Ecliptic coordinate system

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 sky over the course of a year. It is also the projection of the Earth's orbital plane onto the celestial sphere. The latitudinal angle is called the ecliptic latitude (denoted β), measured positive towards the north. The longitudinal angle is called the ecliptic longitude (denoted λ), measured eastwards from 0° to 360°. Like right ascension in the equatorial coordinate system, 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, so that always a reference epoch should be specified. Usually epoch 2000 is taken, but the instantaneous equinox of the day is possible too.

This coordinate system can be particularly useful for charting solar system objects. Most planets (except Mercury and Pluto) and many of the asteroids have orbits with small inclinations to the ecliptic plane, and therefore their ecliptic latitude β is always small.

Conversion between celestial coordinate systems

In the formulas below Harris, Jason. Astroinfo (included with KStars, a Desktop Planetarium for Linux/KDE. See Kstars
*λ and β are the ecliptic longitude and latitude, respectively;
*α and δ are the right ascension and declination, respectively;
*ε = 23.439 281° is the Earth's axial tilt.

Conversion to equatorial coordinates

Declination δ and right ascension α are obtained from:

sin δ = sin ε sin λ cos β + cos ε sin β

cos α cos δ = cos λ cos β

sin α cos δ = cos ε sin λ cos β - sin ε sin β

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

Conversion to ecliptic coordinates

:sin β = cos ε sin δ - sin α cos δ sin ε

cos λ cos β = cos α cos δ

sin λ cos β = sin ε sin δ + sin α cos δ cos ε

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 α in the first set, one could divide out the cos δ leaving one expression for tan α only. Or, one may try to discard the last one equation altogether, only using the second in the form cos α = cos λ cos β / cos δ. 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 α 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 α as it is the same as λ (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 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

References

Explanatory supplement to the Astronomical ephemeris and nautical almanac