# Axial tilt

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In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. [1] It differs from orbital inclination.

## Contents

At an obliquity of 0 degrees, the two axes point in the same direction; i.e., the rotational axis is perpendicular to the orbital plane. Earth's obliquity oscillates between 22.1 and 24.5 degrees [2] on a 41,000-year cycle. Based on a continuously updated formula (here Laskar, 1986, though since 2006 the IMCCE and the IAU recommend the P03 model), Earth's mean obliquity (without taking into account the nutation in obliquity) is currently about 23°26′11.5″ (or 23.43653°) and decreasing; according to P03 astronomical model, its value (without taking into account the nutation in obliquity) was 23° 26' 11,570" (23.4365472133°) on January 1st 2021, 0 TT.

Over the course of an orbital period, the obliquity usually does not change considerably, and the orientation of the axis remains the same relative to the background of stars. This causes one pole to be directed more toward the Sun on one side of the orbit, and the other pole on the other side—the cause of the seasons on Earth.

## Standards

The positive pole of a planet is defined by the right-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.

There are two standard methods of specifying tilt. The International Astronomical Union (IAU) defines the north pole of a planet as that which lies on Earth's north side of the invariable plane of the Solar System; [3] under this system, Venus is tilted 3° and rotates retrograde, opposite that of most of the other planets. [4] [5]

The IAU also uses the right-hand rule to define a positive pole [6] for the purpose of determining orientation. Using this convention, Venus is tilted 177° ("upside down").

## Earth

Earth's orbital plane is known as the ecliptic plane, and Earth's tilt is known to astronomers as the obliquity of the ecliptic, being the angle between the ecliptic and the celestial equator on the celestial sphere. [7] It is denoted by the Greek letter ε .

Earth currently has an axial tilt of about 23.44°. [8] This value remains about the same relative to a stationary orbital plane throughout the cycles of axial precession. [9] But the ecliptic (i.e., Earth's orbit) moves due to planetary perturbations, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 46.8″ [10] per century (see details in Short term below).

### History

Earth's obliquity may have been reasonably accurately measured as early as 1100 BC in India and China. [11] The ancient Greeks had good measurements of the obliquity since about 350 BC, when Pytheas of Marseilles measured the shadow of a gnomon at the summer solstice. [12] About 830 AD, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years. [13] In 1437, Ulugh Beg determined the Earth's axial tilt as 23°30′17″ (23.5047°). [14]

It was widely believed, during the Middle Ages, that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known as trepidation of the equinoxes. Perhaps the first to realize this was incorrect (during historic time) was Ibn al-Shatir in the fourteenth century [15] and the first to realize that the obliquity is decreasing at a relatively constant rate was Fracastoro in 1538. [16] The first accurate, modern, western observations of the obliquity were probably those of Tycho Brahe from Denmark, about 1584, [17] although observations by several others, including al-Ma'mun, al-Tusi, [18] Purbach, Regiomontanus, and Walther, could have provided similar information.

### Seasons

Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in its orbit). This means that one pole (and the associated hemisphere 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's seasons. Summer occurs in the Northern hemisphere when the north pole is directed toward the Sun. Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-term climatic change (also see Milankovitch cycles).

### Oscillation

#### Short term

The exact angular value of the obliquity is found by observation of the motions of Earth and planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, and from these ephemerides various astronomical values, including the obliquity, are derived.

Annual almanacs are published listing the derived values and methods of use. Until 1983, the Astronomical Almanac's angular value of the mean obliquity for any date was calculated based on the work of Newcomb, who analyzed positions of the planets until about 1895:

ε = 23° 27′ 8.26″ − 46.845″ T − 0.0059″ T2 + 0.00181T3

where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question. [19]

From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated:

ε = 23° 26′ 21.448″ − 46.8150″ T − 0.00059″ T2 + 0.001813T3

where hereafter T is Julian centuries from J2000.0. [20]

JPL's fundamental ephemerides have been continually updated. For instance, according to IAU resolution in 2006 in favor of the P03 astronomical model, the Astronomical Almanac for 2010 specifies: [21]

ε = 23° 26′ 21.406″ − 46.836769T0.0001831T2 + 0.00200340T3 − 5.76″ × 10−7T4 − 4.34″ × 10−8T5

These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps ± several centuries. [22] J. Laskar computed an expression to order T10 good to 0.02″ over 1000 years and several arcseconds over 10,000 years.

ε = 23° 26′ 21.448″ − 4680.93″ t − 1.55″ t2 + 1999.25″ t3 − 51.38″ t4 − 249.67″ t5 − 39.05″ t6 + 7.12″ t7 + 27.87″ t8 + 5.79″ t9 + 2.45″ t10

where here t is multiples of 10,000 Julian years from J2000.0. [23]

These expressions are for the so-called mean obliquity, that is, the obliquity free from short-term variations. Periodic motions of the Moon and of Earth in its orbit cause much smaller (9.2 arcseconds) short-period (about 18.6 years) oscillations of the rotation axis of Earth, known as nutation, which add a periodic component to Earth's obliquity. [24] [25] The true or instantaneous obliquity includes this nutation. [26]

#### Long term

Using numerical methods to simulate Solar System behavior, long-term changes in Earth's orbit, and hence its obliquity, have been investigated over a period of several million years. For the past 5 million years, Earth's obliquity has varied between 22° 2′ 33″ and 24° 30′ 16″, with a mean period of 41,040 years. This cycle is a combination of precession and the largest term in the motion of the ecliptic. For the next 1 million years, the cycle will carry the obliquity between 22° 13′ 44″ and 24° 20′ 50″. [27]

The Moon 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 to orbital resonances and chaotic behavior of the Solar System, reaching as high as 90° in as little as a few million years (also see Orbit of the Moon). [28] [29] However, more recent numerical simulations [30] 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°. [31] The Moon's stabilizing effect will continue for less than 2 billion years. As the Moon continues to recede from Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity. [32]

Long-term obliquity of the ecliptic. Left: for the past 5 million years; note that 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. (Source: Berger, 1976).

## Solar System bodies

All four of the innermost, rocky planets of the Solar 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. [33] The rate varies due to tidal dissipation and core-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. [33]

Mercury and Venus have most likely been stabilized by the tidal dissipation of the Sun. Earth was stabilized by the Moon, as mentioned above, but before its formation, 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 on perturbations of the planets. [28] [34] 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. [4] [35] 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. [36] [37]

The obliquities of the outer planets are considered relatively stable.

Axis and rotation of selected Solar System bodies
Body NASA, J2000.0 [38] IAU, 0 January 2010, 0h TT [39]
Axial tilt
(degrees)
North PoleRotation
(hours)
Axial tilt
(degrees)
North PoleRotation
(deg/day)
R.A. (degrees) Dec. (degrees) R.A. (degrees) Dec. (degrees)
Sun 7.25286.1363.87609.12B7.25A286.1563.8914.18
Mercury 0.03281.0161.421407.60.01281.0161.456.14
Venus 2.64272.7667.16−5832.62.64272.7667.16−1.48
Earth 23.440.0090.0023.9323.44undef.90.00360.99
Moon 6.68655.731.54C270.0066.5413.18
Mars 25.19317.6852.8924.6225.19317.6752.88350.89
Jupiter 3.13268.0564.499.93D3.12268.0664.50870.54D
Saturn 26.7340.6083.5410.66D26.7340.5983.54810.79D
Uranus 82.23257.43−15.10−17.24D82.23257.31−15.18−501.16D
Neptune 28.32299.3643.4616.11D28.33299.4042.95536.31D
Pluto E57.47(312.99)(6.16)−153.2960.41312.996.16−56.36
A with respect to the ecliptic of 1850
B at 16° latitude; the Sun's rotation varies with latitude
C with respect to the ecliptic; the Moon's orbit is inclined 5.16° to the ecliptic
D from the origin of the radio emissions; the visible clouds generally rotate at different rate
E NASA lists the coordinates of Pluto's positive pole; values in (parentheses) have been reinterpreted to correspond to the north/negative pole.

## Extrasolar planets

The stellar obliquity ψs, 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. But for 49 stars as of today, the sky-projected spin-orbit misalignment λ has been observed, [40] which serves as a lower limit to ψs. Most of these measurements rely on the Rossiter–McLaughlin effect. So far, it has not been possible to constrain the obliquity of an extrasolar planet. But the rotational flattening of the planet and the entourage of moons and/or rings, which are traceable with high-precision photometry, e.g. by the space-based Kepler space telescope, could provide access to ψp in the near future.

Astrophysicists have applied tidal theories to predict the obliquity of extrasolar planets. It has been shown that the obliquities of exoplanets in the habitable zone around low-mass stars tend to be eroded in less than 109 years, [41] [42] which means that they would not have seasons as Earth has.

## Related Research Articles

The ecliptic is the plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars. The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system.

Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called free nutation or Euler nutation. A pure nutation is a movement of a rotational axis such that the first Euler angle is constant. In spacecraft dynamics, precession is sometimes referred to as nutation.

In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects. The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance between bodies with similar orbital radii causes large solar system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.

The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions and orbits of Solar System objects. Because most planets and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the vernal (March) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.

Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object.

Tidal locking, in the best-known case, occurs when an orbiting astronomical body always has the same face toward the object it is orbiting. This is known as synchronous rotation: the tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon.

In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In particular, it can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 26,000 years. This is similar to the precession of a spinning top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.

The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract projection of the terrestrial equator into outer space. Due to Earth's axial tilt, the celestial equator is currently inclined by about 23.44° with respect to the ecliptic, but has varied from about 22.0° to 24.5° over the past 5 million years due to perturbation from other planets.

Lunar precession is a term used for three different precession motions related to the Moon. First, it can refer to change in orientation of the lunar rotational axis with respect to a reference plane, following the normal rules of precession followed by spinning objects. In addition, the orbit of the Moon undergoes two important types of precessional motion: apsidal and nodal.

Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term is named for Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypothesized that variations in eccentricity, axial tilt, and precession resulted in cyclical variation in the solar radiation reaching the Earth, and that this orbital forcing strongly influenced the Earth's climatic patterns.

Jack Wisdom is a Professor of Planetary Sciences at the Massachusetts Institute of Technology. He received his B.S. from Rice University in 1976 and his Ph.D. from California Institute of Technology in 1981. His research interests are the dynamics of the Solar System.

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Astronomical nutation is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Although they are caused by the same effect operating over different timescales, astronomers usually make a distinction between precession, which is a steady long-term change in the axis of rotation, and nutation, which is the combined effect of similar shorter-term variations.

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