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The ecliptic coordinate system is a celestial coordinate system
Celestial coordinate system

In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on the surface of the Earth....
 that uses the ecliptic
Ecliptic

The ecliptic is the apparent path that the Sun traces out in the sky during the year. As it appears to move in the sky in relation to the stars, the apparent path aligns with the planets throughout the course of the year....
 for its fundamental plane
Fundamental plane

The fundamental plane may refer to:* The fundamental plane which divides a spherical coordinate system.* The fundamental plane which shows an empirical relationship between mean surface brightness, velocity dispersion and effective radius of an elliptical galaxy....
. The ecliptic is the path that the sun
Sun

The Sun , a G V star, is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 98.6% of the Solar System's mass....
 appears to follow across the sky over the course of a year. It is also the projection
Projection

Projection can be any of:* The display of an image by devices such as:**Movie projector**Video projector**Overhead projector**Slide projector...
 of the Earth's orbital plane
Orbital plane (astronomy)

The orbital plane of an object orbiting another is the geometrical Plane in which the orbit is embedding. Three points in space suffice to define the orbital plane....
 onto the celestial sphere
Celestial sphere

In astronomy and navigation, the celestial sphere is an imagination rotation sphere of "gigantic radius", concentric spheres and coaxial with the Earth....
. The latitudinal
Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
 angle is called the ecliptic latitude
Ecliptic latitude

Ecliptic latitude, or Celestial latitude, is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
 or celestial latitude (denoted ß), measured positive towards the north. The longitudinal
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 angle is called the ecliptic longitude
Ecliptic longitude

Ecliptic longitude is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
 or celestial longitiude (denoted ?), measured eastwards from 0° to 360°.






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The ecliptic coordinate system is a celestial coordinate system
Celestial coordinate system

In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on the surface of the Earth....
 that uses the ecliptic
Ecliptic

The ecliptic is the apparent path that the Sun traces out in the sky during the year. As it appears to move in the sky in relation to the stars, the apparent path aligns with the planets throughout the course of the year....
 for its fundamental plane
Fundamental plane

The fundamental plane may refer to:* The fundamental plane which divides a spherical coordinate system.* The fundamental plane which shows an empirical relationship between mean surface brightness, velocity dispersion and effective radius of an elliptical galaxy....
. The ecliptic is the path that the sun
Sun

The Sun , a G V star, is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 98.6% of the Solar System's mass....
 appears to follow across the sky over the course of a year. It is also the projection
Projection

Projection can be any of:* The display of an image by devices such as:**Movie projector**Video projector**Overhead projector**Slide projector...
 of the Earth's orbital plane
Orbital plane (astronomy)

The orbital plane of an object orbiting another is the geometrical Plane in which the orbit is embedding. Three points in space suffice to define the orbital plane....
 onto the celestial sphere
Celestial sphere

In astronomy and navigation, the celestial sphere is an imagination rotation sphere of "gigantic radius", concentric spheres and coaxial with the Earth....
. The latitudinal
Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
 angle is called the ecliptic latitude
Ecliptic latitude

Ecliptic latitude, or Celestial latitude, is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
 or celestial latitude (denoted ß), measured positive towards the north. The longitudinal
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 angle is called the ecliptic longitude
Ecliptic longitude

Ecliptic longitude is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
 or celestial longitiude (denoted ?), measured eastwards from 0° to 360°. Like right ascension
Right ascension

Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system....
 in the equatorial coordinate system
Equatorial coordinate system

The equatorial coordinate system is probably the most widely used celestial coordinate system, whose equatorial coordinates are:* declination ...
, the origin for ecliptic longitude is the vernal equinox. This choice makes the coordinates of the fixed stars subject to shifts due to the precession
Precession of the equinoxes

In astronomy, precession refers to a gravitationally-induced slow but continuous change in an astronomical body's rotational axis or orbital path....
, so that always a reference epoch should be specified. Usually epoch J2000.0 is taken, but the instantaneous equinox of the day (called the epoch of date) is possible too.

This coordinate system can be particularly useful for charting solar system
Solar System

The Solar System consists of the Sun and those Astronomical object bound to it by gravity: the eight planets and five dwarf planets, their 173 known Natural satellite, and billions of Small Solar System body....
 objects. Most planets (except Mercury
Mercury (planet)

Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 88 days. The orbit of Mercury has the highest Orbital eccentricity of all the Solar System planets, and it has the smallest axial tilt....
), dwarf planet
Dwarf planet

A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be rounded by its own gravity but has not Clearing the neighbourhood of planetesimals and is not a natural satellite....
s, and many small solar system bodies have orbits with small inclinations to the ecliptic plane, and therefore their ecliptic latitude ß is always small. Because of the planets' small deviation from the plane of the ecliptic, ecliptic coordinates were used historically to compute their positions.

Conversion between celestial coordinate systems

In the formulas below
  • ? and ß are the ecliptic longitude
    Ecliptic longitude

    Ecliptic longitude is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
     and ecliptic latitude
    Ecliptic latitude

    Ecliptic latitude, or Celestial latitude, is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
    , respectively;
  • a and d are the right ascension
    Right ascension

    Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system....
     and declination
    Declination

    In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle....
    , respectively;
  • e = 23.439 281° is the Earth
    Earth

    Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
    's axial tilt
    Axial tilt

    In astronomy, axial tilt is the inclination angle of a planet axis of rotation in relation to its Orbital plane . It is also called axial inclination or obliquity....
    .


Conversion from ecliptic coordinates to equatorial coordinates

Declination d and right ascension a are obtained from:

sin d = sin e sin ? cos ß + cos e sin ß


cos a cos d = cos ? cos ß


sin a cos d = cos e sin ? cos ß - sin e sin ß


All three equations must in general be satisfied because cos and sin do not specify their argument uniquely.

Conversion from equatorial coordinates to ecliptic coordinates

sin ß = cos e sin d - sin a cos d sin e


cos ? cos ß = cos a cos d


sin ? cos ß = sin e sin d + sin a cos d cos e


Caution

One may be tempted to 'simplify' the last two equations in each set, but in general this is not a wise policy because cos and sin do not specify their argument uniquely, while standard implementations of inverse trigonometric functions assume the angle to be in a restricted range. For example, to obtain a in the first set, one could divide out the cos d leaving one expression for tan a only. Or, one may try to discard the last one equation altogether, only using the second in the form cos a = cos ? cos ß / cos d. While this works in some straightforward cases, it can be misleading in general. For example cos-1 gives angles between 0° and 180° in most implementations, while a can take on all angles up to 360°. sin-1 and tan-1 are also limited to a 180° range. All these functions are also very prone to rounding errors near their limits.

In practice, for bodies close to the ecliptic, one can infer the right quadrant of a as it is the same as ? (but beware exceptions near the poles!). This, however, is manual tweaking, and not easily programmed for more general applications.

An algorithm

If the calculation is to be done with an electronic pocket calculator, it is best to use a rectangular to polar (R->P) and polar to rectangular (P->R) function, which are found on most scientific calculators. They avoid all the above problems and give us an extra sanity check as well.

The algorithm for the ecliptic to equatorial transformation then becomes as follows.
  • Calculate the terms right of the = sign of the 3 equations given above
  • Apply a R->P conversion taking the cos a cos d as the X value and the sin a cos d as the Y value
  • The angle part of the answer is the right ascension, an angle over the full range of 0° to 360° (or -180° to +180° etc.), which after division by 15 gives the hours.
  • Apply a second R->P conversion taking the radius part of the last answer as the X and the sin d of the first equation as the Y value
  • The angle part of the answer is the declination, an angle between -90° and +90°
  • The radius part of the answer must be 1 exactly, if not you have made an error.
Similarly for the equatorial to ecliptic transformation

See also


  • Ecliptic latitude
    Ecliptic latitude

    Ecliptic latitude, or Celestial latitude, is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....
  • Ecliptic longitude
    Ecliptic longitude

    Ecliptic longitude is one of the co-ordinates which can be used to define the location of an astronomical object on the celestial sphere in the ecliptic coordinate system....