Axial tilt

# Axial tilt

Overview

In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, axial tilt (also called obliquity) is the angle between an object's rotational axis, and a line perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to its orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...

. It differs from inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

.

To measure obliquity, use the right hand grip rule for both the rotation and the orbital motion, i.e.: the line from the vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

at the object's centre to its north pole
Poles of astronomical bodies
The poles of astronomical bodies are determined based on their axis of rotation in relation to the celestial poles of the celestial sphere.-Geographic poles:...

(above which the object appears to rotate counter-clockwise); and the line drawn from the vertex in the direction of the normal to its orbital plane, (above which the object moves counter-clockwise in its orbit).
Discussion

Encyclopedia

In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, axial tilt (also called obliquity) is the angle between an object's rotational axis, and a line perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to its orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...

. It differs from inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

.

To measure obliquity, use the right hand grip rule for both the rotation and the orbital motion, i.e.: the line from the vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

at the object's centre to its north pole
Poles of astronomical bodies
The poles of astronomical bodies are determined based on their axis of rotation in relation to the celestial poles of the celestial sphere.-Geographic poles:...

(above which the object appears to rotate counter-clockwise); and the line drawn from the vertex in the direction of the normal to its orbital plane, (above which the object moves counter-clockwise in its orbit). At zero degrees, these lines point in the same direction.

The planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...

has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

. The north pole of Venus is pointed 'downward' (our southward). The planet Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

. Hence the axial tilt of Uranus is 97°.

Over the course of an orbit, while the angle of the axial tilt does not change, the orientation of a planet's axial tilt moves through 360 degrees (one complete orbit around the Sun), relative to a line between the planet and the Sun, causing season
Season
A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...

s on Earth.

## Obliquity

In the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, the Earth's orbital plane is known as the ecliptic plane, and so the Earth's axial tilt is officially called the obliquity of the ecliptic. It is denoted by the Greek letter ε
Epsilon
Epsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a close-mid front unrounded vowel . In the system of Greek numerals it has a value of 5. It was derived from the Phoenician letter He...

.

The Earth currently has an axial tilt of about 23.5°. The axis remains tilted in the same direction towards the stars throughout a year and this means that when a hemisphere (a northern or southern half of the earth) is pointing away from the Sun at one point in the orbit then half an orbit later (half a year later) this hemisphere will be pointing towards the Sun. This effect is the main cause of the season
Season
A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...

s (see effect of sun angle on climate
Effect of sun angle on climate
The amount of heat energy received at any location on the globe is a direct effect of sun angle on climate, as the angle at which sunlight strikes the Earth varies by location, time of day, and season due to the Earth's orbit around the sun and the Earth's rotation around its tilted axis...

). Whichever hemisphere is currently tilted toward the Sun experiences more hours of sunlight
Sunlight
Sunlight, in the broad sense, is the total frequency spectrum of electromagnetic radiation given off by the Sun. On Earth, sunlight is filtered through the Earth's atmosphere, and solar radiation is obvious as daylight when the Sun is above the horizon.When the direct solar radiation is not blocked...

each day, and the sunlight at midday also strikes the ground at an angle nearer the vertical
Vertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...

and thus delivers more energy per unit surface area.

Lower obliquity causes polar regions to receive less seasonally contrasting solar radiation
Insolation
Insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter or kilowatt-hours per square meter per day...

, producing conditions more favorable to glaciation. Like changes in precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

and eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

, changes in tilt influence the relative strength of the seasons, but the effects of the tilt cycle are particularly pronounced in the high latitudes where the great ice ages began. Obliquity is a major factor in glacial/interglacial fluctuations (see Milankovitch cycles
Milankovitch cycles
Milankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...

).

The obliquity of the ecliptic is not a fixed quantity but changing over time in a cycle with a period of 41,000 years (see below). Note that the obliquity and the precession of the equinoxes are calculated from the same theory and are thus related to each other. A smaller ε means a larger p (precession in longitude) and vice versa. Yet the two movements act independent from each other, going in mutually perpendicular directions.

## Measurement

Knowledge of the obliquity of the ecliptic (ε) is critical for astronomical calculations and observations from the surface of the Earth (Earth-based, positional astronomy).

To quickly grasp an idea of its numerical value one can look at how the Sun's angle above the horizon varies with the seasons. The measured difference between the angles of the Sun above the horizon at noon on the longest and shortest days of the year gives twice the obliquity.

To an observer on the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

standing all year long looking above, the sun will be directly overhead at noon on the March Equinox, then swing north until it is over the Tropic of Cancer
Tropic of Cancer
The Tropic of Cancer, also referred to as the Northern tropic, is the circle of latitude on the Earth that marks the most northerly position at which the Sun may appear directly overhead at its zenith...

, 23° 26’ away from the equator on the Northern Solstice. On the September Equinox it will be back overhead, then swing south until it is over the Tropic of Capricorn
Tropic of Capricorn
The Tropic of Capricorn, or Southern tropic, marks the most southerly latitude on the Earth at which the Sun can be directly overhead. This event occurs at the December solstice, when the southern hemisphere is tilted towards the Sun to its maximum extent.Tropic of Capricorn is one of the five...

, 23° 26’ away from the equator on the Southern Solstice.

Example: an observer at 50° latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

(either north or south) will see the Sun 63° 26’ above the horizon at noon on the longest day of the year, but only 16° 34’ on the shortest day. The difference is 2ε = 46° 52’, and so ε = 23° 26’.

(90° - 50°) + 23.4394° = 63.4394° when measuring angles from the horizon
(90° - 50°) − 23.4394° = 16.5606°

At the Equator, this would be 90° + 23.4394° = 113.4394° and 90° − 23.4394° = 66.5606° (measuring always from the southern horizon
Horizon
The horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...

).

Abu-Mahmud Khojandi measured the Earth's axial tilt in the 10th century using this principle with a giant sextant
Sextant
A sextant is an instrument used to measure the angle between any two visible objects. Its primary use is to determine the angle between a celestial object and the horizon which is known as the altitude. Making this measurement is known as sighting the object, shooting the object, or taking a sight...

and noted that his value was lower than those of earlier astronomers, thus discovering that the axial tilt is not constant.

## Values

The Earth's axial tilt varies between 22.1° and 24.5° (but see below), with a 41,000 year period, and at present, the tilt is decreasing. In addition to this steady decrease there are much smaller short term (18.6 years) variations, known as nutation
Nutation
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 behavior of a mechanism...

, mainly due to the changing plane of the moon's orbit. This can shift the Earth's axial tilt by plus or minus 0.005 degree.

Simon Newcomb
Simon Newcomb
Simon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of...

's calculation at the end of the nineteenth century for the obliquity of the ecliptic gave a value of 23° 27’ 8.26” (epoch of 1900), and this was generally accepted until improved telescopes allowed more accurate observations, and electronic computers permitted more elaborate models to be calculated. Lieske developed an updated model in 1976 with ε equal to 23° 26’ 21.448” (epoch of 2000), which is part of the approximation formula recommended by the International Astronomical Union
International Astronomical Union
The International Astronomical Union IAU is a collection of professional astronomers, at the Ph.D. level and beyond, active in professional research and education in astronomy...

in 2000:

ε = 84381.448 − 46.84024T − (59 × 10−5)T2 + (1.813 × 10−3)T3, measured in seconds of arc, with T being the time in Julian centuries (that is, 36,525 days) since the ephemeris
Ephemeris
An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...

epoch
Epoch (reference date)
In the fields of chronology and periodization, an epoch is an instance in time chosen as the origin of a particular era. The "epoch" then serves as a reference point from which time is measured...

of 2000 (which occurred on Julian day 2,451,545.0). A straight application of this formula to 1900 (T=-1) returns 23° 27’ 8.29”, which is very close to Newcomb's value.

With the linear term in T being negative, at present the obliquity is slowly decreasing. It is implicit that this expression gives only an approximate value for ε and is only valid for a certain range of values of T. If not, ε would approach infinity as T approaches infinity. Computations based on a numerical model of the solar system
Numerical model of solar system
A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model established the more general field of celestial mechanics. The results of this simulation can be...

show that ε has a period of about 41,000 years, the same as the constants of the precession p of the equinoxes (although not of the precession itself).

Other theoretical models may come with values for ε expressed with higher powers of T, but since no (finite) polynomial can ever represent a periodic function, they all go to either positive or negative infinity for large enough T. In that respect one can understand the decision of the International Astronomical Union to choose the simplest equation which agrees with most models. For up to 5,000 years in the past and the future all formulas agree, and up to 9,000 years in the past and the future, most agree to reasonable accuracy. For eras farther out discrepancies get too large.

## Long period variations

Nevertheless extrapolation of the average polynomials gives a fit to a sine curve with a period of 41,013 years, which, according to Wittmann, is equal to:

ε = A + B sin(C(T + D)); with A = 23.496932° ± 0.001200°, B = − 0.860° ± 0.005°, C = 0.01532 ± 0.0009 radian/Julian century, D = 4.40 ± 0.10 Julian centuries, and T, the time in centuries from the epoch of 2000 as above.

This means a range of the obliquity from 22° 38’ to 24° 21’, the last maximum was reached in 8700 BC, the mean value occurred around 1550 and the next minimum will be in 11800. This formula should give a reasonable approximation for the previous and next million years or so. Yet it remains an approximation in which the amplitude of the wave remains the same, while in reality, as seen from the results of the Milankovitch cycles
Milankovitch cycles
Milankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...

, irregular variations occur. The quoted range for the obliquity is from 21° 30’ to 24° 30’, but the low value may have been a one-time overshot of the normal 22° 30’.

Over the last 5 million years, the obliquity of the ecliptic (or more accurately, the obliquity of the Equator on the moving ecliptic of date) has varied from 22.0425° to 24.5044°, but for the next one million years, the range will be only from 22.2289° to 24.3472°.

Other planets may have a variable obliquity, too; for example, on Mars
Astronomy on Mars
The Astronomy on Mars article presents information and images about viewing astronomical phenomena from the planet Mars. In many cases these are the same or similar to those seen from Earth but sometimes they can be quite different...

, the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets. The relatively small range for the Earth is due to the stabilizing influence of the Moon, but it will not remain so. According to W.R. Ward, the orbit of the Moon (which is continuously increasing due to tidal effects) will have gone from the current 60 to approximately 66.5 Earth radii in about 1.5 billion years. Once this occurs, a resonance from planetary effects will follow, causing swings of the obliquity between 22° and 38°. Further, in approximately 2 billion years, when the Moon reaches a distance of 68 Earth radii, another resonance will cause even greater oscillations, between 27° and 60°. This would have extreme effects on climate.

## Axial tilt of selected objects in the solar system

Object Axial tilt (°) Axial tilt (radians)
Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

7.25 0.1265
Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

0.0352 0.000614
Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...

177.4 3.096
Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

23.44 0.4091
Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

6.688 0.1167
Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...

25.19 0.4396
Ceres  ~4 ~0.07
Pallas
2 Pallas
Pallas, formally designated 2 Pallas, is the second asteroid to have been discovered , and one of the largest. It is estimated to constitute 7% of the mass of the asteroid belt, and its diameter of 530–565 km is comparable to, or slightly larger than, that of 4 Vesta. It is however 20%...

~60 ~1
Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

3.13 0.0546
Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

26.73 0.4665
Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

97.77 1.7064
Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

28.32 0.4943
Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

119.61 2.0876

† Tilt to its orbit in the Earth-Moon system. Moon's tilt is 1.5424° (0.02692 radians) to ecliptic

• Celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

• Celestial pole
Celestial pole
The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the imaginary rotating sphere of stars called the celestial sphere...

• Ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

• Milankovitch cycles
Milankovitch cycles
Milankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...

• Nutation
Nutation
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 behavior of a mechanism...

• Orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...

• Polar motion
Polar motion
Polar motion of the earth is the movement of Earth's rotational axis across its surface. This is measured with respect to a reference frame in which the solid Earth is fixed...

Retrograde motion is motion in the direction opposite to the movement of something else, and is the contrary of direct or prograde motion. This motion can be the orbit of one body about another body or about some other point, or the rotation of a single body about its axis, or other phenomena such...

• Rotation axis
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

• True polar wander
True polar wander
True polar wander is a solid-body rotation of a planet or moon with respect to its spin axis, causing the geographic locations of the North and South Poles to change, or "wander". In a stable state, the largest moments of inertia axis is aligned with the spin axis, with the smaller two moment of...

• Explanatory supplement to "the Astronomical ephemeris" and the American Ephemeris and Nautical Almanac
American Ephemeris and Nautical Almanac
The American Ephemeris and Nautical Almanac was published for the years 1855 to 1980, containing information necessary for astronomers, surveyors, and navigators...

• A comparison of values predicted by different theories at tenspheres.com
• Berger, A. L. "Obliquity & precession for the last 5 million years". Astronomy & Astrophysics 51, 127 (1976)
• Wittmann, A. "The obliquity of the ecliptic". Astronomy & Astrophysics 73, 129-131 (1979)
• Ward, W. R. "Comments on the long-term stability of the Earth's obliquity". Icarus 1982, 50, 444
• National Space Science Data Center
• Bryant, Jeff. Axial Tilts of Planets, Wolfram Demonstrations Project
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...