Angle between the rotational axis and orbital axis of a body
Thepositive pole of a planet is defined by theright-hand rule: if the fingers of the right hand are curled in the direction of the rotation then the thumb points to the positive pole. The axial tilt is defined as the angle between the direction of the positive pole and the normal to the orbital plane. The angles for Earth, Uranus, and Venus are approximately 23°, 97°, and 177° respectively.
At an obliquity of 0 degrees, the two axes point in the same direction; that is, the rotational axis is perpendicular to the orbital plane.
The rotational axis ofEarth, for example, is the imaginary line that passes through both theNorth Pole andSouth Pole, whereas the Earth's orbital axis is the line perpendicular to the imaginaryplane through which the Earth moves as it revolves around theSun; the Earth's obliquity or axial tilt is the angle between these two lines.
Over the course of anorbital period, the obliquity usually does not change considerably, and theorientation of the axis remains the same relative to thebackground ofstars. This causes one pole to be pointed more toward the Sun on one side of the orbit, and more away from the Sun on the other side—the cause of theseasons on Earth.
There are two standard methods of specifying a planet's tilt. One way is based on the planet'snorth pole, defined in relation to the direction of Earth's north pole, and the other way is based on the planet'spositive pole, defined by theright-hand rule:
The IAU also uses the right-hand rule to define apositive pole[5] for the purpose of determining orientation. Using this convention, Venus is tilted 177° ("upside down") and rotates prograde.
Earth currently has an axial tilt of about 23.44°.[7] This value remains about the same relative to a stationary orbital plane throughout the cycles ofaxial precession.[8] But the ecliptic (i.e., Earth's orbit) moves due to planetaryperturbations, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about46.8″[9] percentury(see details inShort term below).
The ancient Greeks had good measurements of the obliquity since about 350 BCE, whenPytheas of Marseilles measured the shadow of agnomon at the summer solstice.[10] About 830 CE, the CaliphAl-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years.[11] In 1437,Ulugh Beg determined the Earth's axial tilt as 23°30′17″ (23.5047°).[12]
During theMiddle Ages, it was widely believed that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known astrepidation of the equinoxes. Perhaps the first to realize this was incorrect (during historic time) wasIbn al-Shatir in the fourteenth century[13] and the first to realize that the obliquity is decreasing at a relatively constant rate wasFracastoro in 1538.[14] The first accurate, modern, western observations of the obliquity were probably those ofTycho Brahe fromDenmark, about 1584,[15] although observations by several others, includingal-Ma'mun,al-Tusi,[16]Purbach,Regiomontanus, andWalther, could have provided similar information.
An illustration ofaxial parallelism. The axis of Earth remains oriented in the same direction with reference to the background stars regardless of where it is in itsorbit. Northern hemisphere summer occurs at the right side of this diagram, where the north pole (red) is directed toward the Sun, winter at the left.
Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in itsorbit) – this is known asaxial parallelism. This means that one pole (and the associatedhemisphere of Earth) will be directed away from the Sun at one side of the orbit, and half an orbit later (half a year later) this pole will be directed towards the Sun. This is the cause of Earth'sseasons.Summer occurs in theNorthern hemisphere when the north pole is directed toward and the south pole away from the Sun. Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-termclimatic change(also seeMilankovitch cycles).
Relationship between Earth's axial tilt (ε) to the tropical and polar circles
Obliquity of the ecliptic for 20,000 years, fromLaskar (1986). The red point represents the year 2000.
The exact angular value of the obliquity is found by observation of the motions of Earth andplanets over many years. Astronomers produce newfundamental ephemerides as the accuracy ofobservation improves and as the understanding of thedynamics increases, and from these ephemerides various astronomical values, including the obliquity, are derived.
Annualalmanacs are published listing the derived values and methods of use. Until 1983, theAstronomical Almanac's angular value of the mean obliquity for any date was calculated based on thework of Newcomb, who analyzed positions of the planets until about 1895:
JPL's fundamental ephemerides have been continually updated. For instance, according to IAU resolution in 2006 in favor of the P03 astronomical model, theAstronomical Almanac for 2010 specifies:[19]
These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps± several centuries.[20]Jacques Laskar computed an expression to orderT10 good to 0.02″ over 1000 years and severalarcseconds over 10,000 years.
These expressions are for the so-calledmean obliquity, that is, the obliquity free from short-term variations. Periodic motions of the Moon and of Earth in its orbit cause much smaller (9.2arcseconds) short-period (about 18.6 years) oscillations of the rotation axis of Earth, known asnutation, which add a periodic component to Earth's obliquity.[22][23] Thetrue or instantaneous obliquity includes this nutation.[24]
Usingnumerical methods to simulateSolar System behavior over a period of several million years, long-term changes in Earth'sorbit, and hence its obliquity, have been investigated. For the past 5 million years, Earth's obliquity has varied between22°2′33″ and24°30′16″, with a mean period of 41,040 years. This cycle is a combination of precession and the largestterm in the motion of theecliptic. For the next 1 million years, the cycle will carry the obliquity between22°13′44″ and24°20′50″.[25]
TheMoon has a stabilizing effect on Earth's obliquity. Frequency map analysis conducted in 1993 suggested that, in the absence of the Moon, the obliquity could change rapidly due toorbital resonances andchaotic behavior of the Solar System, reaching as high as 90° in as little as a few million years (also seeOrbit of the Moon).[26][27] However, more recent numerical simulations[28] made in 2011 indicated that even in the absence of the Moon, Earth's obliquity might not be quite so unstable; varying only by about 20–25°. To resolve this contradiction, diffusion rate of obliquity has been calculated, and it was found that it takes more than billions of years for Earth's obliquity to reach near 90°.[29] The Moon's stabilizing effect will continue for less than two billion years. As the Moon continues to recede from Earth due totidal acceleration, resonances may occur which will cause large oscillations of the obliquity.[30]
Long-term obliquity of the ecliptic. Left: for the past 5 million years; the obliquity varies only from about 22.0° to 24.5°. Right: for the next 1 million years; note the approx. 41,000-year period of variation. In both graphs, the red point represents the year 1850.[31]
All four of the innermost, rocky planets of theSolar System may have had large variations of their obliquity in the past. Since obliquity is the angle between the axis of rotation and the direction perpendicular to the orbital plane, it changes as the orbital plane changes due to the influence of other planets. But the axis of rotation can also move (axial precession), due to torque exerted by the Sun on a planet's equatorial bulge. Like Earth, all of the rocky planets show axial precession. If the precession rate were very fast the obliquity would actually remain fairly constant even as the orbital plane changes.[32] The rate varies due totidal dissipation andcore-mantle interaction, among other things. When a planet's precession rate approaches certain values,orbital resonances may cause large changes in obliquity. The amplitude of the contribution having one of the resonant rates is divided by the difference between the resonant rate and the precession rate, so it becomes large when the two are similar.[32]
Mercury andVenus have most likely been stabilized by the tidal dissipation of the Sun. Earth was stabilized by the Moon, as mentioned above, but before itsformation, Earth, too, could have passed through times of instability.Mars's obliquity is quite variable over millions of years and may be in a chaotic state; it varies as much as 0° to 60° over some millions of years, depending onperturbations of the planets.[26][33] Some authors dispute that Mars's obliquity is chaotic, and show that tidal dissipation and viscous core-mantle coupling are adequate for it to have reached a fully damped state, similar to Mercury and Venus.[3][34]
The occasional shifts in the axial tilt of Mars have been suggested as an explanation for the appearance and disappearance of rivers and lakes over the course of the existence of Mars. A shift could cause a burst of methane into the atmosphere, causing warming, but then the methane would be destroyed and the climate would become arid again.[35][36]
The obliquities of the outer planets are considered relatively stable.
The stellar obliquityψs, i.e. the axial tilt of a star with respect to the orbital plane of one of its planets, has been determined for only a few systems. By 2012, 49 stars have had sky-projected spin-orbit misalignmentλ has been observed,[39] which serves as a lower limit toψs. Most of these measurements rely on theRossiter–McLaughlin effect. Since the launch of space-based telescopes such asKepler space telescope, it has been made possible to determine and estimate the obliquity of an extrasolar planet. The rotational flattening of the planet and the entourage of moons and/or rings, which are traceable with high-precision photometry provide access to planetary obliquity,ψp. Many extrasolar planets have since had their obliquity determined, such asKepler-186f andKepler-413b.[40][41]
Astrophysicists have applied tidal theories to predict the obliquity ofextrasolar planets. It has been shown that the obliquities of exoplanets in thehabitable zone around low-mass stars tend to be eroded in less than 109 years,[42][43] which means that they would not have tilt-induced seasons as Earth has.
^U.S. Naval Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.).Explanatory Supplement to the Astronomical Almanac. University Science Books. p. 733.ISBN978-0-935702-68-2.
^U.S. Naval Observatory Nautical Almanac Office; U.K. Hydrographic Office; H.M. Nautical Almanac Office (2008).The Astronomical Almanac for the Year 2010. US Government Printing Office. p. M11.ISBN978-0-7077-4082-9.
^"Glossary" inAstronomical Almanac Online. (2023). Washington DC: United States Naval Observatory. s.v. obliquity.
^Sayili, Aydin (1981).The Observatory in Islam. p. 78.
^U.S. Naval Observatory Nautical Almanac Office; H.M. Nautical Almanac Office (1961).Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac.H.M. Stationery Office. Section 2B.
^U.S. Naval Observatory; H.M. Nautical Almanac Office (1989).The Astronomical Almanac for the Year 1990. US Government Printing Office. p. B18.ISBN978-0-11-886934-8.
^See table 8 and eq. 35 inLaskar, J. (1986). "Secular terms of classical planetary theories using the results of general theory".Astronomy and Astrophysics.157 (1):59–70.Bibcode:1986A&A...157...59L. and erratum to articleLaskar, J. (1986). "Erratum: Secular terms of classical planetary theories using the results of general theory".Astronomy and Astrophysics.164: 437.Bibcode:1986A&A...164..437L. Units in article are arcseconds, which may be more convenient.
