Ecliptic coordinate system
Encyclopedia
The ecliptic coordinate system is a celestial coordinate system
that uses the ecliptic
for its fundamental plane
. The ecliptic is the path that the sun
appears to follow across the celestial sphere
over the course of a year. It is also the intersection of the Earth's orbital plane
and the celestial sphere
. The latitudinal
angle is called the ecliptic latitude or celestial latitude (denoted β), measured positive towards the north. The longitudinal
angle is called the ecliptic longitude or celestial longitude (denoted λ), measured eastwards from 0° to 360°. Like right ascension
in the equatorial coordinate system
, 0° ecliptic longitude is pointing towards the Sun from the Earth at the Northern hemisphere vernal equinox. This choice makes the coordinates of the fixed stars subject to shifts due to the precession
, 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
objects. Most planets (except Mercury
), and many small solar system bodies have orbits with small inclinations to the ecliptic plane, and therefore their ecliptic latitude β is always small. Because of the planets' small deviation from the plane of the ecliptic, ecliptic coordinates were used historically to compute their positions. (Aaboe 2001, 17-19)
(Seidelmann 1992, 555–8)
(Seidelmann 1992, 555–8)
Conversely, given equatorial coordinates α and δ, the ecliptic coordinates are:
(Jean Meeus: Astronomical Algorithms, 2nd Edition. ISBN 0-943396-61-1)
The algorithm for the ecliptic to equatorial transformation then becomes:
Similarly for the equatorial to ecliptic transformation.
Celestial coordinate system
In astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.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...
that uses the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
for its fundamental plane
Fundamental plane (spherical coordinates)
The fundamental plane in a spherical coordinate system is a plane which divides the sphere into two hemispheres. The latitude of a point is then the angle between the fundamental plane and the line joining the point to the centre of the sphere....
. The ecliptic is the path that the sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
appears to follow across the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
over the course of a year. It is also the intersection of the Earth's orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...
and the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
. The latitudinal
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
angle is called the ecliptic latitude or celestial latitude (denoted β), measured positive towards the north. The longitudinal
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
angle is called the ecliptic longitude or celestial longitude (denoted λ), 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. The other coordinate is the declination.-Explanation:...
in the equatorial coordinate system
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...
, 0° ecliptic longitude is pointing towards the Sun from the Earth at the Northern hemisphere vernal equinox. This choice makes the coordinates of the fixed stars subject to shifts due to the precession
Precession of the equinoxes
In astronomy, axial precession is a gravity-induced, slow and continuous change in the orientation of an astronomical body's rotational axis. In particular, it refers to the gradual shift in the orientation of Earth's axis of rotation, which, like a wobbling top, traces out a pair of cones joined...
, 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 the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
objects. Most planets (except Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
), and many small solar system bodies have orbits with small inclinations to the ecliptic plane, and therefore their ecliptic latitude β is always small. Because of the planets' small deviation from the plane of the ecliptic, ecliptic coordinates were used historically to compute their positions. (Aaboe 2001, 17-19)
Conversion between celestial coordinate systems
In the formulas below- λ and β are the ecliptic longitude and ecliptic latitude, respectively;
- α and δ are the right ascensionRight ascensionRight ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...
and declinationDeclinationIn astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
, respectively; - ε = 23° 26' 21.406" is the Earth's axial tiltAxial tiltIn astronomy, axial tilt is the angle between an object's rotational axis, and a line perpendicular to its orbital plane...
as of J2000, known as the mean obliquity of the ecliptic. (Astronomical Almanac for the Year 2011, K6)
Conversion from ecliptic coordinates to equatorial coordinates
(Seidelmann 1992, 555–8)
Conversion from equatorial coordinates to ecliptic coordinates
(Seidelmann 1992, 555–8)
Converting angular quantities
Given ecliptic coordinates λ and β, the equatorial coordinates are:- α = atan2Atan2In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
((sin λ cos ε - tan β sin ε), cos λ) - δ = asin(sin ε sin λ cos β + cos ε sin β)
Conversely, given equatorial coordinates α and δ, the ecliptic coordinates are:
- λ = atan2((sin α cos ε + tan δ sin ε), cos α)
- β = asin(sin δ cos ε - cos δ sin α sin ε)
(Jean Meeus: Astronomical Algorithms, 2nd Edition. ISBN 0-943396-61-1)
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:
- Calculate the terms right of the = sign of the 3 equations given above
- Apply a R->P conversion taking the cos α cos δ as the X value and the sin α cos δ 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 δ 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 poleEcliptic poleThe ecliptic pole is the point on the celestial sphere where the sphere meets the imaginary line perpendicular to the ecliptic plane, the path the Earth travels on its orbit around the Sun.There are two ecliptic poles:...
s, where the ecliptic latitude is ±90° - Celestial coordinate systemCelestial coordinate systemIn astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.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...