|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{distinguish|Elliptic coordinates}}
| | Hello! My name is Shelli. I am satisfied that I could join to the whole globe. I live in France, in the MIDI-PYRENEES region. I dream to visit the different nations, to obtain acquainted with appealing individuals.<br><br>My homepage :: [http://soulstarorange.com/_classppost/index.php?showimage=4%FFfree+minecraft+account%FFexternal+nofollow Fifa 15 Coin Generator] |
| | |
| The '''ecliptic coordinate system''' is a [[celestial coordinate system]] commonly used for representing the positions and [[orbit]]s of [[Solar System]] objects. Because most [[planets]] (except [[Mercury (planet)|Mercury]]), and many [[small solar system bodies]] have orbits with small [[inclination]]s to the [[ecliptic]], it is convenient to use it as the
| |
| [[fundamental plane (spherical coordinates)|fundamental plane]]. The system's [[Origin_(mathematics)|origin]] can be either the center of the [[Sun]] or the center of the [[Earth]], its primary direction is towards the vernal (northbound) [[equinox]], and it has a [[Orientation (vector space)|right-handed]] convention. It may be implemented in [[Spherical coordinate system|spherical]] or [[Cartesian coordinate system|rectangular]] coordinates.<ref>
| |
| {{cite book
| |
| | author = Nautical Almanac Office, U.S. Naval Observatory
| |
| | coauthors = H.M. Nautical Almanac Office, Royal Greenwich Observatory
| |
| | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
| |
| | publisher = H.M. Stationery Office, London
| |
| | year = 1961
| |
| |pages=24-27}}</ref>
| |
| | |
| [[File:Ecliptic grid globe.png|thumb|Earth-centered '''Ecliptic coordinates''' as seen from outside the [[celestial sphere]]. Ecliptic longitude (red) is measured along the [[ecliptic]] from the vernal [[equinox]]. Ecliptic latitude (yellow) is measured perpendicular to the ecliptic. A full globe is shown here, although high-latitude coordinates are seldom seen except for certain [[comet]]s and [[asteroid]]s.]]
| |
| | |
| == Primary direction ==
| |
| | |
| [[File:Ecliptic vs equator small.gif|thumb|The apparent motion of the [[Sun]] along the ecliptic (red) as seen on the inside of the [[celestial sphere]]. Ecliptic coordinates appear in (red). The [[celestial equator]] (blue) and the [[Equatorial coordinate system|equatorial coordinates]] (blue), being inclined to the ecliptic, appear to wobble as the Sun advances.]]
| |
| | |
| {{also|Axial precession|Nutation}}
| |
| | |
| The [[celestial equator]] and the [[ecliptic]] are slowly moving due to [[Perturbation_(astronomy)|perturbing forces]] on the [[Earth]], therefore the [[Orientation_(geometry)|orientation]] of the primary direction, their intersection at the [[Northern Hemisphere]] vernal [[equinox]], is not quite fixed. A slow motion of Earth's axis, [[Axial_precession|precession]], causes a slow, continuous turning of the coordinate system westward about the poles of the [[ecliptic]], completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the [[ecliptic]], and a small oscillation of the Earth's axis, [[nutation]].<ref>
| |
| ''Explanatory Supplement'' (1961), pp. 20, 28</ref><ref>
| |
| {{cite book
| |
| | last1 = U.S. Naval Observatory
| |
| | first1=Nautical Almanac Office
| |
| | editor = P. Kenneth Seidelmann
| |
| | title = Explanatory Supplement to the Astronomical Almanac
| |
| | publisher = University Science Books, Mill Valley, CA
| |
| | year = 1992
| |
| | isbn = 0-935702-68-7
| |
| |pages=11-13}}
| |
| </ref>
| |
|
| |
| In order to reference a coordinate system which can be considered as fixed in space, these motions require specification of the [[equinox]] of a particular date, known as an [[Epoch_(astronomy)|epoch]], when giving a position in ecliptic coordinates. The three most commonly used are:
| |
| *Mean equinox of a standard epoch (usually [[Epoch_(astronomy)|J2000.0]], but may include B1950.0, B1900.0, etc.)
| |
| :is a fixed standard direction, allowing positions established at various dates to be compared directly.
| |
| *Mean equinox of date
| |
| :is the intersection of the [[ecliptic]] of "date" (that is, the ecliptic in its position at "date") with the ''mean'' equator (that is, the equator rotated by [[Axial_precession|precession]] to its position at "date", but free from the small periodic oscillations of [[nutation]]). Commonly used in planetary [[orbit]] calculation.
| |
| *True equinox of date
| |
| :is the intersection of the [[ecliptic]] of "date" with the ''true'' equator (that is, the mean equator plus [[nutation]]). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.
| |
| A position in the ecliptic coordinate system is thus typically specified ''true equinox and ecliptic of date'', ''mean equinox and ecliptic of J2000.0'', or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.<ref>
| |
| {{cite book
| |
| | last = Meeus
| |
| | first = Jean
| |
| | title = Astronomical Algorithms
| |
| | publisher = Willmann-Bell, Inc., Richmond, VA
| |
| | year = 1991
| |
| |page=137
| |
| |ISBN=0-943396-35-2 }}</ref>
| |
| | |
| ==Spherical coordinates==
| |
| | |
| {| class="wikitable" style="float:right;"
| |
| |+<br>'''Summary of notation for ecliptic coordinates'''<ref>''Explanatory Supplement'' (1961), sec. 1G</ref>
| |
| | rowspan="2" bgcolor="#89CFF0" |
| |
| | colspan="3" align="center" bgcolor="#89CFF0" | '''spherical'''
| |
| | rowspan="2" align="center" bgcolor="#89CFF0" | '''rectangular'''
| |
| |- bgcolor="#89CFF0" align="center"
| |
| | longitude
| |
| | latitude
| |
| | distance
| |
| |- align="center"
| |
| | bgcolor="#89CFF0" | '''geocentric'''
| |
| | {{math|''λ''}}
| |
| | {{math|''β''}}
| |
| | {{math|''Δ''}}
| |
| |
| |
| |- align="center"
| |
| | bgcolor="#89CFF0" | '''heliocentric'''
| |
| | {{math|''l''}}
| |
| | {{math|''b''}}
| |
| | {{math|''r''}}
| |
| | {{math|''x''}}, {{math|''y''}}, {{math|''z''}}<ref group="note">Occasional use; {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are usually reserved for [[Equatorial coordinate system|equatorial coordinates]].</ref>
| |
| |-
| |
| | colspan="5" | {{Reflist|group="note"}}
| |
| |}
| |
| | |
| '''Ecliptic longitude''' or celestial longitude (symbols: heliocentric <math>l</math>, geocentric <math>\lambda</math>) measures the angular distance of an object along the [[ecliptic]] from the primary direction. Like [[right ascension]] in the [[equatorial coordinate system]], the primary direction (0° ecliptic longitude) points from the Earth towards the Sun at the vernal [[equinox]] of the Northern Hemisphere. Because it is a right-handed system, ecliptic longitude is measured positive eastwards in the fundamental plane (the ecliptic) from 0° to 360°.
| |
| | |
| '''Ecliptic latitude''' or celestial latitude (symbols: heliocentric <math>b</math>, geocentric <math>\beta</math>), measures the angular distance of an object from the [[ecliptic]] towards the north (positive) or south (negative) [[ecliptic pole]]. For example, the [[Ecliptic_pole|north ecliptic pole]] has a celestial latitude of +90°.
| |
| | |
| '''Distance''' is also necessary for a complete spherical position (symbols: heliocentric <math>r</math>, geocentric <math>\mathit\Delta</math>). Different distance units are used for different objects. Within the [[Solar System]], [[astronomical unit]]s are used, and for objects near the [[Earth]], [[Earth_radius|Earth radii]] or [[kilometer]]s are used.
| |
| | |
| === In ancient times ===
| |
| | |
| Historically, ecliptic longitude was measured using twelve ''[[Astrological_sign|signs]]'', each of 30° longitude, a legacy of [[astrology]]. The signs approximately corresponded to the [[constellation]]s crossed by the ecliptic. Longitudes were specified in signs, degrees, minutes, and seconds; for instance a longitude of [[Image:leo.svg|12px]] 19° 55' 58" was 19°.933 east of the start of the sign [[Leo_(astrology)|Leo]], which was 120° from the [[equinox]].<ref>
| |
| {{cite web
| |
| |url=http://books.google.com/books?id=z3gRvA2J3DYC&source=gbs_navlinks_s
| |
| |title=A Compleat System of Astronomy
| |
| |first=Charles
| |
| |last=Leadbetter
| |
| |year=1742
| |
| |publisher=J. Wilcox, London
| |
| |page=94}}, at [http://books.google.com/books Google books]; numerous examples of this notation appear throughout the book.</ref>
| |
| | |
| == Rectangular coordinates == | |
| | |
| [[File:Heliocentric rectangular ecliptic.png|thumb|Heliocentric ecliptic coordinates. The [[Origin_(mathematics)|origin]] is the center of the [[Sun]]. The fundamental [[Plane_(geometry)|plane]] is the plane of the [[ecliptic]]. The primary direction (the ''x'' axis) is the vernal [[equinox]]. A [[right-handed]] convention specifies a ''y'' axis 90° to the east in the fundamental plane; the ''z'' axis points toward the north [[ecliptic pole]]. The reference frame is relatively stationary, aligned with the vernal equinox.]]
| |
| | |
| There is a [[Cartesian coordinate system|rectangular]] variant of ecliptic coordinates often used in [[orbit]]al calculation. It has its [[Origin_(mathematics)|origin]] at the center of the [[Sun]], its fundamental [[Plane_(geometry)|plane]] in the plane of the ecliptic, its primary direction (the <math>x</math> axis) toward the vernal [[equinox]], that is, the place where the [[Sun]] crosses the [[celestial equator]] in a northward direction in its annual apparent circuit around the [[ecliptic]], and a [[right-handed]] convention, specifying a <math>y</math> axis 90° to the east in the fundamental plane and a <math>z</math> axis perpendicular to the <math>x</math>-<math>y</math> plane in a right-handed sense.<ref>
| |
| ''Explanatory Supplement'' (1961), pp. 20, 27</ref>
| |
| | |
| These rectangular coordinates are related to the corresponding spherical coordinates by
| |
| | |
| ::<math>x = r \cos b \cos l</math>
| |
| | |
| ::<math>y = r \cos b \sin l</math>
| |
| | |
| ::<math>z = r \sin b</math>.
| |
| | |
| == Conversion between celestial coordinate systems ==
| |
| {{main|Celestial coordinate system}}
| |
| | |
| === Converting Cartesian vectors ===
| |
| ==== Conversion from ecliptic coordinates to equatorial coordinates ====
| |
| <math>
| |
| \begin{bmatrix}
| |
| x_{equatorial} \\
| |
| y_{equatorial} \\
| |
| z_{equatorial} \\
| |
| \end{bmatrix}
| |
| =
| |
| \begin{bmatrix}
| |
| 1 & 0 & 0 \\
| |
| 0 & \cos \epsilon & -\sin \epsilon \\
| |
| 0 & \sin \epsilon & \cos \epsilon \\
| |
| \end{bmatrix} \! \cdot \!
| |
| \begin{bmatrix}
| |
| x_{ecliptic} \\
| |
| y_{ecliptic} \\
| |
| z_{ecliptic} \\
| |
| \end{bmatrix}
| |
| </math><ref>
| |
| ''Explanatory Supplement'' (1992), pp. 555-558</ref>
| |
| | |
| ==== Conversion from equatorial coordinates to ecliptic coordinates ====
| |
| <math>
| |
| \begin{bmatrix}
| |
| x_{ecliptic} \\
| |
| y_{ecliptic} \\
| |
| z_{ecliptic} \\
| |
| \end{bmatrix}
| |
| =
| |
| \begin{bmatrix}
| |
| 1 & 0 & 0 \\
| |
| 0 & \cos \epsilon & \sin \epsilon \\
| |
| 0 & -\sin \epsilon & \cos \epsilon \\
| |
| \end{bmatrix} \! \cdot \!
| |
| \begin{bmatrix}
| |
| x_{equatorial} \\
| |
| y_{equatorial} \\
| |
| z_{equatorial} \\
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| where <math>\epsilon</math> is the [[obliquity of the ecliptic]].
| |
| | |
| == See also ==
| |
| * [[Celestial coordinate system]]
| |
| *[[Ecliptic]]
| |
| * [[Ecliptic pole]]s, where the ecliptic latitude is ±90°
| |
| *[[Equinox]]
| |
| | |
| == External links ==
| |
| * [http://www.dur.ac.uk/john.lucey/users/solar_year.html The Ecliptic: the Sun's Annual Path on the Celestial Sphere] Durham University Department of Physics
| |
| * [http://stars.astro.illinois.edu/celsph.html MEASURING THE SKY A Quick Guide to the Celestial Sphere] James B. Kaler, University of Illinois
| |
| | |
| ==Notes and references==
| |
| {{reflist}}
| |
| | |
| {{Celestial coordinate systems}}
| |
| | |
| [[Category:Celestial coordinate system]]
| |