Axial tilt

Standards

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; under this system, Venus is tilted 3° and spins retrograde, opposite that of most of the other planets.

The IAU also uses the right-hand rule to define a positive pole 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. It is denoted by the Greek letter ε.

Earth currently has an axial tilt of about 23.4°. This value remains about the same relative to a stationary orbital plane throughout the cycles of axial precession. 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 47″ 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. 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. 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.

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 and the first to realize that the obliquity is decreasing at a relatively constant rate was Fracastoro in 1538. The first accurate, modern, western observations of the obliquity were probably those of Tycho Brahe from Denmark, about 1584, although observations by several others, including al-Ma'mun, al-Tusi, 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 climate change (also see Milankovitch cycles).

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:

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

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:

where hereafter T is Julian centuries from J2000.0.

JPL's fundamental ephemerides have been continually updated. For instance, the Astronomical Almanac for 2010 specifies:

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

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

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. The true or instantaneous obliquity includes this nutation.

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″.

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 can 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). However, more recent numerical simulations 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°. The Moon's stabilizing effect will continue for less than 2 billion years. If the Moon continues to recede from Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity.

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. Like Earth, all of the rocky planets have a small precessional rotation of their spin axis. This rate varies due to, among other things, tidal dissipation and core-mantle interaction. When each planet reaches certain values of precession, orbital resonances may cause very large, chaotic changes in obliquity. Mercury and Venus have most likely been stabilized by the tidal dissipation of the Sun. Earth was stabilized by the Moon, as above, but before its capture, Earth, too, could have passed through times of instability. Mars's obliquity is currently in a chaotic state; it varies as much as 0° to 60° over some millions of years, depending on perturbations of the planets. The obliquities of the outer planets are considered relatively stable. 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.

Extrasolar planets

The stellar obliquity ψ_s, 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, which serves as a lower limit to ψ_s. 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 spacecraft, could provide access to ψ_p 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, which means that they would not have seasons as Earth has.