## Solar System Ephemerides
Ephemeris (plural: ephemerides) comes from the Greek " ## 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. 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. 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. 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. ## 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.
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).
( ## Precession
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. 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. ## Nutation
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). 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 OffsetThe 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. |

Cover << Sail Away << Celestial Navigation << . | last updated: 23-Feb-2006 |