The X-Y plane of the ephemeris rectangular 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 toward RA = 6 hours, Dec = 0, and the Z axis is in the direction of the north celestial pole. The position components are in kilometers, and the velocity components are in kilometers per day. The time independent variable for the equations is Terrestrial Dynamic Time in fractional Julian days.

The origin of the rectangular coordinates is at the solar system barycenter (center of mass) for the sun and all planets, except the earth. The solar system barycentric positions of the earth and moon may be derived from the ephemeris specifications of the position and velocity of the earth-moon barycenter and the geocentric position and velocity of the moon.

Since the orientation of the earth's equator and pole are is continuously changing, the coordinate system must be specified for a particular epoch. The DE100-series ephemerides use the B1950 equator and equinox. The DE200 series uses the J2000 system [ref 1]. The most recent DE400 series are also in the J2000 system, but its equator and equinox are defined by the reference frame of the International Earth Rotation Service (IERS), which is more precisely tied to distant celestial objects than is the IAU J2000 standard.

The solar system is not static. Apparent directions are significantly affected by the light travel time between two objects as explained below.

To get the instantaneous topocentric (observatory's) position of the
moon we add vectors from the observatory to the earth's center, and from
the earth's center to the moon's center. The latter is supplied by the
ephemeris. If we let `(x,y,z)` be geocentric and `(a,b,c)`
be topocentric vector coordinates, the vector equation is

Moon(a,b,c) = Moon(x,y,z) - Observatory(x,y,z)Using available ephemeris information, a planet's topocentric position is the vector sum:

observatory->(earth's center) +

(earth's center)->(earth-moon barycenter) +

(earth-moon barycenter)->(solar system barycenter) +

(solar system barycenter)->planet.

If we let `(X,Y,Z)` be solar system barycentric coordinates,
`EMB` represent the earth-moon barycenter, and `EMrat` be the
earth/moon mass ratio, then

Planet(a,b,c) = Planet(X,Y,Z) - EMB(X,Y,Z) + Moon(x,y,z) / (1 + EMrat) - Observatory(x,y,z)where the third term quotient on the right hand side of the equation is the geocentric position of the earth-moon barycenter.

Positions from any point in the solar system relative to any other point may be computed in a similar manner as long as the locations are given in the ephemeris.

Neglecting the light travel time correction for the moment, conversion of a topocentric vector to right ascension and declination in the ephemeris' reference frame is simply

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

Things are always as they appear, but it depends on whom you ask. When
we look at a planet we are actually seeing the planet where it was when its
light left the planet. This could be minutes or even hours before the
current time. The procedure for compensating for this time delay is to
compute the distance to the planet at the time of observation. From
this compute the light travel time, recompute the planet's position for
current time minus light travel time, and use this earlier planet's
position with the current observatory, moon, and earth-moon
barycenter positions in the `Planet(a,b,c)` equation above.

The planet's position obtained from this procedure is its "astrometric" position in the reference frame of the solar system ephemeris. That is, it is the position of the planet as it would appear on the background of stars as plotted in this frame, J2000 for example.

The second correction to the apparent direction of a planet, due to the finite speed of light, comes from the motion of the observer. The same correction needs to made to star positions, where it is called "aberration of star light." The time-worn analogy is of a person running in the rain. If the person is standing still, the rain appears to be coming straight down, but, if the person is moving, the rain appears to be coming from the direction of motion. The aberration correction to the apparent direction of a star or planet, in radians, is 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 so the annual aberration can be much as 30 / 300,000 = .0001 radians or about 20 arcseconds.

One might ask "velocity with respect to what?" when computing
aberration. In the case of stars we dodge the question by computing only
*differential* aberration for different times of day and year,
hence, the terms "diurnal aberration" and "annual aberration." For a
planet we can use the observer's lateral velocity with respect to the
planet, but this correction will include the time-of-flight correction for
the speed of light outlined above. Since the time-of-flight correction
used the planet's velocity with respect to the solar system barycenter, we
can add the aberration correction using the earth's velocity with respect
to the barycenter. Another, slightly more rigorous approach is to compute
the sum of both corrections together by computing the direction of the
planet using the positions of both the planet and observer at the current
time minus the light travel time.

The Astronomical Almanac is not entirely consistent on how it lists planetary positions. Pluto's position is tabulated as astrometric. In other words, it is corrected for the light's time-of-flight but not for aberration, and the coordinate frame is J2000 as given by the DE200 ephemeris. All other planetary positions are listed as "apparent", the position with respect to the current equator and equinox including the correction for aberration. In other words, these positions are corrected for precession, nutation, aberration, and time-of-flight.

The B1950 and J2000 equatorial coordinate systems are defined by the mean orientation of the earth's equator and ecliptic at the beginning of the years 1950 and 2000. The assumed orientation of the earth on these two dates is more a matter of definition than actual, since the 'mean' orientation does not include short term motions of the earth's spin axis (nutation and smaller effects). Of course, he coordinates of celestial objects in B1950 and J2000 differ by many arcminutes.

The B1950 system is tied to the sky by star coordinates in the FK4 catalog, and J2000 is tied to FK5. Because star catalogs tend to be weighted toward nearby stars, they are subject to stellar proper motions and the assumption that the motions of many stars will average out. Since the FK4 catalog was published, offsets and drift rates have been determined for its average stellar positions at the level of a few hundredths of an arcsecond as summarized in [ref 2]. These corrections are insignificant for telescope pointing, but they are important for VLBI and pulsar observations.

The IERS reference frame is essentially the J2000 system except that it is tied to the sky by the published positions of 228 radio sources, roughly 23 of which are monitored by several VLBI networks to determine the day to day changes in the orientation of the earth [ref 3]. These radio sources are very distant compared to stars and should not suffer any of the proper motion problems. The DE403 and IERS celestial frames are tied together to an accuracy of a few milliarcseconds [ref 5]. The IERS and FK5 systems are consistent to the accuracy of the FK5 catalog.

The JPL ephemerides themselves define a celestial reference frame since they publish the positions of solar system objects in space in a coordinate system which is presumably fixed with respect to distant celestial objects. Although every effort has been made to tie the ephemeris dynamical frame to a stellar frame, there will always be an offset at some level. You must be aware that, when working with more than one ephemeris or coordinate system, you need to take into account the accuracy with which these systems are known to be tied. Avoid switching systems unnecessarily, and define your assumptions and choices clearly for other observers using your data.

Pulsar timing also requires a relativistic gravitational correction due to the earth's motion in the solar system's potential well. This correction can be as large as 1.6 milliseconds. The standard pulsar timing reference is the solar system barycenter using Barycentric Dynamic Time.

The geometric timing correction is simply the speed of light divided into the distance from the observer to the solar system barycenter times the cosine of the angle between the observer-barycenter line and the direction to the pulsar. In vector language this is the dot product of the observer's barycentric vector and the pulsar direction unit vector divided by the speed of light. Using the ephemeris vector notation above, the observer's barycentric vector is

Observer(X,Y,Z) = EMB(X,Y,Z) + Moon(x,y,z) / (1 + EMrat) + Observatory(x,y,z)and the unit vector components of the pulsar's direction are

Pulsar_unit_X = cosine( RA ) cosine( Dec ) Pulsar_unit_Y = sine( RA ) cosine( Dec ) Pulsar_unit_Z = sine( Dec )The main caution is to keep the pulsar's position in the same coordinate frame as the ephemeris. A 0.1 arcsecond pulsar direction error can produce a timing error as high as 240 microseconds, so all of the reference frame caveats of the previous section apply here, too. Uncertainties involved with converting from B1950 to J2000 coordinates, for instance, can have effects at the 10 to 100 microsecond level.

Equations for the relativistic time correction are given in the Explanatory Supplement to the Astronomical Almanac [ref 4].

Last updated January, 24, 1996.