Axial Precession - Values

Values

Simon Newcomb's calculation at the end of the 19th century for general precession (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. Modern techniques such as VLBI and LLR allowed further refinements, and the International Astronomical Union adopted a new constant value in 2000, and new computation methods and polynomial expressions in 2003 and 2006; the accumulated precession is:

p_A = 5,028.796195×T + 1.1054348×T2 + higher order terms,

in arcseconds, with T, the time in Julian centuries (that is, 36,525 days) since the epoch of 2000.

The rate of precession is the derivative of that:

p = 5,028.796195 + 2.2108696×T + higher order terms.

The constant term of this speed corresponds to one full precession circle in 25,772 years.

The precession rate is not a constant, but is (at the moment) slowly increasing over time, as indicated by the linear (and higher order) terms in T. 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 values. In reality, more elaborate calculations on the numerical model of the 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 + b sin (2πT/P), where P is the 410-century period.

Theoretical models may calculate the proper constants (coefficients) corresponding to the higher powers of T, but since it is impossible for a (finite) polynomial to match a periodic function over all numbers, the error in all such approximations will grow without bound as T increases. In that respect, the International Astronomical Union chose the best-developed available theory. For up to a few centuries in the past and the future, all formulas do not diverge very much. For up to a few thousand years in the past and the future, most agree to some accuracy. For eras farther out, discrepancies become too large — the exact rate and period of precession may not be computed using these polynomials even for a single whole precession period.

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.

Precession exhibits a secular decrease due to tidal dissipation from 59"/a to 45"/a (a = annum = Julian year) during the 500 million year period centered on the present. After short-term fluctuations (tens of thousands of years) are averaged out, the long-term trend can be approximated by the following polynomials for negative and positive time from the present in "/a, where T is in milliards of Julian years (Ga):

p− = 50.475838 − 26.368583T + 21.890862T2

p+ = 50.475838 − 27.000654T + 15.603265T2

Precession will be greater than p+ by the small amount of +0.135052"/a between +30 Ma and +130 Ma. The jump to this excess over p+ will occur in only 20 Ma beginning now because the secular decrease in precession is beginning to cross a resonance in Earth's orbit caused by the other planets.

According to Ward, when, in about 1,500 million years, the distance of the Moon, which is continuously increasing from tidal effects, has increased from the current 60.3 to approximately 66.5 Earth radii, 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 with wild swings in the obliquity of the ecliptic as well. Ward, however, 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. Long before that time (about 2,100 million years from now), due to the gradually increasing luminosity of the Sun, the oceans of the Earth will have vaporized, which will reduce tidal effects significantly.