Precession of the equinoxes
The
precession of the equinoxes refers to the
precession of
Earth's axis of rotation with respect to
inertial space.
It was discovered by
Hipparchus that the positions of the
equinoxes move westward along the
ecliptic compared to the fixed stars on the
celestial sphere. Currently, this annual motion is about 50.3 seconds of arc per year or 1 degree every 71.6 years. The process is slow but cumulative. A complete precession cycle covers a period of approximately 25,700 years, (the so called
great Platonic year), during which time the equinox regresses over a full 360°. Precessional movement is also the determining factor in the length of an
Astrological_Age.
|
Precession of Earth's axis around the north ecliptical pole |
|
Precession of Earth's axis around the south ecliptical pole |
A consequence of the precession is a changing pole star. Currently
Polaris is extremely well-suited to mark the position of the north celestial pole, as Polaris is moderately bright star (visual
magnitude is 2.1 (variable)), and it is located within a half degree of the pole. On the other hand,
Thuban in the
constellation Draco, which was the pole star in 3000 BC is much less conspicious at magnitude 3.67 (one-fifth as bright as Polaris); today it is all but invisible in light-polluted urban skies. The brilliant
Vega in the constellation
Lyra is often touted as the best north star (when it fulfilled that role around 12000 BC and will do so again around the year AD 14000), however, it never comes closer than 5° to the pole.
When Polaris becomes the north star again around 27800 AD, due to its
proper motion it will be farther away from the pole then than it is now, while in 23600 BC it came closer to the pole.
It is more difficult to find the south celestial pole in the sky at this moment, as that area is a particularly bland portion of the sky, and the nominal south pole star is
Sigma Octantis, which with magnitude 5.5 is barely visible even under ideal conditions. However that will change from the eightieth to the nintieth centuries, when the south celestial pole travels through the
False Cross.
It is also seen from a starmap that the south pole, which has been nicely pointed to by the
Southern cross for the last 2,000 years or so, is moving towards that constellation. By consequence it is now no longer visible from subtropical northern latitudes, as it was in the time of the
ancient Greeks.
Still pictures like these, found in many astronomy books, are only first approximations as they do not take into account the variable speed of the precession, the variable obliquity of the ecliptic, the planetary precession (which makes not the ecliptic pole the centre, but a circle about 6° away from it) and the proper motions of the stars.
|
Precessional movement as seen from 'outside' the celestial sphere |
|
Same picture as above but now from (near) Earth perspective |
It might not be directly clear to the non-astronomer what the shift of the equinoxes has to do with the precession of the rotation axis of the Earth. The figures to the right try to explain that.
The rotation axis of the Earth describes, over a period of 25,700 years, a small circle (blue) among the stars, centred around the
ecliptic northpole (the blue
E) and with an angular radius of about 23.4°, an angle known as the
obliquity of the ecliptic.
The orange axis was the Earth's rotation axis 5,000 years ago, when it pointed to the star Thuban. The yellow axis, pointing to Polaris, is the situation now. When the
celestial sphere is seen from the outside (as it is in the first drawing, although such a perspective is impossible), the constellations appear in mirror image. Furthermore, the daily rotation of the Earth around its axis is opposite to the precessional rotation.
When the polar axis
precesses from one direction to another, the equatorial plane of the Earth (indicated by the circular grid around the equator) and the associated celestial equator moves too. Where the celestial equator intersects the ecliptic (red line) there are the equinoxes. As seen from the the orange grid, 5,000 years ago, the
vernal equinox was close to the star
Aldebaran of
Taurus. Now, as seen from the yellow grid, it has shifted (indicated by the red arrow) to somewhere in the constellation of
Pisces.
This is why the equinoctal shift is a consequence of the precession of the rotation axis of the Earth, as well as vice versa. The second drawing shows the perspective of a near-Earth position as seen through a very wide angle lens (from which the apparent distortion).
|
The precession as a consequence of the torque exerted on Earth by differential gravitation. |
The precession of the equinoxes is caused by the differential gravitational forces of the
Sun and the
Moon on the Earth.
In popular science books, precession is often explained by an analogy to a spinning top. While, the physical effect is the same, however, some crucial details differ. In a spinning top, gravity causes the top to wobble which, in turn, causes precession. The applied force is thus in the first instance parallel to the rotation axis. But for the Earth the applied forces of the Sun and the Moon are in the first instance perpendicular to it.
Thus, the Sun and the Moon do not work on the rotation axis. Instead they work on the equatorial bulge; due to its own rotation, the Earth is not a perfect sphere but an
oblate spheroid, with an equatorial diameter about 43 kilometers larger than its polar diameter. If the Earth were a perfect sphere, there would be no precession.
The figure explains how this works. The Earth is given as a perfect sphere (so that all gravitational forces working on it can be taken equal as one force working on its centre), and the bulge is approximated to be a torus of blue mass around its equator. The green arrows indicate the gravitational forces from the Sun on some extreme points. These forces are not parallel as they all point toward the center of the Sun. Therefore, the forces working on the northernmost and southernmost parts of the equatorial bulge have a component perpendicular on the ecliptical plane and directed toward it. We find them, the small cyan arrows, when the average gravitational force on the center of the Earth is subtracted, because this force will be used as centripetal force for the Earth in its orbit around the Sun. In all cases, in addition to these tangential components, there will be also radial components, but they are not shown as they do not contribute to precession (they contribute to the tides). These tangential forces create a
torque (orange), and this torque added to the rotation (magenta) shifts the rotational axis to a slightly new position (yellow). Over time, the axis precesses along the white circle, which is centered around the ecliptic pole.
This torque is always in the same direction, perpendicular to the direction in which the rotation axis is tilted away from the ecliptic pole, so that it does not change the axial tilt itself. The magnitude of the torque from the sun (or the moon) varies with the gravitational object's alignment with the earth's spin axis and approaches zero when it is orthogonal.
Although the above explanation involved the Sun, the same explanation holds true for any object moving around the Earth along or close to the ecliptic, notably, the Moon. The combined action of the Sun and the Moon is called the lunisolar precession. In addition to the steady progressive motion (resulting in a full circle in 25,700 years) the Sun and Moon also cause small periodic variations, due to their changing positions. These oscillations, in both precessional speed and axial tilt, are known as the
nutation. The most important term has a period of 18.6 years and an amplitude of less than 20 seconds of arc.
In addition to lunisolar precession, the actions of the other planets of the solar system cause the whole ecliptic to slowly rotate around an axis which has an ecliptic longitude of about 174° measured on the instantaneous ecliptic. This planetary precession shift is only 0.47 seconds of arc per year (more than a hundred times smaller than lunisolar precession), and takes place along the instantaneous equator.
The sum of the two precessions is known as the general precession.
|
Effects of axial precession on the seasons |
This figure illustrates the effects of axial precession on the seasons, relative to
perihelion and
aphelion. The precession of the equinoxes contributes to periodic
climate change, and is known as the
Milankovitch cycle.
Hipparchus estimated the Earth's precession in around
130 BC, adding his own observations to those of Babylonian astronomers in the preceding centuries. In particular, they measured the distance of
stars like
Spica to the Moon and the Sun during
lunar eclipses, and because he could compute the distance of the Moon and the 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 time for the Earth to return to the same position with respect to the stars. This results in a slow change (one day every 71 calendar years) in the position of the Sun with respect to the stars at an equinox.
|
The steady westward shift of the vernal equinox among the stars is evident over the millennia. |
Simon Newcomb's calculation at the end of the nineteenth century for general precession (known as
p) in longitude gave a value of 5,025.64 arcseconds per tropical century, and was the generally accepted value until artificial satellites delivered more accurate observations and electronic computers allowed more elaborate models to be calculated.
Lieske developed an updated theory in 1976, where
p equals 5,029.0966 arcseconds per Julian century, which with some amendments became the officially approved theory by the
International Astronomical Union in 2000:
p = 5,028.79695 + 1.11113
T âˆ' (6 × 10
-6)T² in arcseconds per Julian century, with
T, the time in Julian centuries (that is, 36,525 days) since
the epoch of 2000. The constant term of this speed corresponds to one full precession circle in 25,772 years.
The precession is not a constant but slowly increasing over time because of the linear term in
T. Still, this increase is diminishing due to the quadratic term. In any case it must be stressed that this formula is only valid over a
limited time period. It is clear that if
T gets large enough (far in the future or far in the past), the
T² term will dominate and
p will go to very large negative values. In reality, more elaborate calculations on the
numerical model of solar system show that the precessional
constants have a period of about 41,000 years, the same as the obliquity of the ecliptic. Note that the
constants mentioned here are the linear and all higher terms of the formula above, not the precession itself. That is,
p =
A +
BT +
CT² + … is an approximation of
p =
A +
Bsin (2Ï€
T/
P), where
P is the 410-century period.
Other theoretical models may calculate values for
p that have higher powers of
T, but since no (finite) polynomial can ever represent a periodic function, they all go to either positive or negative infinity for large enough
T. In that respect, the International Astronomical Union chose the simplest equation which agrees with most models. For up to 2,000 years in the past and the future, all formulas agree. For up to 4,000 years in the past and the future, most agree to some accuracy. For eras farther out, discrepanies become too large.
The precession of 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 on a daily basis. Note that although the precession and the tilt of Earth's axis (the obliquity of the ecliptic) are calculated from the same theory and thus are related to each other, the two movements act independently of each other, moving in mutually perpendicular directions.
Over longer time periods, that is millions of years, it appears that precession is quasiperiodic at around 25,700 years. However, it will not remain so. According to Ward, when the distance of the Moon, which is continuously increasing from tidal effects, will have gone from the current 60.3 to approximately 66.5 Earth radii in about 1,500 million years, resonances from planetary effects will push precession to 49,000 years at first and then, when the Moon reaches 68 Earth radii in about 2,000 million years, to 69,000 years. This will be associated by wild swings in the obliquity of the ecliptic as well. However, Ward used the abnormally large modern value for tidal dissipation. Using the 620-million year average provided by
tidal rhythmites of about half the modern value, these resonances will not be reached until about 3,000 and 4,000 million years, respectively. However, long before that time (about 2,100 million years from now), due to the increasing luminosity of the Sun, the oceans of the Earth will have boiled away, which will alter tidal effects significantly (see
Earth's future).
*
Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac*
Precession and the Obliquity of the Ecliptic has a comparison of values predicted by different theories
* A.L. Berger, "Obliquity & precession for the last 5 million years",
Astronomy & astrophysics 51 (1976) 127
* W.R. Ward, "Comments on the long-term stability of the earth's obliquity",
Icarus 50 (1982) 444
