# Apsidal precession

Each planet orbiting the Sun follows an elliptic orbit that gradually rotates over time (apsidal precession). This figure illustrates positive apsidal precession (advance of the perihelion), with the orbital axis turning in the same direction as the planet's orbital motion. The eccentricity of this ellipse and the precession rate of the orbit are exaggerated for visualization. Most orbits in the Solar System have a much lower eccentricity and precess at a much slower rate, making them nearly circular and stationary.
The main orbital elements (or parameters). The line of apsides is shown in blue, and denoted by ω. The apsidal precession is the rate of change of ω through time, dω/dt.
Animation of Moon's orbit around Earth - Polar view
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In celestial mechanics, apsidal precession (or apsidal advance)[1] is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its primary body. The apsidal precession is the first time derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°.[2]

## History

The ancient Greek astronomer Hipparchos noted the apsidal precession of the Moon's orbit;[3] it is corrected for in the Antikythera Mechanism (circa 80 BCE) with the very accurate value of 8.88 years per full cycle, correct to within 0.34% of current measurements.[4] The precession of the solar apsides was discovered in the eleventh century by al-Zarqālī.[5] The lunar apsidal precession was not accounted for in Claudius Ptolemy's Almagest, and as a group these precessions, the result of a plethora of phenomena, remained difficult to account for until the 20th century when the last unidentified part of Mercury's precession was precisely explained.

## Calculation

A variety of factors can lead to periastron precession such as general relativity, stellar quadrupole moments, mutual star–planet tidal deformations, and perturbations from other planets.[6]

ωtotal = ωGeneral Relativity + ωquadrupole + ωtide + ωperturbations

For Mercury, the perihelion precession rate due to general relativistic effects is 43″ (arcseconds) per century. By comparison, the precession due to perturbations from the other planets in the Solar System is 532″ per century, whereas the oblateness of the Sun (quadrupole moment) causes a negligible contribution of 0.025″ per century.[7][8]

From classical mechanics, if stars and planets are considered to be purely spherical masses, then they will obey a simple 1/r2 inverse-square law, relating force to distance and hence execute closed elliptical orbits according to Bertrand's theorem. Non-spherical mass effects are caused by the application of external potential(s): the centrifugal potential of spinning bodies like the spinning of pizza dough causes flattening between the poles and the gravity of a nearby mass raises tidal bulges. Rotational and net tidal bulges create gravitational quadrupole fields (1/r3) that lead to orbital precession.

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

ωtotal = ωtidal perturbations + ωGeneral Relativity + ωrotational perturbations + ωrotational * + ωtidal *

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. The good resulting approximation of the tidal bulge is useful for understanding the interiors of such planets. For the shortest-period planets, the planetary interior induces precession of a few degrees per year. It is up to 19.9° per year for WASP-12b.[9][10]

## Newton's theorem of revolving orbits

Newton derived an early theorem which attempted to explain apsidal precession. This theorem is historically notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995.[11] Newton proposed 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 are 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. Additionally, the rate of apsidal precession calculated via Newton's theorem of revolving orbits is not as accurate as it is for newer methods such as by perturbation theory.

change in orbit over time
Effects of apsidal precession on the seasons with the eccentricity and ap/peri-helion in the orbit exaggerated for ease of viewing. The seasons shown are in the northern hemisphere and the seasons will be reverse in the southern hemisphere at any given time during orbit. Some climatic effects follow chiefly due to the prevalence of more oceans in the Southern Hemisphere.

## General relativity

An apsidal precession of the planet Mercury was noted by Urbain Le Verrier in the mid-19th century and accounted for by Einstein's general theory of relativity.

Einstein showed that for a planet, the major semi-axis of its orbit being a, 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

${\displaystyle \varepsilon =24\pi ^{3}{\frac {a^{2}}{T^{2}c^{2}\left(1-e^{2}\right)}}}$

where c is the speed of light.[12] In the case of Mercury, half of the greater axis is about 5.79×1010 m, the eccentricity of its orbit is 0.206 and the period of revolution 87.97 days or 7.6×106 s. From these and the speed of light (which is ~3×108 m/s), it can be calculated that the apsidal precession during one period of revolution is ε = 5.028×10−7 radians (2.88×10−5 degrees or 0.104″). In one hundred years, Mercury makes approximately 415 revolutions around the Sun, and thus in that time, the apsidal perihelion due to relativistic effects is approximately 43″, which corresponds almost exactly to the previously unexplained part of the measured value.

## Long-term climate

Earth's apsidal precession slowly increases its argument of periapsis; it takes about 112,000 years for the ellipse to revolve once relative to the fixed stars.[13] Earth's polar axis, and hence the solstices and equinoxes, precess with a period of about 26,000 years in relation to the fixed stars. These two forms of 'precession' combine so that it takes between 20,800 and 29,000 years (and on average 23,000 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).[14]

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 to the right 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.

• Axial precession
• Nodal precession
• Hypotrochoid
• Rosetta (orbit)
• Spirograph

## Notes

1. ^ Bowler, M. G. (2010). "Apsidal advance in SS 433?". Astronomy and Astrophysics. 510 (1): A28. arXiv:0910.3536. Bibcode:2010A&A...510A..28B. doi:10.1051/0004-6361/200913471. S2CID 119289498.
2. ^ Hilditch, R. W. (2001). An Introduction to Close Binary Stars. Cambridge astrophysics series. Cambridge University Press. p. 132. ISBN 9780521798006.
3. ^ Jones, A., Alexander (September 1991). "The Adaptation of Babylonian Methods in Greek Numerical Astronomy" (PDF). Isis. 82 (3): 440–453. Bibcode:1991Isis...82..441J. doi:10.1086/355836. S2CID 92988054.
4. ^ Freeth, Tony; Bitsakis, Yanis; Moussas, Xenophon; Seiradakis, John. H.; Tselikas, A.; Mangou, H.; Zafeiropoulou, M.; Hadland, R.; et al. (30 November 2006). "Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism" (PDF). Nature. 444 Supplement (7119): 587–91. Bibcode:2006Natur.444..587F. doi:10.1038/nature05357. PMID 17136087. S2CID 4424998. Archived from the original (PDF) on 20 July 2015. Retrieved 20 May 2014.
5. ^ Toomer, G. J. (1969), "The Solar Theory of az-Zarqāl: A History of Errors", Centaurus, 14 (1): 306–336, Bibcode:1969Cent...14..306T, doi:10.1111/j.1600-0498.1969.tb00146.x, at pp. 314–317.
6. ^ David M. Kipping (8 August 2011). The Transits of Extrasolar Planets with Moons. Springer. pp. 84–. ISBN 978-3-642-22269-6. Retrieved 27 August 2013.
7. ^ 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. arXiv:1208.4115. Bibcode:2012ApJ...757..105K. doi:10.1088/0004-637x/757/1/105. S2CID 54193207.
8. ^ Richard Fitzpatrick (30 June 2012). An Introduction to Celestial Mechanics. Cambridge University Press. p. 69. ISBN 978-1-107-02381-9. Retrieved 26 August 2013.
9. ^ Ragozzine, D.; Wolf, A. S. (2009). "Probing the interiors of very hot Jupiters using transit light curves". The Astrophysical Journal. 698 (2): 1778–1794. arXiv:0807.2856. Bibcode:2009ApJ...698.1778R. doi:10.1088/0004-637x/698/2/1778. S2CID 29915528.
10. ^ Michael Perryman (26 May 2011). The Exoplanet Handbook. Cambridge University Press. pp. 133–. ISBN 978-1-139-49851-7. Retrieved 26 August 2013.
11. ^ Chandrasekhar, p. 183.
12. ^ Hawking, Stephen (2002). On the Shoulders of Giants : the Great Works of Physics and Astronomy. Philadelphia, Pennsylvania, USA: Running Press. pp. der Physik. ISBN 0-7624-1348-4.
13. ^ van den Heuvel, E. P. J. (1966). "On the Precession as a Cause of Pleistocene Variations of the Atlantic Ocean Water Temperatures". Geophysical Journal International. 11 (3): 323–336. Bibcode:1966GeoJ...11..323V. doi:10.1111/j.1365-246X.1966.tb03086.x.
14. ^ The Seasons and the Earth's Orbit, United States Naval Observatory, retrieved 16 August 2013

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SVG replacement for File:Spaceship and the Sun.jpg. A stylized illustration of a spaceship and the sun, based on the description of the emblem of the fictional Galactic Empire in Isaac Asimov's Foundation series ("The golden globe with its conventionalized rays, and the oblique cigar shape that was a space vessel"). This image could be used as a icon for science-fiction related articles.
Crab Nebula.jpg
This is a mosaic image, one of the largest ever taken by NASA's Hubble Space Telescope, of the Crab Nebula, a six-light-year-wide expanding remnant of a star's supernova explosion. Japanese and Chinese astronomers recorded this violent event in 1054 CE, as did, almost certainly, Native Americans.

The orange filaments are the tattered remains of the star and consist mostly of hydrogen. The rapidly spinning neutron star embedded in the center of the nebula is the dynamo powering the nebula's eerie interior bluish glow. The blue light comes from electrons whirling at nearly the speed of light around magnetic field lines from the neutron star. The neutron star, like a lighthouse, ejects twin beams of radiation that appear to pulse 30 times a second due to the neutron star's rotation. A neutron star is the crushed ultra-dense core of the exploded star.

The Crab Nebula derived its name from its appearance in a drawing made by Irish astronomer Lord Rosse in 1844, using a 36-inch telescope. When viewed by Hubble, as well as by large ground-based telescopes such as the European Southern Observatory's Very Large Telescope, the Crab Nebula takes on a more detailed appearance that yields clues into the spectacular demise of a star, 6,500 light-years away.

The newly composed image was assembled from 24 individual Wide Field and Planetary Camera 2 exposures taken in October 1999, January 2000, and December 2000. The colors in the image indicate the different elements that were expelled during the explosion. Blue in the filaments in the outer part of the nebula represents neutral oxygen, green is singly-ionized sulfur, and red indicates doubly-ionized oxygen.
He1523a.jpg
Author/Creator: ESO, European Southern Observatory, Licence: CC BY 4.0
Artist's impression of "the oldest star of our Galaxy": HE 1523-0901
• About 13.2 billion years old
• Approximately 7500 light years far from Earth
• Published as part of Hamburg/ESO Survey in the May 10 2007 issue of The Astrophysical Journal
Perihelion precession.svg
Perihelion precession. Blue circle - mooving perihelion, dotted line - semi-major axis
Precessing Kepler orbit 280frames e0.6 smaller.gif
Author/Creator: WillowW, Licence: CC BY 3.0
Made by me on 18 January 2008 using Blender. A gradually precessing elliptical orbit, such as might result from a 1/r3 force, as occurs in general relativity.
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Author/Creator: Phoenix7777, Licence: CC BY-SA 4.0
Animation of Moon orbit around Earth - Polar view
Moon ·
Orbit1.svg
(c) Lasunncty at the English Wikipedia, CC-BY-SA-3.0
Digram illustrating and explaining various terms in relation to Orbits of Celestial bodies.
Earth-moon.jpg
This view of the rising Earth greeted the Apollo 8 astronauts as they came from behind the Moon after the fourth nearside orbit. Earth is about five degrees above the horizon in the photo. The unnamed surface features in the foreground are near the eastern limb of the Moon as viewed from Earth. The lunar horizon is approximately 780 kilometers from the spacecraft. Width of the photographed area at the horizon is about 175 kilometers. On the Earth 240,000 miles away, the sunset terminator bisects Africa.
Precession and seasons.svg
Author/Creator: Krishnavedala, Licence: CC BY-SA 3.0
Effects of precession on the seasons (using the Northern Hemisphere terms).
Solar system.jpg
This is a montage of planetary images taken by spacecraft managed by the Jet Propulsion Laboratory in Pasadena, CA. Included are (from top to bottom) images of Mercury, Venus, Earth (and Moon), Mars, Jupiter, Saturn, Uranus and Neptune. The spacecraft responsible for these images are as follows:
• the Mercury image was taken by Mariner 10,
• the Venus image by Magellan,
• the Earth and Moon images by Galileo,
• the Mars image by Mars Global Surveyor,
• the Jupiter image by Cassini, and
• the Saturn, Uranus and Neptune images by Voyager.
• Pluto is not shown as it is no longer a planet, and no spacecraft has yet visited it when this montage was taken. The inner planets (Mercury, Venus, Earth, Moon, and Mars) are roughly to scale to each other; the outer planets (Jupiter, Saturn, Uranus, and Neptune) are roughly to scale to each other. PIA 00545 is the same montage with Neptune shown larger in the foreground. Actual diameters are given below:
• Sun (to photosphere) 1,392,684 km
• Mercury 4,879.4 km
• Venus 12,103.7 km
• Earth 12,756.28 km
• Moon 3,476.2 km
• Mars 6,804.9 km
• Jupiter 142,984 km
• Saturn 120,536 km
• Uranus 51,118 km
• Neptune 49,528 km