# Apsidal precession

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In celestial mechanics, **perihelion precession**, **apsidal precession** or **orbital precession** is the precession (rotation) of the orbit of a celestial body. More precisely it is the gradual rotation of the line joining the apsides of an orbit, which are the points of closest and farthest approach. Perihelion is the closest point to the Sun.

## History

The ancient Greek astronomersTemplate:Who noted the apsidal precession of the Moon's orbit; the precession of the solar apsides was discovered in the eleventh century by al-Zarqālī.^{[1]} These remained difficult to account for until the 19th century.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Calculation

There are a variety of factors which can lead to periastron precession, such as general relativity, stellar quadrupole moments, mutual star–planet tidal deformations, and perturbations from other planets.^{[2]}

˙ωtotal = ˙ωGR + ˙ωquad + ˙ωtide + ˙ωpert,

For Mercury, the perihelion precession rate due to general relativistic effects is 43'' per century. By comparison, the precession due to perturbations from the other solar system planets is 532" per century while the oblateness of the Sun (quadrupole moment) causes a negligible contribution of 0".025 per century.^{[3]}^{[4]}

From classical mechanics, if stars and planets are considered to be purely spherical masses, then they will obey a simple r^{-2} force law and hence execute closed elliptical orbits. Non-spherical mass effects are caused by the application of external potential(s): the centrifugal potential of spinning bodies causes rotational flattening and the tidal potential of a nearby mass raises tidal bulges. Rotational and tidal bulges create gravitational quadrupole fields (r^{-3}) that lead to orbital precession

Total apsidal precession broadly in order of importance for isolated very hot Jupiters is (considering only lowest order effects)

ωtot= ̇ωtid,p+ ̇ωGR+ ̇ωrot,p+ ̇ωrot,∗+ ̇ωtid,∗

with planetary tidal bulge being the dominant term, exceeding the effects of general relativity and the stellar quadrupole by more than an order of magnitude and thus can help us in understanding their interiors. For the shortest-period planets, the planetary interior induces precession of a few degrees per year and up to 19◦.9 per year for WASP-12b.^{[5]}^{[6]}

## Newton's theorem of revolving orbits

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Newton derived an intriguing theorem showing that variations in the angular motion of a particle can be accounted for by the addition of a force that varies as the inverse cube of distance, without affecting the radial motion of a particle. Using a forerunner of the Taylor series, Newton generalized his theorem to all force laws provided that the deviations from circular orbits is small, which is valid for most planets in the Solar System. However, his theorem did not account for the apsidal precession of the Moon without giving up the inverse-square law of Newton's law of universal gravitation.

## Perturbation theory

The expected rate of apsidal precession can be calculated more accurately using the methods of perturbation theory.

## General relativity

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An apsidal precession of the planet Mercury was noted by Urbain Le Verrier in the mid-19th century and accounted for by Einstein's theory of general relativity. To first approximation, this theory adds a central force that varies as the inverse fourth power of the distance.

Einstein showed that for a planet, the major semi-axis of its orbit being , the eccentricity of the orbit *e* and the period of revolution *T*, then the apsidal precession due to relativistic effects, during one period of revolution in radians, is

where *c* is the speed of light.^{[7]} In the case of Mercury, half of the greater axis is circa 57,9 million kilometers or 57,9 · 10^{9} m, the eccentricity of its orbit is 0,206 and the period of revolution 87,97 days, or 7,6 10^{6} s. From these and the speed of light (which is 3 10^{8} m/s), it can be calculated that the apsidial precession during one period of revolution is = 5,028 · 10^{-7} radians, 2,88 · 10^{-5} degrees or 0,104 arc seconds. In one hundred years, Mercury makes circa 415 revolutions around the Sun, and thus in that time, the apsidal perihelion due to relativistic effects is circa 43 arc seconds, which corresponds almost exactly to the previously unexplained part of the measured value.

## Long-term climate

Because of apsidal precession the Earth's argument of periapsis slowly increases; it takes about over 134,000 years for the ellipse to revolve once relative to the fixed stars. The Earth's polar axis, and hence the solstices and equinoxes, precess with a period of about 25,771.4 years in relation to the fixed stars. These two forms of 'precession' combine so that it takes over 21,600 years for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly).^{[8]}

This interaction between the anomalistic and tropical cycle is important in the long-term climate variations on Earth, called the Milankovitch cycles. An equivalent is also known on Mars.

The figure illustrates the effects of precession on the northern hemisphere seasons, relative to perihelion and aphelion. Notice that the areas swept during a specific season changes through time. Orbital mechanics require that the length of the seasons be proportional to the swept areas of the seasonal quadrants, so when the orbital eccentricity is extreme, the seasons on the far side of the orbit may be substantially longer in duration.

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}, at pp. 314-17.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Kane, S. R., Horner, J., & von Braun, K. (2012). Cyclic Transit Probabilities of Long-period Eccentric Planets due to Periastron Precession. The Astrophysical Journal, 757(1), 105.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Ragozzine, D., & Wolf, A. S. (2009). Probing the interiors of very hot Jupiters using transit light curves. The Astrophysical Journal, 698(2), 1778.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}