Monday, November 28, 2005

Precession of Earth's Rotation

Precession of the equinoxes

The Earth goes through one complete precession cycle in a period of approximately 25,800 years, during which the positions of stars as measured in the equatorial coordinate system will slowly change; the change is actually due to the change of the coordinates. Over this cycle the Earth's north axial pole moves from where it is now, within 1° of Polaris, in a circle around the ecliptic pole, with an angular radius of 23 degrees 27 arcminutes [1], or about 23.5 degrees. The shift is 1 degree in 180 years (the angle is taken from the observer, not from the center of the circle).

The explanation of this is: The axis of the Earth undergoes precession due to a combination of the Earth's nonspherical shape (it is an oblate spheroid, bulging outward at the equator) and the gravitational tidal forces of the Moon and Sun applying torque as they attempt to pull the equatorial bulge into the plane of the ecliptic. The portion of the precession due to the combined action of the Sun and the Moon is called lunisolar precession.

A changing north star

Polaris is not particularly well-suited for marking the north celestial pole, as its visual magnitude, which is variable, hovers around 2.1, fairly far down the list of brightest stars in the sky. On the other hand, in 3000 BC the faint star Thuban in the constellation Draco was the pole star; at magnitude 3.67 it is five times fainter than Polaris; today it is all but invisible in light-polluted urban skies. The brightest star known to have been North Star or to be predictable as taking that role in the future is the brilliant Vega in the constellation Lyra, which will be the pole star around the year AD 14,000. When viewed looking down onto the Earth from the north, the direction of precession is clockwise. When standing on Earth looking outward, the axis appears to move counter-clockwise across the sky. This sense of precession, against the sense of Earth's own axial rotation, is opposite to the precession of a top on a table. The reason is that the torques imposed on the Earth by the Sun and Moon act in the sense of trying to align its axis normal to the ecliptic, i.e. to stand up more vertically in regards to the ecliptic plane, while the torque on a top spinning on a hard surface acts in the sense of trying to make the top fall over, rather than to stand up straighter.

Image:precession starchart.jpg

Polaris is not exactly at the pole; any long-exposure unguided photo will show it having a short trail. It is close enough for most practical purposes, though. The south celestial pole precesses too, always remaining exactly opposite the north pole. The south pole is in a particularly bland portion of the sky, and the nominal south pole star is Sigma Octantis, which, while fairly close to the pole, is even weaker than Thuban -- magnitude 5.5, which is barely visible even under a properly dark sky. The precession of the Earth is not entirely regular due to the fact that the Sun and Moon are not in the same plane and move relative to each other, causing the torque they apply to Earth to vary. This varying torque produces a slight irregular motion in the poles called nutation.

Precession of the Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account. Note that precession has no effect on the inclination ("tilt") of the plane of the Earth's equator (and thus its axis of rotation) on its orbital plane. It is 23.5 degrees and precession does not change that. The inclination of the equator on the ecliptic does change due to gravitational torque, but its period is different (main period about 41000 years).

The following figure illustrates the effects of axial precession on the seasons, relative to perihelion and aphelion. The precession of the equinoxes can cause periodic climate change (see Milankovitch cycles), because the hemisphere that experiences summer at perihelion and winter at aphelion (as the southern hemisphere does presently) is in principle prone to more severe seasons than the opposite hemisphere.

Image:precession and seasons.jpg


Hipparchus first estimated Earth's precession around 130 BC, adding his own observations to those of Babylonian and Chaldean astronomers in the preceding centuries. In particular they measured the distance of the stars like Spica to the Moon and Sun at the time of lunar eclipses, and because he could compute the distance of the Moon and Sun from the equinox at these moments, he noticed that Spica and other stars appeared to have moved over the centuries.

Precession causes the cycle of seasons (tropical year) to be about 20.4 minutes less than the period for the earth to return to the same position with respect to the stars as one year previously (sidereal year). This results in a slow change (one day per 58 calendar years) in the position of the sun with respect to the stars at an equinox. It is significant for calendars and their leap year rules.



Demonstration of how the Earth's equatorial bulge causes precession
Gyroscope (a bicycle wheel works well), turntable (optional)
Background and Demonstration:
Due to the Earth's rotation, there is a slight equatorial bulge (and an offsetting depression at the poles). The maximum deviation from a sphere is only about 15 km, or 0.2% of the spherical radius. Nevertheless, since the axis of rotation of the Earth is inclined (tipped) relative to the plane of the Sun, Moon and planets (the ecliptic), the gravitational effect of the Moon (and to a lesser extent the Sun and planets) on the equatorial bulge causes a torque to be applied to the Earth. This is a rotational force in the direction that would decrease the inclination of the rotation axis.

As it spins, the Earth behaves somewhat like a gyroscope; it wants to maintain it's orientation (it is difficult to change it's direction). A bicycle wheel (particularly one with a loaded rim) is excellent as a demonstration gyroscope, although small toy gyroscopes work also. If we apply a torque to the gyroscope (balance the spinning bicycle wheel on one hand, and pull a string attached to top axle with the other), it's axis of rotation does not move toward you (the direction you are pulling), but perpendicular to that so that the axis traces out a circle. This is called precession. Another simple example of precession is the spinning of a top; because of variations in density of the top, its axis of rotation traces out a circle. [Note: in Physics labs, this demonstration is often done by hanging the bicycle wheel from the string, with its axis horizontal. This works well, but I think keeping the spinning axis vertical helps the students to visualize the Earth's coordinate system.]

We can see that the Earth's axis of rotation precesses, since we know that the North Star (Polaris) was not aligned with the rotational pole in the past (it wasn't the "North Star"), and appears to be moving away from that position, so that it won't be aligned in the future. By observing the apparent movement of the stars with respect to the rotational axis, we can determine that Earth's period of precession is almost 25,735 years (it will take 25,735 years for Polaris to become the North Star once again).

The quantity that relates the rate of precession to the amount of torque applied is the moment of inertia. The moment of inertia is related to the distribution of mass about the axis of rotation. If the mass is concentrated near the axis, the moment of inertia will be small, but if the mass is distributed outward, the moment of inertia will be large. For a constant torque, a small moment of inertia will result in a rapid rate of rotation. Ice skaters make use of this principle in their spins, and you can demonstrate this if you have a turntable that you can stand on. Hold your arms out and begin spinning; if you pull your arms in, you will spin much faster (be careful).

Since we have good estimates of the mass and distance of the Moon, and we can observe the rate of precession of the Earth, we may determine the moment of inertia of the Earth. It is 8.07 x 1037 kg-m2, or 0.33 M R2 (where M and R are the Earth's mass and average radius, respectively). A homogeneous sphere would have a moment of inertia of 0.4 M R2, so this indicates that the mass of the Earth is concentrated toward it's center (density increases inward). More importantly, however, the Earth's moment of inertia puts a very tight constraint on how the density increases inward.

Positions From StarCalc of

Current Pole Star
Alp Ursae Minoris (1; Alp UMi; HR 424)
Current apparent equat. coordinates:
Alp: 2h 39m 38.1s Del: +89° 17' 38"
Equat. coordinates (J2000):
Alp: 2h 31m 48.7s Del: +89° 15' 51"
V: 2.02m; Parallax: 0.007"
Spectrum: F7:Ib-IIv; B-V: 0.60m binary.; spec.-binary.

Position of Seven Rishi's

Sage Sayana Position In Hours (Ascention time)
Kratuh 165:55:56 11:03:44
Pulahah 165:27:37 11:01:50.50
Pulastya 178:31:34 11:54:06.06
Atri 183:55:14 12:15:41.41
Angirasa 193:33:48 12:54:15.15
Vasishta 201:01:55 13:24:08.08
Marichi 206:56:06 13:47:44.44

»Pole Star ON

21 March 2141 BC

Alpha Draconis (11; 6546; HR 5291)
Current apparent equat. coordinates:
Alp: 14h 4m 29.4s Del: +64° 20' 58"
Equat. coordinates (J2000):
Alp: 14h 4m 23.3s Del: +64° 22' 33"
V: 3.65m; Parallax: 0.018"
Spectrum: A0III; B-V: -0.05m spec.-binary.


Earth Rotation and Equatorial Coordinates


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.


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].


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 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 )


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 / 298.275
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.


[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 February 5, 1996.

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