9, the 0.021 s light-time and the chord and ellipse chi-square. Its printed albedo equation omits the factor 0.4 that the code applies.
  5. 5peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9F Transits of Mercury, pages 262 to 276 Read in full (OCR text). Node condition and the table of node limits, dates and frequencies (Mercury ascending node about 16 November 238 arcminutes, descending 8 May 108; Venus ascending 9 December 37, descending 7 June 41), Newcomb's heliocentric method and the transit fundamental plane, the light-time rule, the contact solution, least distance, position angles with V, Newcomb's q p w, the B C D reduction and Examples 9.26 to 9.32 for 1960 November 7. The node-limit formula for theta and the cone-angle equation are damaged in the OCR text and are not transcribed.
  6. 6primary NASA (2026). Solar Eclipse of the Heart. Image article, Artemis II, page last updated 2026 April 8 Date of the eclipse the crew saw from Orion: "during a solar eclipse on April 6, 2026". Read 2026-10-19.
  7. 7primary NASA Science (2026). Artemis II Preliminary Lunar Science Report. Feature, 2026 October 7 States that the crew "watched the Sun slip behind the Moon and stay there for 54 minutes", because the Moon "appeared huge to the crew, and it didn't seem to move in the sky". Read 2026-10-19.
  8. 8primary NASA JPL (2022). NASA's Perseverance Rover Captures Video of Solar Eclipse on Mars. News release, 2022 April 20 Mastcam-Z recorded Phobos crossing the Sun on sol 397, April 2; the eclipse lasted "a little over 40 seconds"; Phobos is "about 157 times smaller than Earth's Moon". Read 2026-10-19.
  9. 9primary Chauvenet, W., A Manual of Spherical and Practical Astronomy, Vol. I (1863), chapter on Eclipses, Arts. 288 to 296 Read Arts. 288 to 291 and the cone-angle precepts in OCR text (garbled in places). Original almanac form of Bessel's method: point Z as the Sun seen from the Moon's centre, equations (482) for x y z, principal plane of reference, log k = 9.435000, c = z ± k/sin f, l = c tan f.
  10. 10primary IMCCE, /pub/ephem/satel/phenjupiter/README (2021 September) Read in full. Event codes, TT time scale, mid-event convention, the theory used for each range of years (G-5 to 2014, L1 2015 to 2022, Lainey 2021 after), and the TT-UT table.
  11. 11company Gray, B. (Project Pluto). Assorted celestial events: mutual planetary events, -1000 to +6000 Read. The author documents his own search: VSOP theory, times in TD, Gregorian dates, an event listed if at some point on Earth one disc overlaps another, an average rate of about one per 35 years, and the list including 1818 January 3 21:52 and 2065 November 22 12:45 TD.
  12. 12primary 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.