Earth Rotation and Equatorial Coordinates

Introduction

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-arcsecond accuracy, we need not worry about shape and surface feature changes, 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 and declination 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 catalogs, we have agreed to use the position of the earth's pole and equator at specified times, essentially snapshots of the RA and Dec coordinates at those times. January 1, 1950 and 2000 are the most common coordinate epochs.

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 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. 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 coodinate 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 catalogs list mostly nearby stars so any definition of coordinates tied to these catalogs 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 arcseconds per century, quite a bit by VLBI standards.

Currently, the most stable definition of J2000 coordinates is one based on about 400 extragalactic objects in the Radio Optical Reference Frame. This is heavily biased toward VLBI radio sources, but it will soon be tied to many more optical objects by the HIPPARCOS satellite. The RORF is stable to at least 0.020 arcseconds per century, and this is improving with better observations and a longer time base. The positional accuracy of the ensemble of 400 objects is about 0.0005 arcseconds.

For partly historical and partly practical reasons, the time variablity 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.

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 [ref 1]. 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 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 arcseconds per year, and the ecliptic pole is moving about 0.5 arcseconds 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 = RA(2000) + (3.075 + 1.336 * sin(RA) * tan(Dec)) * y
	Dec = Dec(2000) + 20.04 * cos(RA) * y
where y is the time from January 1, 2000 in fractional years, and the offsets in RA and Dec are in seconds of time and arcseconds, respectively. Very accurate telescope pointing calculations should use the exact equations given on pages 104 and 105 of ref [1].

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).

Normally, the corrections for precession and nutation in right ascension and declination will be handled by the telescope control computer. But, if you are stuck in the wilderness with a hand held calculator, or you want to check a position, the following approximation for nutation are good to about an arcsecond [ref 2].

	delta RA = (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, pages B24-B31, 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 arcseconds.

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 on the order of 0.03 arcseconds.

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 arcseconds 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.

Observing Station Coordinates

There is quite a variety of local and globle coordinate systems that may be used to describe locations on the surface of the earth. The three of most importance in astronomy, geocentric, geodetic, and astronomical, are briefly described here. See Chapter 4 of the Explanatory Supplement to the Astronomical Almanac [ref 3] for a more complete discussion of terrestrial coordinates.

Geocentric

Geocentric coordinates are most useful for VLBI and pulsar timing where the observer's three-dimensional location in space is important. The reference planes are the equator, the Greenwich Meridian, and the plane through the earth's axis and perpendicular to the Greenwich Meridian, call it the east-west plane. A telescope's rectangular cordinate components (x,y,z) are
	x = distance from the east-west plane, Greewich being positive x
	y = eastward distance from the Greenwich Merdian
	z = northward distance from the equator
For example, the coordinates for the 140-ft telescope from VLBI are
	x =   882880.0208 meters
	y = -4924482.4385 meters
	z =  3944130.6438 meters
Geocentric latitude and longitude are not commonly used, but they are defined by
	latitude  = arctan( z / sqrt( x^2 + y^2 ) )
	longitude = arctan( y / x )

Geodetic

The closest simple approximation to the shape of the surface of the earth is an ellipse rotated around the earth's axis, an ellipsoid. The difference between the long and short axes of this ellipse is about 0.3%. The value of flattening [ref 4], adopted by the IERS in 1989 is [ref 5]
	f = ( a - b ) / a = 1.0 / 298.257
where a is the equatorial axis, and b is the polar axis of the ellipsoid.

Geodetic coordinates are a measure of the direction of the line perpendicular to the ideal ellipsoid at the observer's location on the earth. Geodetic longitude is the same as geocentric longitude because they share the same reference meridian and axis. Geodetic and geocentric latitude can differ by as much as 10 arcminutes at mid latitudes. The ellipsoid is mathematical concept so you cannot measure from it directly, but it differs from mean sea level, also called the geoid, by less than 100 meters and more typically by less than 20 or 30 meters [ref 7].

Observatory longitude and latitude given in Section J of the Astronomical Almanac can be considered geodetic to the accuracy of the significant figures listed. Observatory elevation are listed above mean sea level. Until 1984 the Almanac gave the height displacement between the reference ellipsoid and mean sea level for a number observatories. This has been discontinued, and the definition of the reference ellipsoid has been refined in the meantime.

If geocentric coordinates for an observatory are not available directly they may be derived from geodetic coordinates using the equations given in ref [6].

Astronomical or Geographic

The observatory coordinates that can be measured with only local information are astronomical or geographic longitude and latitude. They are defined by the local gravity vector and the direction of the celestial pole. Since the gravity vector is influenced by the local distribution of mass and density near the observatory, the difference between astronomical and geodetic coordinates can be as much as an arcminute.

For the purpose of pointing a telescope, astronomical coordinates are often sufficient. The conversion from altitude and azimuth to celestial coordinates can be made perfectly accurately using astronomical longitude and latitude and the sidereal time consistent with this longitude.

However, pointing corrections to most telescopes are on the order of minutes of arc and are determined from observations of celestial objects. Hence, there is no particular advantage to using astronomical coordinates. The local vertical and any known corrections are good starting points for determining telescope pointing. Once an altitude/azimuth coordinate system is defined on the basis of celestial measurements, it can be defined to be consistent with coordinates as defined in three dimensions by VLBI or some other technique.

References

[1] Hohenkerk, C.Y., Yallop, B.D., Smith, C.A., Sinclair, A.T., 1992, "Celestial Reference Systems", Chapter 3, p. 96, Explanatory Supplement to the Astronomical Almanac, Seidelmann, P.K., Ed., U. S. Naval Observatory, University Science Books, Mill Valley, CA.

[2] ibid. p. 120

[3] Archinal, B.A., 1992, "Terrestrial Coordinates and the Rotation of the Earth", Chapter 4, p. 199, Explanatory Supplement to the Astronomical Almanac, Seidelmann, P.K., Ed., U. S. Naval Observatory, University Science Books, Mill Valley, CA.

[4] ibid. p. 203

[5] ibid. p. 220

[6] ibid. p. 206

[7] Seidelmann, P.K., Wilkins, G.A., 1992, "Introduction to Positional Astronomy", Chapter 16, p. 199, Explanatory Supplement to the Astronomical Almanac, Seidelmann, P.K., Ed., U. S. Naval Observatory, University Science Books, Mill Valley, CA.

Last updated August 2, 2010.

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