Ecliptic coordinate system

Spherical coordinates

Ecliptic longitude

Ecliptic longitude or celestial longitude (symbols: heliocentric ℓ, geocentric λ) 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 March equinox. Because it is a right-handed system, ecliptic longitude is measured positive eastwards in the fundamental plane (the ecliptic) from 0°–360° . Because of axial precession, the ecliptic longitude of most "fixed stars" (referred to the equinox of date) increases by about 50.3 arcseconds per year, or 83.8 arcminutes per century, the speed of general precession. However, for stars near the ecliptic poles, the rate of change of ecliptic longitude is dominated by the slight movement of the ecliptic (that is, of the plane of the Earth's orbit), so the rate of change may be anything from minus infinity to plus infinity depending on the exact position of the star.

Ecliptic latitude

Ecliptic latitude or celestial latitude (symbols: heliocentric b, geocentric β ), measures the angular distance of an object from the ecliptic towards the north (positive) or south (negative) ecliptic pole. For example, the north ecliptic pole has a celestial latitude of +90°. Ecliptic latitude for "fixed stars" is not affected by precession.

Distance

Distance is also necessary for a complete spherical position (symbols: heliocentric r, geocentric Δ). Different distance units are used for different objects. Within the Solar System, astronomical units (au) are used, and for objects near the Earth, Earth radii, Lunar distances, or thousands of kilometers are used.

Historical use

From antiquity through the 18th century, ecliptic longitude was commonly measured using twelve zodiacal signs, each of 30° longitude, a practice that continues in modern astrology. The signs are approximately the positions on the ecliptic, relative to the position of the Sun on the March equinox, where constellations were on the ecliptic in the year 200 ce ; however the stars of those constellations have precessed eastward and the stars are now shifted by about one-and-a-half 30° signs from the locations of their nominal zodiacal signs. Longitudes were specified in signs, degrees, minutes, and seconds.

For an example of historical use, a longitude of ♌ 19°55′58″ is 19.933° east of the start of the sign Leo (♌). Since Leo begins 120° from the March equinox, the longitude in modern form is 139° 55′ 58″.

In China, ecliptic longitude is measured using 24 Solar terms, each of 15° longitude, and are used by Chinese lunisolar calendars to stay synchronized with the seasons, which is crucial for agrarian societies.

Rectangular coordinates

A rectangular variant of ecliptic coordinates is often used in orbital calculations and simulations. It has its origin at the center of the Sun (or at the barycenter of the Solar System), its fundamental plane on the ecliptic plane, and the x-axis toward the March equinox. The coordinates have a right-handed convention, that is, if one extends their right thumb upward, it simulates the z-axis, their extended index finger the x-axis, and the curl of the other fingers points generally in the direction of the y-axis.

These rectangular coordinates are related to the corresponding spherical coordinates by

${\displaystyle \begin{array}{cc}x& =r\cos b\cos \ell \\ y& =r\cos b\sin \ell \\ z& =r\sin b\end{array}}$

Conversion between celestial coordinate systems

Converting Cartesian vectors

Conversion from ecliptic coordinates to equatorial coordinates

$${\displaystyle \left[\begin{array}{c}{x}_{\text{equatorial}}\\ y_{\text{equatorial}}\\ z_{\text{equatorial}}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \epsilon & -\sin \epsilon \\ 0& \sin \epsilon & \cos \epsilon \end{array}\right]\left[\begin{array}{c}{x}_{\text{ecliptic}}\\ y_{\text{ecliptic}}\\ z_{\text{ecliptic}}\end{array}\right]}$$

Conversion from equatorial coordinates to ecliptic coordinates

$${\displaystyle \left[\begin{array}{c}{x}_{\text{ecliptic}}\\ y_{\text{ecliptic}}\\ z_{\text{ecliptic}}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \epsilon & \sin \epsilon \\ 0& -\sin \epsilon & \cos \epsilon \end{array}\right]\left[\begin{array}{c}{x}_{\text{equatorial}}\\ y_{\text{equatorial}}\\ z_{\text{equatorial}}\end{array}\right]}$$ where ε is the obliquity of the ecliptic.