Ecliptic coordinate system

Not to be confused with Elliptic coordinates.

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)

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 ascension and declination, respectively;
• ε = 23° 26' 21.406" is the Earth's axial tilt as of J2000, known as the mean obliquity of the ecliptic. (Astronomical Almanac for the Year 2011, K6)

Converting Cartesian vectors

Conversion from ecliptic coordinates to equatorial coordinates

$\begin{bmatrix} x_{equatorial} \\ y_{equatorial} \\ z_{equatorial} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \epsilon & -\sin \epsilon \\ 0 & \sin \epsilon & \cos \epsilon \\ \end{bmatrix} \! \cdot \! \begin{bmatrix} x_{ecliptic} \\ y_{ecliptic} \\ z_{ecliptic} \\ \end{bmatrix}$

(Seidelmann 1992, 555–8)

Conversion from equatorial coordinates to ecliptic coordinates

$\begin{bmatrix} x_{ecliptic} \\ y_{ecliptic} \\ z_{ecliptic} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \epsilon & \sin \epsilon \\ 0 & -\sin \epsilon & \cos \epsilon \\ \end{bmatrix} \! \cdot \! \begin{bmatrix} x_{equatorial} \\ y_{equatorial} \\ z_{equatorial} \\ \end{bmatrix}$

(Seidelmann 1992, 555–8)

Converting angular quantities

Given ecliptic coordinates λ and β, the equatorial coordinates are:

α = atan2((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 poles, where the ecliptic latitude is ±90°
• Celestial coordinate system

References

• Astronomical Almanac for the Year 2011. 2010. Washington: US Government Printing Office. ISBN 978-0-7077-41031
• Aaboe, Asger. 2001 Episodes from the Early History of Astronomy. New York: Springer-Verlag.
• Seidelmann, P. Kenneth, ed. 1992. Explanatory Supplement to the Astronomical Almanac. Sausalito, CA: University Science Books. ISBN 0-935702-68-7