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It differs from [[orbital inclination]].
It differs from [[orbital inclination]].


[[File:Right-hand grip rule.svg|thumb|right|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#Rotation|planet's rotation]], the thumb points in the direction of the north [[Geographical_pole|pole]].]]
[[File:Right-hand grip rule.svg|thumb|right|150px|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#Rotation|planet's rotation]], the thumb points in the direction of the north [[Geographical_pole|pole]].]]


[[File:Planet axis comparison.png|thumb|right|300px|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 [[Geographical_pole|pole]] (red) is the tilt. The arrows (green) show the direction of the [[Planet#Rotation|planet's rotation]].]]
[[File:Planet axis comparison.png|thumb|right|300px|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 [[Geographical_pole|pole]] (red) is the tilt. The arrows (green) show the direction of the [[Planet#Rotation|planet's rotation]].]]


== Introduction ==
Orientation of the axes is established by the [[right hand rule]] for both the [[Planet#Rotation|rotation]] and the [[orbit|orbital motion]]. When the fingers of the right hand curl around in the direction of the object's rotation, the thumb points in the direction of its [[poles of astronomical bodies|north pole]] (from which, looking back at the object, it appears to rotate [[counter-clockwise]]). Similarly, when the fingers of the right hand curl around in the direction of the object's orbital motion, the thumb points in the direction of the north pole of the orbit (from which the object appears to move counter-clockwise in its orbit). At an obliquity of 0°, these axes point in the same direction.
Orientation of the axes is established by the [[right hand rule]] for both the [[Planet#Rotation|rotation]] and the [[orbit|orbital motion]]. When the fingers of the right hand curl around in the direction of the object's rotation, the thumb points in the direction of its [[poles of astronomical bodies|north pole]] (from which, looking back at the object, it appears to rotate [[counter-clockwise]]). Similarly, when the fingers of the right hand curl around in the direction of the object's orbital motion, the thumb points in the direction of the north pole of the orbit (from which the object appears to move counter-clockwise in its orbit). The angle between these two poles is the obliquity. At an obliquity of 0°, the axes point in the same direction.


Because the [[planet]] [[Venus]] has an axial tilt of 177° its rotation can be considered [[Retrograde_motion|retrograde]], opposite that of most of the other planets. 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.<ref>
Because the [[planet]] [[Venus]] has an axial tilt of 177° its rotation can be considered [[Retrograde_motion|retrograde]], opposite that of most of the other planets. 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.<ref>
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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 [[season]]s on the [[Earth]].
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 [[season]]s on the [[Earth]].


==Obliquity==
== Obliquity of the ecliptic (Earth's axial tilt) ==
[[Image:AxialTiltObliquity.png|thumb|right|380px|Earth's axial tilt is 23.44°.<ref name=IERS/>]]
[[Image:AxialTiltObliquity.png|thumb|right|380px|Earth's axial tilt is 23.44°.]]
{{main|Ecliptic}}
The [[Earth]]'s [[orbit|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]].<ref>
{{cite book
| author = U.S. Naval Observatory Nautical Almanac Office
| first =Nautical Almanac Office
| coauthors = U.K. Hydrographic Office, H.M. Nautical Almanac Office
| title = The Astronomical Almanac for the Year 2010
| publisher = U.S. Govt. Printing Office
| year = 2008
| isbn = 970-0-7077-40829}}
, p. M11</ref>
It is denoted by the [[Greek letter]] ''[[Epsilon|ε]]''.


The Earth currently has an axial tilt of about 23.44°,<ref>
In the [[solar system]], 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]] [[Epsilon|ε]].
''Astronomical Almanac 2010'', p. B52</ref>
This value remains approximately the same relative to a stationary orbital plane throughout the cycles of [[precession]].<ref>
{{cite book
| last = Chauvenet
| first = William
| title = A Manual of Spherical and Practical Astronomy
| publisher = J.B. Lippincott Co., Philadelphia
| year = 1906
| volume = I
|url=http://books.google.com/books?id=yobvAAAAMAAJ&dq=editions:QmrAl-CNghIC&source=gbs_navlinks_s}}
, art. 365-367, p. 694-695, at Google books</ref>
However, because the ecliptic (i.e. the Earth's orbit) moves due to planetary [[Perturbation_(astronomy)|perturbations]], the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about [[Minute_of_arc|47"]] per [[century]] (see below).


=== Short term ===
The Earth currently has an axial tilt of about 23.5°.<ref name=IERS>{{cite web
The exact angular value of the obliquity is found by observation of the motions of the [[Earth]] and [[solar system|planets]] over many years. Astronomers produce new [[fundamental ephemeris|fundamental ephemerides]] as the accuracy of [[Observational astronomy|observation]] improves and as the understanding of the [[Analytical dynamics|dynamics]] increases, and from these ephemerides various astronomical values, including the obliquity, are derived.
| author=Staff | date=2007-08-07
| url=http://hpiers.obspm.fr/eop-pc/models/constants.html
| title=Useful Constants | publisher=International Earth Rotation and Reference Systems Service (IERS)
| accessdate=2008-09-23 }}</ref> <ref> [http://www.neoprogrammics.com/obliquity_of_the_ecliptic/ ''daily updated obliquity of the ecliptic (Eps Mean)'']</ref> 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]]s (see [[effect of sun angle on climate]]). Whichever hemisphere is currently tilted toward the Sun experiences more hours of [[sunlight]] each day, and the sunlight at midday also strikes the ground at an angle nearer the [[Vertical direction|vertical]] and thus delivers more energy per unit surface area.


[[File:Obliquity of the ecliptic laskar.PNG|thumb|Obliquity of the ecliptic for 20,000 years, from Laskar (1986). The red point represents the year 2000.]]
Lower obliquity causes polar regions to receive less seasonally contrasting [[insolation|solar radiation]], producing conditions more favorable to [[glaciation]]. Like changes in [[precession]] and [[orbital eccentricity|eccentricity]], 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.<ref>[http://www.ncdc.noaa.gov/paleo/slides/slideset/11/11_187_slide.html Paleo Slide Sets<!-- Bot generated title -->]</ref> Obliquity is a major factor in glacial/interglacial fluctuations (see [[Milankovitch cycles]]).


Annual [[almanac]]s 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 [[Newcomb's Tables of the Sun|work of Newcomb]], who analyzed positions of the planets until about 1895:
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.


{{math|''&epsilon;'' {{=}} 23&deg; 27&prime; 08&Prime;.26 &minus; 46&Prime;.845 ''T'' &minus; 0&Prime;.0059 ''T''<sup>2</sup> + 0&Prime;.00181 ''T''<sup>3</sup>}}
==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).


where {{math|''&epsilon;''}} is the obliquity and {{math|''T''}} is [[Tropical year|tropical centuries]] from [[Epoch (astronomy)|B1900.0]] to the date in question.<ref>
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.
{{cite book
| author = U.S. Naval Observatory Nautical Almanac Office
| coauthors = H.M. Nautical Almanac Office
| title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
| publisher = H.M. Stationery Office, London
| year = 1961}}, sec. 2B</ref>


From 1984, the [[Jet Propulsion Laboratory Development Ephemeris|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:
To an observer on the [[equator]] 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]], 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]], 23° 26’ away from the equator on the [[Southern Solstice]].


{{math|''&epsilon;'' {{=}} 23&deg; 26&prime; 21&Prime;.45 &minus; 46&Prime;.815 ''T'' &minus; 0&Prime;.0006 ''T''<sup>2</sup> + 0&Prime;.00181 ''T''<sup>3</sup>}}
Example: an observer at 50° [[latitude]] (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’.


where hereafter {{math|''T''}} is [[Julian day|Julian centuries]] from [[Epoch (astronomy)|J2000.0]].<ref>
(90° - 50°) + 23.4394° = 63.4394° when measuring angles from the horizon
{{cite book
(90° - 50°) − 23.4394° = 16.5606°
| last = U.S. Naval Observatory
| first =Nautical Almanac Office
| coauthors = H.M. Nautical Almanac Office
| title = The Astronomical Almanac for the Year 1990
| publisher = U.S. Govt. Printing Office
| year = 1989
| isbn = 0-11-886934-5}}
, p. B18</ref>


JPL's fundamental ephemerides have been continually updated. For instance, the ''Astronomical Almanac'' for 2010 specifies:<ref>''Astronomical Almanac 2010'', p. B52</ref>
At the Equator, this would be 90° + 23.4394° = 113.4394° and 90° − 23.4394° = 66.5606° (measuring always from the southern [[horizon]]).


{{math|''&epsilon;'' {{=}} 23&deg; 26&prime; 21&Prime;.406 &minus; 46&Prime;.836769 ''T'' &minus; 0&Prime;.0001831 ''T''<sup>2</sup> + 0&Prime;.00200340 ''T''<sup>3</sup> &minus; 0&Prime;.576&times;10<sup>&minus;6</sup> ''T''<sup>4</sup> &minus; 4&Prime;.34&times;10<sup>&minus;8</sup> ''T''<sup>5</sup>}}
[[Abu-Mahmud Khojandi]] measured the Earth's axial tilt in the 10th century using this principle with a giant [[sextant]] and noted that his value was lower than those of earlier astronomers, thus discovering that the axial tilt is not constant.<ref>[http://www.encyclopedia.com/doc/1G2-2830902295.html Al-Khujandī, Abū Maḥmūd Ḥāmid Ibn Al-Khiḍr], ''Complete Dictionary of Scientific Biography'', 2008</ref>


These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps {{math|&plusmn;}} several centuries.<ref>
[[File:Earth tilt animation.gif|thumb|right|The [[rotation|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.]]
{{cite book
|url = http://books.google.com/books?id=CAxDAAAAIAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
|title = A Compendium of Spherical Astronomy
|last = Newcomb
|first = Simon
|year = 1906
|publisher=MacMillan Co., New York}}
, p. 226-227, at Google books</ref>
J. Laskar computed an expression to order {{math|''T''<sup>10</sup>}} good to {{math|0&Prime;.02}} over 1000 years and several [[Minute of arc|arcseconds]] over 10,000 years:


{{math|''&epsilon;'' {{=}} 23&deg; 26&prime; 21&Prime;.448 &minus; 4680&Prime;.93 ''T'' &minus; 1&Prime;.55 ''T''<sup>2</sup> + 1999&Prime;.25 ''T''<sup>3</sup> &minus; 51&Prime;.38 ''T''<sup>4</sup> &minus; 249&Prime;.67 ''T''<sup>5</sup> &minus; 39&Prime;.05 ''T''<sup>6</sup> + 7&Prime;.12 ''T''<sup>7</sup> + 27&Prime;.87 ''T''<sup>8</sup> + 5&Prime;.79 ''T''<sup>9</sup> + 2&Prime;.45 ''T''<sup>10</sup>}}
==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]], 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.


where here {{math|''T''}} is multiples of 10,000 [[Julian day|Julian years]] from [[Epoch (astronomy)|J2000.0]].<ref name="laskar">
[[Simon Newcomb]]'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]] in 2000:
{{cite web
|url = http://articles.adsabs.harvard.edu/full/1986A%26A...157...59L
|title = Secular Terms of Classical Planetary Theories Using the Results of General Relativity
|last = Laskar
|first = J.
|year = 1986}}
, table 8 and eq. (35), at [http://www.adsabs.harvard.edu/ SAO/NASA ADS]</ref>


{{main|Nutation}}
ε = 84381.448 − 46.84024''T'' − (59 × 10<sup>−5</sup>)''T''<sup>2</sup> + (1.813 × 10<sup>−3</sup>)''T''<sup>3</sup>, measured in seconds of arc, with ''T'' being the time in Julian centuries (that is, 36,525 days) since the [[ephemeris]] [[Epoch (reference date)|epoch]] 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.
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<ref>
''Explanatory Supplement'' (1961), sec. 2C</ref>
much smaller (a few [[Minute of arc|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.
The ''true'' or instantaneous obliquity includes this nutation.<ref>
{{cite book
| last = Meeus
| first = Jean
| title = Astronomical Algorithms
| publisher = Willmann-Bell, Inc., Richmond, VA
| year = 1991
|ISBN=0-943396-35-2 }}
, chap. 21</ref>


[[File:Obliquity berger -5000000 to 0.png|thumb|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.]]
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 solar system|numerical model of the solar system]] 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).
[[File:Obliquity berger 0 to 1000000.png|thumb|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 ===
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.


{{main|Formation and evolution of the Solar System}}
==Long period variations==
{{See also2|[[Orbit of the Moon#Tidal evolution of the lunar orbit|Orbit of the Moon (Tidal evolution of the lunar orbit)]]}}


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.
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:
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 [[Axial_precession_(astronomy)|precession]] and the largest [[Term_(mathematics)|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".<ref>
{{cite web
|url=http://adsabs.harvard.edu/abs/1976A%26A....51..127B
|title=Obliquity and Precession for the Last 5000000 Years
|last=Berger
|first=A.L.
|work=Astronomy and Astrophysics '''51'''
|page=127-135
|year=1976}}, at [http://www.adsabs.harvard.edu/ SAO/NASA ADS]</ref>


{{main|Orbit of the Moon}}
'''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.


The [[Moon]] has a stabilizing effect on Earth's obliquity. In the absence of the Moon, the obliquity can change rapidly due to [[orbital resonance]]s and [[Stability_of_the_Solar_System|chaotic behavior]] of the [[Solar System]], reaching as high as 90° in as little as a few million years.<ref>
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]], 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’.{{citation needed|date=December 2010}}
{{cite web
|url=http://bugle.imcce.fr/fr/presentation/equipes/ASD/person/Laskar/misc_files/Laskar_Robutel_1993.pdf
|title=The Chaotic Obliquity of the Planets
|author1=Laskar, J.
|author2=Robutel, P.
|year=1993
|work=Nature '''361'''
|page=608-612}}</ref><ref>
{{cite web
|url=http://www.imcce.fr/Equipes/ASD/person/Laskar/misc_files/Laskar_Joutel_Robutel_1993.pdf
|title=Stabilization of the Earth's Obliquity by the Moon
|author1=Laskar, J.
|author2=Joutel, F.
|author3=Robutel, P.
|year=1993
|work=Nature '''361'''
|page=615-617}}</ref>
This stabilization 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.<ref>
{{cite web
|url=http://adsabs.harvard.edu/abs/1982Icar...50..444W
|title=Comments on the Long-Term Stability of the Earth's Obliquity
|author1=Ward, W.R.
|year=1982
|work=Icarus '''50'''
|page=444-448}}</ref>


== Earth's seasons ==
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°.{{citation needed|date=December 2010}}
[[File:Earth tilt animation.gif|thumb|right|The [[rotation|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.]]
{{main|Season}}
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 [[Hemispheres_of_the_Earth|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 [[season]]s.

Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-term [[climate change]].<ref>
See references at [[Milankovitch cycles]].</ref>

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

Example: an observer at 50° [[latitude]] (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]]).

[[Abu-Mahmud Khojandi]] measured the Earth's axial tilt in the 10th century using this principle with a giant [[sextant]] and noted that his value was lower than those of earlier astronomers, thus discovering that the axial tilt is not constant.<ref>[http://www.encyclopedia.com/doc/1G2-2830902295.html Al-Khujandī, Abū Maḥmūd Ḥāmid Ibn Al-Khiḍr], ''Complete Dictionary of Scientific Biography'', 2008</ref>


Other planets may have a variable obliquity, too; for example, on [[astronomy on Mars|Mars]], the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets.<ref>{{cite journal | last = Touma | first = Jihad | coauthors = Wisdom, Jack | bibcode=1993Sci...259.1294T | title = The Chaotic Obliquity of Mars | journal = Science | volume = 259 | pages = 1294–7 | year = 1993 | doi = 10.1126/science.259.5099.1294 | pmid = 17732249 | issue = 5099}}</ref> 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==
== Other objects of the Solar System ==
Other planets may have a variable obliquity, too; for example, on [[astronomy on Mars|Mars]], the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets.<ref>{{cite journal | last = Touma | first = Jihad | coauthors = Wisdom, Jack | bibcode=1993Sci...259.1294T | title = The Chaotic Obliquity of Mars | journal = Science | volume = 259 | pages = 1294–7 | year = 1993 | doi = 10.1126/science.259.5099.1294 | pmid = 17732249 | issue = 5099}}</ref>


{|class="wikitable"
{|class="wikitable"
Line 140: Line 258:


==External links==
==External links==
{{No footnotes|date=April 2008}}
* Explanatory supplement to "the Astronomical ephemeris" and the ''[[American Ephemeris and Nautical Almanac]]''
* [http://www.tenspheres.com/researches/precession.htm A comparison of values predicted by different theories] at tenspheres.com
* [http://www.tenspheres.com/researches/precession.htm 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
* [http://nssdc.gsfc.nasa.gov/planetary/ National Space Science Data Center]
* [http://nssdc.gsfc.nasa.gov/planetary/ National Space Science Data Center]
* Bryant, Jeff. ''[http://demonstrations.wolfram.com/AxialTiltsOfPlanets/ Axial Tilts of Planets]'', [[Wolfram Demonstrations Project]]
* Bryant, Jeff. ''[http://demonstrations.wolfram.com/AxialTiltsOfPlanets/ Axial Tilts of Planets]'', [[Wolfram Demonstrations Project]]

Revision as of 19:49, 6 April 2012

"Obliquity" redirects here. For the book, see Obliquity (book).

In astronomy, axial tilt, known to astronomers 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.

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 arrows (green) show the direction of the planet's rotation.

Introduction

Orientation of the axes is established by the right hand rule for both the rotation and the orbital motion. When the fingers of the right hand curl around in the direction of the object's rotation, the thumb points in the direction of its north pole (from which, looking back at the object, it appears to rotate counter-clockwise). Similarly, when the fingers of the right hand curl around in the direction of the object's orbital motion, the thumb points in the direction of the north pole of the orbit (from which the object appears to move counter-clockwise in its orbit). The angle between these two poles is the obliquity. At an obliquity of 0°, the axes point in the same direction.

Because the planet Venus has an axial tilt of 177° its rotation can be considered retrograde, opposite that of most of the other planets. 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.[2]

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.

Obliquity of the ecliptic (Earth's axial tilt)

Earth's axial tilt is 23.44°.
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.[3] It is denoted by the Greek letter ε.

The Earth currently has an axial tilt of about 23.44°,[4] This value remains approximately the same relative to a stationary orbital plane throughout the cycles of precession.[5] 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.[6]

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.[7]

JPL's fundamental ephemerides have been continually updated. For instance, the Astronomical Almanac for 2010 specifies:[8]

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.[9] 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.[10]

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[11] much smaller (a few 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. The true or instantaneous obliquity includes this nutation.[12]

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

Main article: Formation and evolution of the Solar System

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".[13]

Main article: Orbit of the Moon

The Moon has a stabilizing effect on Earth's obliquity. 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.[14][15] This stabilization 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.[16]

Earth's seasons

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.
Main article: Season

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.[17]

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

Example: an observer at 50° latitude (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).

Abu-Mahmud Khojandi measured the Earth's axial tilt in the 10th century using this principle with a giant sextant and noted that his value was lower than those of earlier astronomers, thus discovering that the axial tilt is not constant.[18]


Other objects of the Solar System

Other planets may have a variable obliquity, too; for example, on Mars, the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets.[19]

Object Axial tilt (°) Axial tilt (radians)
Sun 7.25 0.1265
Mercury 0.0352 0.000614
Venus 177.4 3.096
Earth 23.44 0.4091
Moon 6.688† 0.1167
Mars 25.19 0.4396
Ceres ~4 ~0.07
Pallas ~60 ~1
Jupiter 3.13 0.0546
Saturn 26.73 0.4665
Uranus 97.77 1.7064
Neptune 28.32 0.4943
Pluto 119.61 2.0876

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

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,[20] 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,[21][22] 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, Mill Valley, CA. ISBN 0-935702-68-7. , p. 733
  2. ^ Planetary Fact Sheet at http://nssdc.gsfc.nasa.gov
  3. ^ U.S. Naval Observatory Nautical Almanac Office, Nautical Almanac Office (2008). The Astronomical Almanac for the Year 2010. U.S. Govt. Printing Office. ISBN 970-0-7077-40829. {{cite book}}: Check |isbn= value: invalid prefix (help); Unknown parameter |coauthors= ignored (|author= suggested) (help) , p. M11
  4. ^ Astronomical Almanac 2010, p. B52
  5. ^ Chauvenet, William (1906). A Manual of Spherical and Practical Astronomy. Vol. I. J.B. Lippincott Co., Philadelphia. , art. 365-367, p. 694-695, at Google books
  6. ^ U.S. Naval Observatory Nautical Almanac Office (1961). Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. H.M. Stationery Office, London. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help), sec. 2B
  7. ^ U.S. Naval Observatory, Nautical Almanac Office (1989). The Astronomical Almanac for the Year 1990. U.S. Govt. Printing Office. ISBN 0-11-886934-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) , p. B18
  8. ^ Astronomical Almanac 2010, p. B52
  9. ^ Newcomb, Simon (1906). A Compendium of Spherical Astronomy. MacMillan Co., New York. , p. 226-227, at Google books
  10. ^ Laskar, J. (1986). "Secular Terms of Classical Planetary Theories Using the Results of General Relativity". , table 8 and eq. (35), at SAO/NASA ADS
  11. ^ Explanatory Supplement (1961), sec. 2C
  12. ^ Meeus, Jean (1991). Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA. ISBN 0-943396-35-2. , chap. 21
  13. ^ Berger, A.L. (1976). "Obliquity and Precession for the Last 5000000 Years". Astronomy and Astrophysics 51. p. 127-135. {{cite web}}: Italic or bold markup not allowed in: |work= (help), at SAO/NASA ADS
  14. ^ Laskar, J.; Robutel, P. (1993). "The Chaotic Obliquity of the Planets" (PDF). Nature 361. p. 608-612. {{cite web}}: Italic or bold markup not allowed in: |work= (help)
  15. ^ Laskar, J.; Joutel, F.; Robutel, P. (1993). "Stabilization of the Earth's Obliquity by the Moon" (PDF). Nature 361. p. 615-617. {{cite web}}: Italic or bold markup not allowed in: |work= (help)
  16. ^ Ward, W.R. (1982). "Comments on the Long-Term Stability of the Earth's Obliquity". Icarus 50. p. 444-448. {{cite web}}: Italic or bold markup not allowed in: |work= (help)
  17. ^ See references at Milankovitch cycles.
  18. ^ Al-Khujandī, Abū Maḥmūd Ḥāmid Ibn Al-Khiḍr, Complete Dictionary of Scientific Biography, 2008
  19. ^ Touma, Jihad (1993). "The Chaotic Obliquity of Mars". Science. 259 (5099): 1294–7. Bibcode:1993Sci...259.1294T. doi:10.1126/science.259.5099.1294. PMID 17732249. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  20. ^ Heller, René. "Holt-Rossiter-McLaughlin Encyclopaedia". René Heller. Retrieved 24 February 2012.
  21. ^ Heller, René (2011). "Tidal obliquity evolution of potentially habitable planets". Astronomy and Astrophysics. 528: A27. Bibcode:2011A&A...528A..27H. doi:10.1051/0004-6361/201015809. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
  22. ^ Heller, René (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. Retrieved 25 February 2012. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)CS1 maint: bibcode (link)

External links

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