Axial tilt

"Obliquity" redirects here. For the book, see Obliquity (book).
To understand axial tilt, we employ the right-hand rule. When the fingers of the right hand are curled around in the direction of the planet's rotation, the thumb points in the direction of the north pole.
The axial tilt of three planets: Earth, Uranus, and Venus. Here, a vertical line (black) is drawn perpendicular to the plane of each planet's orbit. The angle between this line and the planet's north pole (red) is the tilt. The surrounding arrows (green) show the direction of the planet's rotation.

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.

Introduction

In astronomy, axial tilt (also called obliquity) is the angle between an object's rotational axis and the perpendicular to its Orbital plane, both oriented by the right hand rule. At an obliquity of 0°, these lines point in the same direction i.e. the rotational axis is perpendicular to the orbital plane. Axial tilt differs from inclination. Because the planet Venus has an axial tilt of 177° its rotation can be considered retrograde, opposite that of most of the other planets.[2][3] The north pole of Venus is "upside down" relative to its orbit. The planet Uranus has a tilt of 97°, hence it rotates "on its side", its north pole being almost in the plane of its orbit.[4]

Over the course of an orbit, the angle of the axial tilt does not change, and the orientation of the axis remains the same relative to the background stars. This causes one pole to be directed toward the Sun on one side of the orbit, and the other pole on the other side, the cause of the seasons on the Earth.

Two standards

Note that there are two standard methods of specifying tilt. The International Astronomical Union (IAU) defines the north pole as that which lies on the north side of the invariable plane of the Solar System;[5] under this system Venus' tilt is 3°, it rotates retrograde, and the right hand rule does not apply. NASA defines the north pole with the right hand rule, as above;[4] under this system, Venus is tilted 177° ("upside down") and rotates direct. The results are equivalent and neither system is more correct.

Obliquity of the ecliptic (Earth's axial tilt)

Earth's axial tilt is currently about 23.4°.
Main article: Ecliptic

The Earth's orbital plane is known as the ecliptic plane, and the 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.[6] It is denoted by the Greek letter ε.

The Earth currently has an axial tilt of about 23.4°.[7] This value remains approximately the same relative to a stationary orbital plane throughout the cycles of precession.[8] However, because the ecliptic (i.e. the Earth's orbit) moves due to planetary perturbations, the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 47" per century (see below).

Short term

The exact angular value of the obliquity is found by observation of the motions of the 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.

Obliquity of the ecliptic for 20,000 years, from Laskar (1986). The red point represents the year 2000.

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

ε = 23° 27′ 08.26″ − 46.845″ T − 0.0059″ T2 + 0.00181″ T3

where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question.[9]

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′ 21.45″ − 46.815″ T − 0.0006″ T2 + 0.00181″ T3

where hereafter T is Julian centuries from J2000.0.[10]

JPL's fundamental ephemerides have been continually updated. For instance, the Astronomical Almanac for 2010 specifies:[7]

ε = 23° 26′ 21.406″ − 46.836769″ T − 0.0001831″ T2 + 0.00200340″ T3 − 0.576″×10−6 T4 − 4.34″×10−8 T5

These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps ± several centuries.[11] 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.[12]

Main article: Nutation

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 the Earth in its orbit cause much smaller (9.2 arcseconds) short-period (about 18.6 years) oscillations of the rotation axis of the Earth, known as nutation, which add a periodic component to Earth's obliquity.[13][14] The true or instantaneous obliquity includes this nutation.[15]

Obliquity of the ecliptic for the past 5 million years, from Berger (1976). Note that the obliquity varies only from about 22.0° to 24.5°. The red point represents the year 1850.
Obliquity of the ecliptic for the next 1 million years, from Berger (1976). Note the approx. 41,000 year period of variation. The red point represents the year 1850.

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° 02' 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".[16]

Main article: Orbit of the Moon

The Moon has a stabilizing effect on Earth's obliquity. Frequency map analysis suggests 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.[17][18] However, more recent numerical simulations[19] suggest that even in the absence of the Moon, Earth's obliquity could be considerably more stable; 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 the Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity.[20]

Earth's seasons

Main article: Season
The axis of a planet remains oriented in the same direction with reference to the background stars regardless of where it is in its orbit. 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.

The Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (throughout its entire orbit). This means that one pole (and the associated hemisphere of the 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 the Earth's seasons.

Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-term climate change.[21]

History

Earth's obliquity may have been reasonably accurately measured as early as 1100 BC in India and China.[22] 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.[23] About 830 AD, the Calif Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years.[24]

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 and that the obliquity is decreasing at a relatively constant rate (during historic time) was Fracastoro in 1538.[25]

The first accurate, modern, western observations of the obliquity were probably those of Tycho Brahe, about 1584,[26] although observations by several others, including Purbach, Regiomontanus, and Walther, could have provided similar information.

Other objects of the Solar System

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. The Earth was stabilized by the Moon, as above, but before its capture, the 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.[17][27] 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.[2][28]

Axis and rotation of selected Solar System objects
  NASA, J2000.0[4] IAU, 0 Jan 2010, 0h TT[29]
Axial tilt North Pole Rotation Axial tilt North Pole Rotation
( ° ) R.A. ( ° ) Dec. ( ° ) ( hours ) ( ° ) R.A. ( ° ) Dec. ( ° ) ( ° / day )
Sun 7.25 286.13 63.87 609.12B 7.25A 286.15 63.89 14.18
Mercury ~0 281.01 61.45 1407.6 0.01 281.01 61.45 6.14
VenusE 177.36 (92.76) (-67.16) (5832.5) 2.64 272.76 67.16 -1.48
Earth 23.4 0.00 90.00 23.93 23.4 undef. 90.00 360.99
Moon 6.68 655.73 1.54C 270.00 66.54 13.18
Mars 25.19 317.68 52.89 24.62 25.19 317.67 52.88 350.89
Jupiter 3.13 268.05 64.49 9.93D 3.12 268.06 64.50 870.54D
Saturn 26.73 40.60 83.54 10.66D 26.73 40.59 83.54 810.79D
UranusE 97.77 (77.43) (15.10) (17.24)D 82.23 257.31 -15.18 -501.16D
Neptune 28.32 299.36 43.46 16.11D 28.33 299.40 42.95 536.31D
PlutoE 122.53 (133.02) (-9.09) (153.29) 60.41 312.99 6.16 -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's listed tilt is inconsistent with their listed north pole and rotation for these planets; values in (parenthesis) have been reinterpreted to match their listed tilt

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,[30] which serves as a lower limit to ψs. Most of these measurements rely on the so-called 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 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 1 Gyr,[31][32] which means that they would not have seasons as Earth has.

See also

References

  1. ↑ U.S. Naval Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann, ed. Explanatory Supplement to the Astronomical Almanac. University Science Books. p. 733. ISBN 0-935702-68-7.
  2. ↑ 2.0 2.1 Correia, Alexandre C. M.; Laskar, Jacques; de Surgy, Olivier Néron (May 2003). "Long-term evolution of the spin of Venus I. theory" (PDF). Icarus 163 (1): 1–23. Bibcode:2003Icar..163....1C. doi:10.1016/S0019-1035(03)00042-3.
  3. ↑ Correia, A. C. M.; Laskar, J. (2003). "Long-term evolution of the spin of Venus: II. numerical simulations" (PDF). Icarus 163 (1): 24–45. Bibcode:2003Icar..163...24C. doi:10.1016/S0019-1035(03)00043-5.
  4. ↑ 4.0 4.1 4.2 Planetary Fact Sheets, at http://nssdc.gsfc.nasa.gov
  5. ↑ Explanatory Supplement 1992, p. 384
  6. ↑ 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. ISBN 978-0-7077-4082-9.
  7. ↑ 7.0 7.1 Astronomical Almanac 2010, p. B52
  8. ↑ Chauvenet, William (1906). A Manual of Spherical and Practical Astronomy 1. J. B. Lippincott. pp. 694–695.
  9. ↑ 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.
  10. ↑ U.S. Naval Observatory; H.M. Nautical Almanac Office (1989). The Astronomical Almanac for the Year 1990. US Government Printing Office. p. B18. ISBN 0-11-886934-5.
  11. ↑ Newcomb, Simon (1906). A Compendium of Spherical Astronomy. MacMillan. pp. 226–227.
  12. ↑ See table 8 and eq. 35 in Laskar, J. (1986). "Secular Terms of Classical Planetary Theories Using the Results of General Relativity". Bibcode:1986A&A...157...59L.
  13. ↑ Explanatory Supplement (1961), sec. 2C
  14. ↑ "Basics of Space Flight, Chapter 2". Jet Propulsion Laboratory. Jet Propulsion Laboratory/NASA. 2013-10-29. Retrieved 2015-03-26.
  15. ↑ Meeus, Jean (1991). "Chapter 21". Astronomical Algorithms. Willmann-Bell. ISBN 0-943396-35-2.
  16. ↑ Berger, A.L. (1976). "Obliquity and Precession for the Last 5000000 Years". Astronomy and Astrophysics 51: 127–135. Bibcode:1976A&A....51..127B.
  17. ↑ 17.0 17.1 Laskar, J.; Robutel, P. (1993). "The Chaotic Obliquity of the Planets" (PDF). Nature 361 (6413): 608–612. Bibcode:1993Natur.361..608L. doi:10.1038/361608a0.
  18. ↑ Laskar, J.; Joutel, F.; Robutel, P. (1993). "Stabilization of the Earth's Obliquity by the Moon" (PDF). Nature 361 (6413): 615–617. Bibcode:1993Natur.361..615L. doi:10.1038/361615a0.
  19. ↑ Lissauer, J.J.; Barnes, J.W.; Chambers, J.E. (2011). "Obliquity variations of a moonless Earth" (PDF). Icarus 217: 77–87. Bibcode:2012Icar..217...77L. doi:10.1016/j.icarus.2011.10.013.
  20. ↑ Ward, W.R. (1982). "Comments on the Long-Term Stability of the Earth's Obliquity". Icarus 50: 444–448. Bibcode:1982Icar...50..444W. doi:10.1016/0019-1035(82)90134-8.
  21. ↑ See references at Milankovitch cycles.
  22. ↑ See Wittmann, A. (1979). "The Obliquity of the Ecliptic". Astronomy and Astrophysics 73: 129–131. Bibcode:1979A&A....73..129W.
  23. ↑ Gore, J. E. (1907). Astronomical Essays Historical and Descriptive. p. 61.
  24. ↑ Marmery, J. V. (1895). Progress of Science. p. 33.
  25. ↑ Dreyer, J. L. E. (1890). Tycho Brahe. p. 355.
  26. ↑ Dreyer (1890), p. 123
  27. ↑ Touma, J.; Wisdom, J. (1993). "The Chaotic Obliquity of Mars" (PDF). Science 259 (5099): 1294–1297. Bibcode:1993Sci...259.1294T. doi:10.1126/science.259.5099.1294. PMID 17732249.
  28. ↑ Correia, Alexandre C.M; Laskar, Jacques (2009). "Mercury's capture into the 3/2 spin-orbit resonance including the effect of core-mantle friction". Icarus 201 (1): 1. arXiv:0901.1843. Bibcode:2009Icar..201....1C. doi:10.1016/j.icarus.2008.12.034.
  29. ↑ Astronomical Almanac 2010, p. B52, C3, D2, E3, E55
  30. ↑ Heller, R. "Holt-Rossiter-McLaughlin Encyclopaedia". René Heller. Retrieved 24 February 2012.
  31. ↑ Heller, R.; Leconte, J.; Barnes, R. (2011). "Tidal obliquity evolution of potentially habitable planets". Astronomy and Astrophysics 528: A27. arXiv:1101.2156. Bibcode:2011A&A...528A..27H. doi:10.1051/0004-6361/201015809.
  32. ↑ Heller, R.; Leconte, J.; Barnes, R. (2011). "Habitability of Extrasolar Planets and Tidal Spin Evolution". Origins of Life and Evolution of Biospheres: 37. Bibcode:2011OLEB..tmp...37H. doi:10.1007/s11084-011-9252-3.

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