Computing Eclipses, Occultations and Transits

How eclipses, occultations and transits are computed

workingupdated 2026-10-18
  • Every event is one body cutting a light path. Light travels from a source, the Sun or a star, to an observer, either directly or by reflecting off a body such as the Moon or a planet. An event happens when another body cuts that path.
  • Where it cuts the path decides what happens. Cutting the light that reaches a body darkens it for everyone. Cutting the light that reaches the observer hides something from that observer only. Cutting the observer's view of the source itself does both.
  • The shadow's shape follows from the source's size. The Sun casts a cone with an umbra, a penumbra and an antumbra, which give total, partial and annular events. A star casts a cylinder, so it vanishes at once 1 2.
  • The names come from history, and the geometry is shared. The Explanatory Supplement itself notes that a solar eclipse is really an occultation 3. The same construction computes them all.

The light path

Light leaves a source and reaches an observer. Sometimes it travels straight there, as when we look at the Sun or a star. Sometimes it first lights a body, such as the Moon, a planet or a satellite, and reaches us by reflection. The light path therefore has up to three legs, and a body that gets in the way cuts one of them.

Three places a body can cut the light pathA light source on the left sends light along three straight legs: to a body at the top that reflects it, from that body down to an observer on the right, and straight from the source to the observer. A small dark disc sits on each leg, numbered 1 to 3, with a short label saying what an observer notices when a body cuts that leg.Sourcethe Sun, or a starA body lit by the sourceObserver1Darkenedthe lit body goes darkfor everyone2Hiddenthe observer losessight of the lit body3Boththe observer stands in a shadowand sees the source hiddenSchematic, not to scale

Darkened: a body goes dark for everyone

When a body cuts the sunlight reaching another body, that body 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. 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 4.

  • A lunar eclipse is the Moon in the Earth's shadow 4.
  • An eclipse of a satellite is, for example, Io in Jupiter's shadow 5.
  • A shadow transit is a satellite's shadow falling on its planet's cloud tops, a patch of the planet in eclipse 5.
  • A mutual eclipse is one satellite in another satellite's shadow 5.
Darkened: a body cuts the sunlight reaching another bodyThree panels. Left, lunar eclipse: the Sun, the Earth and its shadow cone, with the Moon inside the shadow; an observer on the Earth's night side looks at it. Middle, eclipse of a satellite: the Sun, Jupiter and its shadow, with Io inside; the observer looks from elsewhere. Right, shadow transit: the Sun, Io between the Sun and Jupiter, and Io's small shadow falling on Jupiter's disc as a dark spot.Schematic, not to scaleEarthLunar eclipsethe Moon in the Earth's shadowsame instant for everyoneJupiterIoEclipse of a satelliteIo in Jupiter's shadowseen from the EarthIoJupiterShadow transitIo's shadow on Jupiter'scloud tops
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.umbrapenumbraStill frame: totality. The Moon is in the umbra and lit only by sunlight bent through the Earth's atmosphere.Schematic, not to scale

Hidden: something disappears for one observer

When a body cuts the light on its way to the observer, something disappears for that observer only. 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.. A body nearby, or one looking from somewhere else, may see nothing happen, so the time depends on where the observer stands 2.

  • The Moon occults a star or a planet 2.
  • An asteroid or a trans-Neptunian object occults a star. Only observers inside a narrow band on the Earth see it 6.
  • A planet occults its own satellite, and a satellite transits across its planet 5.
  • One satellite occults another, a mutual occultation 5.
  • One planet occults another, which last happened in 1818 7.

A star is so far away that its light arrives in parallel lines. The body then casts a "shadow" in starlight that is a cylinder exactly as wide as the body. That shadow darkens nothing, but it is how these events are computed: an occultation is predicted as the path of that cylinder across the Earth 2 6.

Hidden: a body cuts the light reaching the observerThree panels. Left, lunar occultation of a star: parallel lines of starlight reach the Moon, and the observer behind the Moon cannot see the star. Middle, occultation of a satellite: Io half hidden behind Jupiter's limb as the observer sees it. Right, occultation by an asteroid: the asteroid's shadow in starlight crosses the Earth as a straight band, and observers inside the band see the star vanish.Schematic, not to scaleMoonLunar occultation of a starthe star vanishes at once:a point has no penumbraJupiterIoOccultation of a satelliteIo passing behind Jupiter,as the observer sees itshadow pathEarthOccultation by an asteroidits shadow path is as wideas the asteroid
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.Jupiter's shadowhidden from the EarthJupiterto the Sunto the Earthshadow transittransitoccultationeclipseIoSchematic, not to scale; angles exaggerated

Both: the observer stands in a shadow

When a body cuts the observer's view of the source itself, both things happen at once. The observer stands in the body's shadow, and the observer sees the source hidden by the body. These are the same fact described two ways. This is why the Supplement says a solar eclipse is really an occultation, and why it describes 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. of Mercury or Venus as a planet's shadow falling on the Earth 3.

  • A solar eclipse: the observer stands in the Moon's shadow 1.
  • A transit of Mercury or Venus: the observer stands in the planet's shadow, which is far too narrow to cover the Sun 8.
Both: the observer stands in a shadow and sees the Sun hiddenThree panels. Left, solar eclipse: the Sun, the Moon and its shadow cone touching the Earth, with an observer inside it; an inset shows the Sun covered. Middle, transit of Venus: the Sun, Venus and its long narrow shadow, which ends far short of the Earth, so the observer is in the antumbra; an inset shows a small dark dot on the Sun. Right, the general rule: being inside a body's shadow and seeing the source hidden by that body are the same thing.Schematic, not to scaleMoonEarthSolar eclipsethe observer in the Moon'sumbra sees the Sun coveredVenusantumbraTransit of Venusthe observer in the antumbrasees a dot on the SunInside a body's shadow,the source is hiddenbehind that body.A total eclipse is the umbra,an annular eclipse or atransit the antumbra,a partial eclipse the penumbra.The same event, two ways
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 on the EarthobserverWhat the observer seesstill frame: the Sun covered, totalitySchematic, not to scale

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., 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., 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., 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 8. 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, each linked to a small inset of the source disc: wholly covered, 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.Extended sourcethe SunBodyumbraantumbrapenumbraumbra: totalpenumbra: partialantumbra: annularStar: a point at infinity, so its light arrives in parallel linescylinder: no penumbrastar 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 on it as a point. An event begins and ends when the point crosses the circle 1 2.

Which events fall where

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

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 is the darkened case. Its occultation and transit are the hidden case, told apart only by which body looks larger. A solar eclipse and a transit of a planet across the Sun are the case where both happen at once.

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 these cases.
  • 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. 4primary 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.
  5. 5primary 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.
  6. 6peer-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.
  7. 7company 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.
  8. 8peer-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.