Astronomical Algorithms (Jean Meeus)

The book

Astronomical Algorithms by the Belgian astronomer Jean Meeus was first published by Willmann-Bell in 1991, with a second edition in 1998. It collects, in a single volume, the methods needed to compute the positions of the Sun, Moon and planets, the times of astronomical events and the transformations between coordinate systems, each with worked numerical examples.

The book grew out of Meeus' earlier Astronomical Formulae for Calculators (1979), written when programmable calculators opened precise astronomy to amateurs. What makes it a reference is not new theory but its selection and presentation. Each chapter explains the accuracy of the method, and each gives a worked example with intermediate values, so a programmer can check their code line by line. The asteroid 2213 Meeus is named in his honour.

What it covers

• Time: Julian Day, calendars and the date of Easter, Dynamical Time and ΔT, sidereal time.
• Coordinates: transformations between equatorial, ecliptic, horizontal and galactic coordinates, angular separations and parallactic angles.
• Corrections: precession, nutation, aberration, parallax and atmospheric refraction.
• The Sun: its position, the equinoxes and solstices and the equation of time.
• The planets: positions from the VSOP87 theory, elliptic motion and Kepler's equation, and the dates of conjunctions, oppositions and elongations.
• The Moon: its position from an abridged ELP-2000/82 theory, phases, perigee and apogee, nodes and the libration.
• Events: rising, transit and setting, eclipses of the Sun and the Moon, and planetary phenomena.
• Physical data: illuminated fraction, magnitudes, apparent diameters (semidiameter), Jupiter's satellites, Saturn's rings and binary star orbits.

From a date to a point in your sky

Every sky map on AstroAlgos follows the same chain of calculations, each step coming from a chapter of the book. For a planet seen from your location:

• Convert the date and time to a Julian Day, then to Julian Ephemeris Day with ΔT.
• Compute the heliocentric ecliptic longitude, latitude and distance of the Earth and of the planet with the VSOP87 series.
• Subtract the Earth's position to get geocentric coordinates, correcting for the time light takes to travel from the planet to the Earth.
• Apply nutation and aberration to obtain the apparent ecliptic position, then rotate it to right ascension and declination using the true obliquity of the ecliptic.
• Compute the apparent sidereal time at your longitude and the hour angle of the planet.
• Correct for parallax, then convert to altitude and azimuth for your latitude. The planet is drawn on the sky disk at that position.

For stars the starting point is a catalogue position at epoch J2000.0. It is brought to the date of observation by precession, nutation and aberration before the last two steps.

How accurate is it?

Meeus chose methods that balance accuracy and computing effort. His abridged VSOP87 tables give planetary positions to a few arcseconds for dates within a few thousand years of the present, and his lunar theory gives the Moon's position to about 10″ in longitude. That is far below what the eye or a small telescope can distinguish, and good enough to predict the time of an occultation to within a few seconds.

Professional ephemerides such as JPL's DE440 are more precise still, but they are based on numerical integration and distributed as large data files. The Meeus approach works entirely from formulas, which is why it can run in a web browser in a fraction of a second.

Explore it on AstroAlgos

• Sky disk — the whole chain of calculations, live
• Heliocentric view — VSOP87 planetary positions in 3D
• Julian Day — the first step of every calculation
• Equatorial coordinates — right ascension and declination
• Horizontal coordinates — the last step