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Solar System Ephemerides

Ephemeris (plural: ephemerides) comes from the Greek "ephémeros", which means "daily. Traditionally, an ephemeris was a table providing the daily positions of the Sun, the Moon, the planets, asteroids or comets in the sky. Astronomers would use these tables to find the celestial objects in the sky. In 1554, Johannes Stadius published his "Ephemerides novae at auctae" that attempted to give accurate planetary positions on a daily basis. Besides the astronomers, also navigators became interested in ephemerides and in keeping track of planetary positions because this could help them to find their position at sea.

The first printed ephemerides however, were a controversial innovation. Many 16th century astrologers felt that those who couldn't calculate planetary positions probably couldn't calculate - or read - a chart, either. Also, early hand calculated ephemerides were often inaccurate and in fact not very much suited for navigation. But the navy - especially the British Royal Navy - recognized the potential for navigation and encouraged the development of accurate ephemerides suitable for celestial navigation. This eventually led to the establishment of the Royal Observatory in Greenwich in 1675.
In 1766 the first "Nautical Almanac and Astronomical Ephemeris" was published by the Astronomer Royal of England containing ephemeris data for the year 1767. Initially the almanacs provided the data required for the method of lunar distances, a technically demanding and mathematically complex method of determining longitude before the invention of accurate clocks for shipboard use. The common availability of precise chronometers on ships beginning in the early 1800s, and the development of methods of "sight reduction" by Sumner, St.-Hilaire and others, provided an easier procedure for navigators to determine their position at sea. The almanacs provided the necessary data for these methods.

Utilizing the predicted position of stars and planets from the nautical almanac to determine his position and being astonished over the precision of the results, Joshua Slocum writes 1896 in his "Sailing around the World alone": ... I realized the mathematical truth of their motions, so well known that astronomers compile tables of their positions through the years and the days, and the minutes of a day, with such precision that one coming along over the sea even five years later may, by their aid, find the standard time of any given meridian on the earth ... .

Today, the computational power of modern computers, not only allows the calculation of planetary ephemeris with high precision, but also the tabulation of planetary positions on an hourly basis (rather than the traditionally daily basis). The hourly based planetary ephemerides are the primary part of today's nautical almanacs.

Scientific ephemeris

The ephemeris of a celestial object, describes the position (and velocity) of the object as a function of time. The ephemeris data is obtained by solving the fundamental equation of motion of the body, which is obtained by applying the fundamental laws of motion postulated by Kepler and Newton.

There are two basic ways to solve the fundamental equation of motion to obtain ephemerides: an analytical and a numerical way.

  • Analytical ephemeris are based on closed-form algebraic expressions which yield the object's position and velocity components for a given instance of time. These expressions must be derived from an algebraic solution to the equation of motion for the object. The primary benefit of analytical ephemerides is that they express the position and velocity components as explicit functions of time. When mutual gravitational perturbations and relativistic effects are taken into account, the expressions necessarily become quite complicated and analytical solutions may become infeasible. In general, for applications requiring a certain degree of precision, analytical ephemerides are no longer used.

  • Numerical ephemeris rely on a numerical solution to the equation of motion. The output of such a computation is a table of numbers giving the position and velocities at a desired instances of time. A potential drawback of this method is the sheer size of the tables when the object's position and velocity components are required for a large number of times. In practice, the numerically generated position tables (and velocities) may be compressed by "fitting" them with a mathematical function, which can replicate the original table values to within a very small tolerance for any desired instance of time. This is the approach taken e.g. in the production of the Jet Propulsion Laboratory (JPL) planetary ephemerides. Notice that the "fitting" function, which allows - a part of the - ephemerides to be calculated in an algebraic way is not the same as the algebraic solution to the fundamental equation of motion!

The planetary positions obtained from the ephemerides are generally not in a form for use by Earth-based observers. Several transformations (shifting the coordinate reference point from the solar system barycenter to the location of the observer) and corrections (to account for atmospheric effects as well as for "time delay" effects due to the limited speed of light) will be necessary to reduce the obtained ephemeris data to a form which can be used to compare it to the observed values.

Coordinate system of the Ephemeris Data

The ephemeris data of a celestial object, describes the tree-dimensional position and velocity vectors of the object as a function of time in a rectangular coordinate system. The origin of the rectangular coordinate system is at the barycenter (center of mass) of the solar system. The Z-axis is in the direction of the north celestial pole and the X-Y plane of the ephemeris coordinate system is parallel to the Earth's equatorial plane. The X-axis is in the direction of the vernal equinox, the Y-axis points towards RA=6 hours, Dec=0.

Since the orientation of the Earth's polar axis - and thus the orientation of the north celestial pole as well as the plane of the Equator - is changing continuously, the coordinate system must be specified for a particular instance of time or epoch. The most recent numerical ephemeris series available from the Jet Propulsion Laboratory (JPL) use the "J2000" equator and equinox. J2000 stands for the 1. of January in the year 2000 following the Julian calendar.

In Jan 1998 the "International Celestial Reference Frame (ICRF) was adopted by the International Astronomical Union as a new reference frame. This ICRF is based on the radio positions of over two hundred extragalactic sources distributed over the entire sky. The positional accuracy of these sources is better than 1 milli-arc-second in both RA and Dec coordinates. The final orientation of the frame axis has been obtained by a rotation of the positions into the system of the International Celestial Reference System (ICRS) and is consistent with the FK5 J2000.0 optical system.

Positions of Celestial Objects in the Sky

Ephemeris data refers to the solar system barycenter, with the cartesian X,Y,Z axis fixed in space as follows: The X and Y axis are in the equatorial celestial plane; The Z axis is perpendicular to this plane and points to the celestial North Pole. The X axis is fixed in the equatorial plane by the vernal equinox. Since this vernal equinox moves over time, these coordinate systems are bound to a date e.g. January 1st 2000 for the J2000 system. In such an orthogonal X,Y,Z coordinate system, any object in space can be represented with a set of 3 coordinates (X,Y,Z) representing the components of the position vector along the orthogonal X,Y,Z axis.

In order for an Earth bound astronomer or navigator to be able to compare his observations with the calculated ephemeris data, a transformation of the planetary coordinates according to the different origins (solar center barycenter and the position of the observer on the Earth) must be performed.

With both the position of the Earth and the position of the celestial objects known, first geo-centric positions (as seen from the center of the Earth) may be computed:

  Object(x,y,z) = Object(X,Y,Z) - Earth(X,Y,Z)

Here the coordinates (X,Y,Z) refer to the original rectangular coordinate system of the ephemeris data, whereas the (x,y,z) coordinates refer to the "translated" the geocentric coordinate system. In a next step, the geo-centric coordinates must be translated to the location of the observer obtaining the topo-centric coordinates (a,b,c):

  Object(a,b,c) = Object(x,y,z) - Observer(x,y,z)

Remember that (x,y,z) are the geocentric coordinates. Finally, the topo-centric coordinates (a,b,c) will be transformed into a more useful grid such as the Right Ascension - Declination coordinate system:

  Dec = atan(c / sqrt(a*a + b*b) )
  RA  = atan(a / b)

This is the grid, which is used by astronomers to describe the position of celestial objects in the sky. It is also used to point a telescope to an object in the sky, which (RA,Dec) coordinates are known.

Time Delay and Apparent Direction

The solar system is not static. Apparent directions of far-away objects are significantly affected by the the fact that it takes some time for the light to travel from the object to the Earth-bound observer. When we look at a planet in the sky, we are actually seeing the planet on the position where it was when its light left the planet. This could be minutes or even several hours before the moment of observation. The procedure for compensating for this "time delay" is to iteratively compute the distance to the planet at the time of observation, calculate the "light travel time" and recompute the planets position at the current time minus this "light travel time". Then recompute the distance to the planet and continue the "time" corrections until the iterations converge to the correct position.

A second correction to the apparent direction of a planet, due to the finite speed of light, comes from the motion of the observer. This correction is called "aberration of light" and - in a first approximation - is equal to the ratio of "the velocity component of the observer's motion perpendicular to the line of sight" to "the speed of light". The Earth's orbital velocity is about 30 km/s whereas the tangential velocity of an observer at the Equator is less than 0.5 km/s. This yields an aberration effect in the order of 0.3 minutes-of-arc.

Corrections to the Coordinate System

By the standards of modern astrometry, the Earth is quite a wobbly platform from which to observe the sky. The Earth's rotation rate is not uniform, its axis of rotation is not fixed in space, and even its shape and relative positions of its surface locations are not fixed. For the purposes of pointing a telescope to one arc-second accuracy, the irregular shape and surface are not an issue, but changes in the orientation of the Earth's rotation axis are very important.

In a sense, equatorial sky coordinates are a compromise between an Earth-based system and one fixed with respect to distant stars. Right Ascension (RA) and Declination (Dec) are quite analogous to Longitude and Latitude on the Earth's surface. They share the same polar axis and Equator, but the sky coordinate grid does not rotate with the Earth's daily spin. However, apparent Right Ascension and Declination are not fixed with respect to the stars because their coordinate frame follows the motion the Earth's Pole and Equator. To be able to list star positions in catalogues, there must be a convention to use the position of the Earth's Pole and Equator at accurately specified times, essentially by defining the RA and Dec "zero" points at those specified times. January 1st, 1950 and 2000 are the most common coordinate epochs (for the older B1950 and more recent J2000 respectively).

The zero point of Right Ascension is not assigned to a particular celestial object in the same way that zero longitude is defined to be at Greenwich, England. Zero Right Ascension is the point where the Sun (on the ecliptic path) appears to cross the Celestial Equator on its South to North journey through the sky in the spring. In three dimensions, the vernal equinox is the direction of the line where the plane of the Earth's equator intersects the plane of the Earth's orbit (the ecliptic plane). Since the Earth's orientation is constantly changing with respect to the stars, so does the position of the vernal equinox.

In practice, celestial coordinates are tied to observed objects because the location of the vernal equinox is hard to measure directly. The B1950 coordinate grid location is defined by the publish positions of stars in the fourth "Fundamental-Katalog", FK4, and the J2000 system is based on FK5. These catalogues list mostly nearby stars so any definition of coordinates tied to these catalogues is subject to errors due to motions of the stars on the sky. The FK4 equinox is now known to drift with respect to the FK5 equinox by about 0.085 arc-seconds per century, which is quite large by VLBI (Very-Large Base Interferometry) standards but of no practical implication for applications based on simple optical observations.

Currently, the most stable definition of J2000 coordinates (X,Y,Z axis) is one based on about 400 extragalactic objects in the Radio Optical Reference Frame (RORF). (NOTICE: the Radio Optical Reference Frame was superseded by the "International Celestial Reference Frame (ICRF) in Jan 1998, but it was realized such that - for most practical applications - it is "compatible" with the J2000 system). The RORF is stable to at least 0.020 arc-seconds per century, and this is improving with better observations and a longer time base. The positional accuracy of the complete 400 objects is about 0.0005 arc-seconds.

For partly historical and partly practical reasons, the time variability of the direction of the Earth's rotation axis and an observatory's relation to it are divided into four components: precession, nutation, celestial pole offset, and polar motion. By definition, precession and nutation are mathematically defined through the adoption of the best available equations. Celestial pole offset and polar motion are observed offsets from the mathematical formulae and are not predictable over long periods of time. All four components are described in more detail below.


Neither the plane of the Earth's orbit, the ecliptic, nor the plane of the Earth's equator are fixed with respect to distant objects. The dominant motion is the precession of the Earth's polar axis around the ecliptic pole, mainly due torques on the Earth cause by the Moon and the Sun. The Earth's axis sweeps out a cone of 23.5 degrees half angle in 26,000 years.

The ecliptic pole moves more slowly. If we imagine the motion of the two poles with respect to very distant objects, the Earth's pole is moving about 20 arc-seconds per year, and the ecliptic pole is moving about 0.5 arc-seconds per year. The combined motion and its effect on the position of the vernal equinox are called general precession. The predictable short term deviations of the Earth's axis from its long term precession are called nutation as explained in the next section.

Equations, accurate to one arcsecond, for computing precession corrections to Right Ascension and Declination for a given date within about 20 years of the year 2000 are

	RA  [s] = RA(2000)  + (3.075 + 1.336 * sin(RA) * tan(Dec)) * Tys
	Dec ["] = Dec(2000) + 20.04 * cos(RA) * Tys

where Tys is the time from January 1, 2000 in fractional years, and the offsets in RA and Dec are in seconds of time and arc-seconds, respectively.


Predictable motions of the Earth's rotation axis on time scales less than 300 years are combined under nutation. This can be thought of as a first order correction to precession. The currently standard nutation theory is composed of 106 non-harmonically-related sine and cosine components, mainly due to second-order torque effects from the Sun and Moon, plus 85 planetary correction terms. The four dominant periods of nutation are 18.6 years (precession period of the lunar orbit), 182.6 days (half a year), 13.7 days (half a month), and 9.3 years (rotation period of the Moon's perigee).

The following approximations for nutation are good to about a second and an arcsecond.

	delta RA  [s] = (0.9175 + 0.3978 * sin(RA) * tan(Dec)) * dL - cos(RA) * tan(Dec) * dE
	delta Dec ["] =  0.3978 * cos(RA) * dL + sin(RA) * dE

where delta RA and delta Dec are added to mean coordinates to get apparent coordinates, and the nutations in longitude, dL, and obliquity of the ecliptic, dE, may be found in the Astronomical Almanac, or computed from the two largest terms in the general theory with:

	dL = -17.3 * sin(125.0 - 0.05295 * d) - 1.4 * sin(200.0 + 1.97129 * d)
	dE =   9.4 * cos(125.0 - 0.05295 * d) + 0.7 * cos(200.0 + 1.97129 * d)

where d = Julian Date - 2451545.0, the sine and cosine arguments are in degrees, and dL and dE are in arc-seconds.

Celestial Pole Offset

The celestial pole offset is the unpredictable part of nutation. These offsets are published in IERS Bulletin A as offsets in dL and dE. For telescope pointing they are not important since they are only the order of 0.03 arc-seconds.

Polar Motion

Because of internal motions and shape deformations of the Earth, an axis defined by the locations of a set of observatories on the surface of the Earth is not fixed with respect to the rotation axis which defines the celestial pole. The movement of one axis with respect to the other is called polar motion. For a particular observatory, it has the effect of changing the observatory's effective latitude as used in the transformation from terrestrial to celestial coordinates. The International Earth Rotation Service definition of the terrestrial reference frame axis is called the IERS Reference Pole (IRP) as defined by it's observatory ensemble.

The dominant component of polar motion, called Chandler wobble, is a roughly circular motion of the IRP around the celestial pole with an amplitude of about 0.7 arc-seconds and a period of roughly 14 months. Shorter and longer time scale irregularities, due to internal motions of the Earth, are not predictable and must be monitored by observation. The sum of Chandler wobble and irregular components of polar motion are published weekly in IERS Bulletin A along with predictions for a number of months into the future.

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