Gourt Home::Gourt :: Precession of the equinoxes
Precession of the equinoxes refers to the precession of Earth's axis of rotation in inertial space.
It was discovered by Hipparchus that the positions of the equinoxes regress (move westward) along the ecliptic compared to the fixed stars on the celestial sphere. Currently, this annual motion is about 50.3 arcseconds per year or 1 degree in 71.6 years. The process is subtle but cumulative, and over the centuries adds up to many degrees. A complete precession cycle is a period of approximately 25700 years, (the so called great Platonic year), during which time the equinox regresses over a full circle of 360°.
Changing pole stars
A consequence of the precession is a changing pole star. Currently Polaris is extremily well-suited to mark the position of the north celestial pole, as it is about a half degree away from it and it is a moderately bright star (visual magnitude is 2.1 (variable)). 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 Northstar, when it fulfilled that role around 12000 BC and will do so again around the year AD 14000. In reality it never comes closer than 5° to the pole.
When Polaris will be 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.
To find the south celestial pole in the sky at this moment, one is less lucky, 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 a properly dark sky. However that will change in the 80th to 90th century, when the south celestial pole travels through the False cross.
It is also seen from the starmap that the south pole, nicely pointed to by the Southern cross for the last 2000 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.
Polar shift and equinoxes shift
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 25700 years a small circle (blue) among the stars, centred around the ecliptic northpole (blue E) and with an angular radius of about 23.4°: the angle known as the obliquity of the ecliptic.
The orange axis was the Earth's rotation axis 5000 years ago when it pointed to the star Thuban. The yellow axis, pointing to Polaris is the situation now. Note that when the celestial sphere is seen from outside (as in the first drawing, an impossibilty of course) constellations appear in mirror image. Also note that the daily rotation of the Earth around its axis is opposite to the precessional rotation.
Of course when the polar axis precesses from one direction to another, then the equatorial plane of the Earth (indicated with the circular grid around the equator) and the associated celestial equator will move too. Where the celestial equator intersects the ecliptic (red line) there are the equinoxes. As seen from the drawing, the orange grid, 5000 years ago one intersection of equator and ecliptic, the vernal equinox was close to the star Aldebaran of Taurus. By now (the yellow grid) it has shifted (red arrow) to somewhere in the constellation of Pisces.
This is why the equinoctial shift is a consequence of the precession of the rotation axis of the Earth and the other way around. The second drawing shows the perspective from a near Earth position as seen through a very wide angle lens (from which the apparent distortion).
Explanation
The precession of the equinoxes is caused by the differential gravitation forces of Sun and Moon on Earth.
In popular scientific books one often finds this explained with the analogy of the precession of a spinning top. Indeed it is the same physical effect, however, some crucial details differ. In a spinning top it is gravity which causes the top to wobble which in its 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 Moon are in the first instance perpendicular to it. So how then can they cause it?
The answer is that the forces 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, the equatorial diameter about 43 km larger than the polar. 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 mass (blue) around its equator. Green arrows indicate the gravitational forces from the Sun on some extreme points. These forces are not parallel as they all point towards the centre 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 towards it. We find them (small cyan arrows) when the average gravitation force on the centre of the Earth is substracted (because this force will be used as the 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 the precession (they contribute to the tides). It is now clear how these tangential forces create a torque (orange), and this torque added to the rotation (magenta) shifts the rotation axis slightly to a new position (yellow). Repeat this again and again, and one sees how the axis precesses along the white circle, which is centred around the ecliptic pole.
It is important to note that the torque is always in the same direction, perpendicular onto the direction in which the rotation axis is tilted away from the ecliptic pole, so that it does not change the axial tilt itself. It is also important to note that the torque is everywhere the same, whatever position of the Earth is in its orbit around the Sun. The precession is thus always steadily progressing and does not change with the seasons.
Although the above explanation involved the Sun, the same story holds true for any object moving around the Earth along (or close to) the ecliptic, i.e. 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 25700 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 arcseconds.
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 arcseconds 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.
Climatic effects
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 (see Milankovitch cycles).
History
Hipparchus estimated Earth's precession 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 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 one year previously (sidereal year). 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.
Values
Simon Newcomb's calculation at the end of the nineteenth century for general precession in longitude: p = 5025.64 arcseconds per tropical century, 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: p = 5029.0966 arcseconds per Julian century, which with some amendments became the officially approved theory by the IAU in 2000:
p = 5028.79695 + 1.11113T - 6×10-6T² in arcseconds per Julian century, with T the time in Julian centuries (36525 days) since J2000 (JED = 2451545.0). The constant term of this speed corresponds to one full precession circle in 25772 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 but also 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 shows that the precessional constants have a period of about 41000 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 + B sin (2πT/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 polynomial can ever represent a periodic function, they all go to either plus or minus infinity for T large enough. In that respect one can understand the decision of the IAU to choose the simplest equation which agrees with most models. Up to 2000 years in the past and the future all formulas agree. Up to 4000 years in past and future most agree to some accuracy. For eras farther out discrepanies become too large.
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 independent of each other, moving in mutually perpendicular directions.
Over longer time periods, millions of years, it appears that precession is quasiperiodic around 25700 years. However this is not going to remain so. According to Ward, when the distance of the Moon, which is continuously increasing due to tidal effects, will have gone from the current 60.3 Earth radii to approximately 66.5 in about 1,500,000,000 years from now, resonance from planetary effects will push it up to 49,000 years first and then, when the Moon reaches 68 Earth radii in about 2,000,000,000 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 and 4,000,000,000 years, respectively.
References
- 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
"Precession of the equinoxes"