Editing Axial precession (section)

==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. [[Jay Henry Lieske]] developed an updated theory in 1976, where ''p'' equals 5,029.0966 arcseconds (or 1.3969713 degrees) per Julian century. Modern techniques such as [[VLBI]] and [[Lunar laser ranging|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:<ref name=Capitaine2003>[http://syrte.obspm.fr/iau2006/aa03_412_P03.pdf N. Capitaine ''et al.'' 2003], p. 581 expression 39</ref>

:''p<sub>A</sub>'' = 5,028.796195{{nnbsp}}''T'' + 1.1054348{{nnbsp}}''T''<sup>2</sup> + higher order terms, in arcseconds, with ''T'', the time in Julian centuries (that is, 36,525 days) since [[J2000|the epoch of 2000]].

The '''rate''' of precession is the derivative of that:

:''p'' = 5,028.796195 + 2.2108696{{nnbsp}}''T'' + higher order terms.

The constant term of this speed (5,028.796195 arcseconds per century in above equation) corresponds to one full precession circle in 25,771.57534 years (one full circle of 360 degrees divided by 50.28796195 arcseconds per year)<ref name=Capitaine2003/> although some other sources put the value at 25771.4 years, leaving a small uncertainty.

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 a polynomial expression centred on the J2000 datum, empirically fitted to observational data, not on a deterministic model of the [[Solar System]]. It is clear that if ''T'' gets large enough (far in the future or far in the past), the ''T''² 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 rate has a period of about 41,000 years, the same as the obliquity of the ecliptic. That is,
:''p'' = ''A'' + ''BT'' + ''CT''<sup>2</sup> + …
is an approximation of
:''p'' = ''a'' + ''b'' sin (2π''T''/''P''), where ''P'' is the 41,000-year period.

Theoretical models may calculate the constants (coefficients) corresponding to the higher powers of ''T'', but since it is impossible for a polynomial to match a periodic function over all numbers, the difference in all such approximations will grow without bound as ''T'' increases. Sufficient accuracy can be obtained over a limited time span by fitting a high enough order polynomial to observation data, rather than a necessarily imperfect dynamic numerical model.{{clarify|date=January 2022}} For present flight trajectory calculations of artificial satellites and spacecraft, the polynomial method gives better accuracy. In that respect, the [[International Astronomical Union]] has chosen the best-developed available theory. For up to a few centuries into the past and future, none of the formulas used 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. Although the precession and the tilt of Earth's axis (the obliquity of the ecliptic) are calculated from the same theory and are thus related one to the other, the two movements act independently of each other, moving in opposite directions.{{clarify|date=January 2022}}

Precession rate exhibits a secular decrease due to [[tidal acceleration|tidal dissipation]] from 59"/a to 45"/a (a = [[annus]] = [[Julian year (astronomy)|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 [[1,000,000,000|billion]]s of Julian years (Ga):<ref>{{Cite journal |doi = 10.1051/0004-6361:20041335|title = A long-term numerical solution for the insolation quantities of the Earth|journal = Astronomy & Astrophysics|volume = 428|pages = 261–285|year = 2004|last1 = Laskar|first1 = J.|last2 = Robutel|first2 = P.|last3 = Joutel|first3 = F.|last4 = Gastineau|first4 = M.|last5 = Correia|first5 = A. C. M.|last6 = Levrard|first6 = B.|bibcode = 2004A&A...428..261L|doi-access = free}}</ref>
:''p''{{sup|−}} = 50.475838 − 26.368583{{nnbsp}}''T'' + 21.890862{{nnbsp}}''T''<sup>2</sup>
:''p''{{sup|+}} = 50.475838 − 27.000654{{nnbsp}}''T'' + 15.603265{{nnbsp}}''T''<sup>2</sup>

This gives an average cycle length now of 25,676 years.

Precession will be greater than ''p''{{sup|+}} by the small amount of +0.135052"/a between {{nowrap|+30 Ma}} and {{nowrap|+130 Ma}}. The jump to this excess over ''p''{{sup|+}} will occur in only {{nowrap|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 W. R. Ward, in about 1,500 million years, when 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&nbsp;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.<ref>{{cite journal| first1=W. R.|last1= Ward|year=1982 |title=Comments on the long-term stability of the earth's obliquity |journal=Icarus|volume=50|issue= 2–3|pages=444–448 | bibcode=1982Icar...50..444W|doi = 10.1016/0019-1035(82)90134-8 }}</ref> Using the 620-million year average provided by [[tidal acceleration#Historical evidence|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, due to the gradually increasing luminosity of the Sun, the oceans of the Earth will have vaporized before that time (about 2,100&nbsp;million years from now).