Editing Axial precession (section)

==Equations==
[[Image:Tidal field and gravity field.svg|thumb|Tidal force on Earth due to the Moon or another celestial body. It shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S).]]

The [[tidal force]] on Earth due to a perturbing body (Sun, Moon or planet) is expressed by [[Newton's law of universal gravitation]], whereby the gravitational force of the perturbing body on the side of Earth nearest is said to be greater than the gravitational force on the far side by an amount proportional to the difference in the cubes of the distances between the near and far sides. If the gravitational force of the perturbing body acting on the mass of the Earth as a point mass at the center of Earth (which provides the [[centripetal force]] causing the orbital motion) is subtracted from the gravitational force of the perturbing body everywhere on the surface of Earth, what remains may be regarded as the tidal force. This gives the paradoxical notion of a force acting away from the satellite but in reality it is simply a lesser force toward that body due to the gradient in the gravitational field. For precession, this tidal force can be grouped into two forces which only act on the [[equatorial bulge]] outside of a mean spherical radius. This [[couple (mechanics)|couple]] can be decomposed into two pairs of components, one pair parallel to Earth's equatorial plane toward and away from the perturbing body which cancel each other out, and another pair parallel to Earth's rotational axis, both toward the [[ecliptic]] plane.<ref>[[Ivan I. Mueller]], ''Spherical and practical astronomy as applied to geodesy'' (New York: Frederick Unger, 1969) 59.</ref> The latter pair of forces creates the following [[torque]] [[Euclidean vector|vector]] on Earth's equatorial bulge:<ref name="Williams">{{Cite journal |last1=Williams |first1=James G. |year=1994 |title=Contribution to the Earth's Obliquity Rate, Precession, and Nutation |url=https://articles.adsabs.harvard.edu/pdf/1994AJ....108..711W |journal=The Astronomical Journal |volume=108 |pages=711 |bibcode=1994AJ....108..711W |doi=10.1086/117108|s2cid=122370108 |doi-access=free }}</ref>

:<math>\overrightarrow{T} = \frac{3GM}{r^3}(C - A) \sin\delta \cos\delta \begin{pmatrix}\sin\alpha \\ -\cos\alpha \\ 0\end{pmatrix}</math>

where

:''GM'', [[standard gravitational parameter]] of the perturbing body
:''r'', geocentric distance to the perturbing body
:''C'', [[moment of inertia]] around Earth's axis of rotation
:''A'', moment of inertia around any equatorial diameter of Earth
:''C'' − ''A'', moment of inertia of Earth's equatorial bulge (''C'' > ''A'')
:''δ'', [[declination]] of the perturbing body (north or south of equator)
:''α'', [[right ascension]] of the perturbing body (east from [[March equinox]]).

The three unit vectors of the torque at the center of the Earth (top to bottom) are '''x''' on a line within the ecliptic plane (the intersection of Earth's equatorial plane with the ecliptic plane) directed toward the March equinox, '''y''' on a line in the ecliptic plane directed toward the summer solstice (90° east of '''x'''), and '''z''' on a line directed toward the north pole of the ecliptic.

The value of the three sinusoidal terms in the direction of '''x''' {{nowrap|(sin''δ'' cos''δ'' sin''α'')}} for the Sun is a [[sine squared]] waveform varying from zero at the equinoxes (0°, 180°) to 0.36495 at the solstices (90°, 270°). The value in the direction of '''y''' {{nowrap|(sin''δ'' cos''δ'' (−cos''α''))}} for the Sun is a sine wave varying from zero at the four equinoxes and solstices to ±0.19364 (slightly more than half of the sine squared peak) halfway between each equinox and solstice with peaks slightly skewed toward the equinoxes (43.37°(−), 136.63°(+), 223.37°(−), 316.63°(+)). Both solar waveforms have about the same peak-to-peak amplitude and the same period, half of a revolution or half of a year. The value in the direction of '''z''' is zero.

The average torque of the sine wave in the direction of '''y''' is zero for the Sun or Moon, so this component of the torque does not affect precession. The average torque of the sine squared waveform in the direction of '''x''' for the Sun or Moon is:

:<math>T_x = \frac{3}{2}\frac{GM}{a^3 \left(1 - e^2\right)^\frac{3}{2}}(C - A) \sin\epsilon \cos\epsilon</math>

where

:<math>a</math>, semimajor axis of Earth's (Sun's) orbit or Moon's orbit
:''e'', eccentricity of Earth's (Sun's) orbit or Moon's orbit

and 1/2 accounts for the average of the sine squared waveform, <math>a^3 \left(1 - e^2\right)^\frac{3}{2}</math> accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit,<ref>G. Boué & J. Laskar, "Precession of a planet with a satellite", ''Icarus'' '''185''' (2006) 312–330, p.329.</ref> and ε (the angle between the equatorial plane and the ecliptic plane) is the maximum value of ''δ'' for the Sun and the average maximum value for the Moon over an entire 18.6 year cycle.

Precession is:

:<math>\frac{d\psi}{dt} = \frac{T_x}{C\omega\sin\epsilon}</math>

where ''ω'' is Earth's [[angular velocity]] and ''Cω'' is Earth's [[angular momentum]]. Thus the first order component of precession due to the Sun is:<ref name=Williams/>

:<math>\frac{d\psi_S}{dt} = \frac{3}{2}\left[\frac{GM}{a^3 \left(1 - e^2\right)^\frac{3}{2}}\right]_S \left[\frac{C - A}{C}\frac{\cos\epsilon}{\omega}\right]_E</math>

whereas that due to the Moon is:

:<math>\frac{d\psi_L}{dt} = \frac{3}{2}\left[\frac{GM\left(1 - 1.5\sin^2 i\right)}{a^3 \left(1 - e^2\right)^\frac{3}{2}}\right]_L \left[\frac{C - A}{C}\frac{\cos\epsilon}{\omega}\right]_E</math>

where ''i'' is the angle between the plane of the Moon's orbit and the ecliptic plane. In these two equations, the Sun's parameters are within square brackets labeled S, the Moon's parameters are within square brackets labeled L, and the Earth's parameters are within square brackets labeled E. The term <math>\left(1 - 1.5\sin^2 i\right)</math> accounts for the inclination of the Moon's orbit relative to the ecliptic. The term {{nowrap|(''C'' − ''A'')/''C''}} is Earth's [[geodesy|dynamical ellipticity or flattening]], which is adjusted to the observed precession because Earth's internal structure is not known with sufficient detail. If Earth were [[Homogeneity (physics)|homogeneous]] the term would equal its [[angular eccentricity#Eccentricity|third eccentricity squared]],<ref>George Biddel Airy, ''[https://archive.org/details/mathematicaltra06airygoog Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics]'' (third edition, 1842) 200.</ref>

:<math>e''^2 = \frac{\mathrm{a}^2 - \mathrm{c}^2}{\mathrm{a}^2 + \mathrm{c}^2}</math>

where a is the equatorial radius ({{val|6378137|u=m}}) and c is the polar radius ({{val|6356752|u=m}}), so {{nowrap|1=''e''<sup>2</sup> = 0.003358481}}.

Applicable parameters for [[J2000.0]] rounded to seven significant digits (excluding leading 1) are:<ref name=Simon>{{Cite journal |bibcode = 1994A&A...282..663S|title = Numerical expressions for precession formulae and mean elements for the Moon and the planets|journal = Astronomy and Astrophysics|volume = 282|pages = 663|last1 = Simon|first1 = J. L.|last2 = Bretagnon|first2 = P.|last3 = Chapront|first3 = J.|last4 = Chapront-Touze|first4 = M.|last5 = Francou|first5 = G.|last6 = Laskar|first6 = J.|year = 1994}}</ref><ref name=IERS>Dennis D. McCarthy, ''[http://ilrs.gsfc.nasa.gov/docs/iers_1996_conventions.ps IERS Technical Note 13 – IERS Standards (1992)]'' (Postscript, use [https://www.xconvert.com/convert-ps-to-pdf XConvert]).</ref>
{| class=wikitable
! Sun !! Moon !! Earth
|-
|''GM'' = 1.3271244{{E|20}} m<sup>3</sup>/s<sup>2</sup>
|''GM'' = 4.902799{{E|12}} m<sup>3</sup>/s<sup>2</sup>
|(''C'' − ''A'')/''C'' = 0.003273763
|-
|
|''a'' = 3.833978{{E|8}} m
|''a'' = 1.4959802{{E|11}} m
|-
|
|''e'' = 0.05554553
|''e'' = 0.016708634
|-
|
|''i'' = 5.156690°
|ε = 23.43928°
|-
|
|
|''ω'' = 7.292115{{E|−5}} rad/s
|}

which yield
:''dψ<sub>S</sub>/dt'' = 2.450183{{E|−12}} /s
:''dψ<sub>L</sub>/dt'' = 5.334529{{E|−12}} /s

both of which must be converted to ″/a (arcseconds/annum) by the number of [[arcsecond]]s in 2[[Pi|π]] [[radian]]s (1.296{{E|6}}″/2π) and the number of [[second]]s in one [[annus]] (a [[Julian year (astronomy)|Julian year]]) (3.15576{{E|7}}s/a):
:''dψ<sub>S</sub>/dt'' = 15.948788″/a &nbsp; vs &nbsp; 15.948870″/a from Williams<ref name=Williams/>
:''dψ<sub>L</sub>/dt'' = 34.723638″/a &nbsp; vs &nbsp; 34.457698″/a from Williams.

The solar equation is a good representation of precession due to the Sun because Earth's orbit is close to an ellipse, being only slightly perturbed by the other planets. The lunar equation is not as good a representation of precession due to the Moon because the Moon's orbit is greatly distorted by the Sun and neither the radius nor the eccentricity is constant over the year.