# Astronomical coordinate systems

Coordinate system of the horizon , northern hemisphere
Coordinate system of the local equator , northern hemisphere

Astronomical coordinate systems are used to indicate the position of celestial bodies . In them the two angles of spherical coordinates are used. As a rule, the distance is not used as the third spherical coordinate. Because of the great distances of the celestial bodies from the earth, it is sufficient for the purpose of observation to determine the direction of the objects as star locations z. B. to be specified in star catalogs .

The origin of the coordinates of astronomical systems is the observer (a place on the earth's surface, sight problems ), the center of the earth ( geocentric view of the world ), the sun ( heliocentric view of the world ) or another celestial body (for example a planet to determine the position of its moons relative to it specify yourself), or a spacecraft. It is located in a reference plane to be selected, within which one of the two astronomical angle coordinates is determined. The second angle is measured vertically over the reference plane up to the observed celestial body. The arbitrarily selectable zero points of the angle coordinates depend on the respective application.

## Classification

### Relative coordinate systems

Relative coordinate systems are tied to the observer. They have their reference point at the place of the observer, i.e. on the earth's surface, and are also called local coordinate systems or topocentric coordinate systems.

The horizon system is the most familiar coordinate system to every observer. It is at its origin, the horizon is the reference plane. The angle above the horizon to the celestial body is its height angle h (elevation). The deviation of the point at which the vertical through the celestial body intersects the horizon from the south direction is the azimuth a . When using the horizon system in the southern hemisphere, north is the reference direction. At the equator and on the poles the horizon system is indefinite.

In the case of the local equator system ( equatorial system at rest) , the observer is also in the origin of the coordinates. The reference plane is the celestial equator , in which the hourly angle  τ on the celestial equator is measured from the upper intersection of the local meridian with the celestial equator to the meridian of the celestial body. The angles determined from the surface of the earth differ little from those with the origin of coordinates in the center of the earth due to the small extent of the earth. Exceptions are observations of near-earth objects such as B. neighboring planets. For comparison purposes (e.g. during a Venus transit ), they are converted to coordinates with their origin in the center of the earth.

Astronauts use coordinate systems linked to their missile in space .

### Absolute coordinate systems

Absolute coordinate systems have their origin at a point that is neutral relative to the observer: in the center of the earth, sun or another celestial body or in the galactic center . Your reference plane is also not tied to the observer, so it rotates relative to him.

The rotating equatorial coordinate system emerges from the stationary equatorial (geocentric) coordinate system mentioned above . It has its origin in the center of the earth, the reference point for the angle measurement in the equatorial plane of the sky is the fixed spring point in the sky . The angle given in the equatorial plane is the right ascension  α . The declination  angle δ is identical to the declination angle in the equatorial system at rest.

With the orbital plane called the ecliptic , in which the earth orbits the sun once a year, as a reference plane, two astronomical coordinate systems are defined. The first of the two ecliptical coordinate systems has the origin in the center of the earth (geocentric), the second in the center of the sun (heliocentric). In both cases, the coordinate angles are called ecliptical longitude  λ (reference point is the vernal equinox) and ecliptical latitude  β.

In addition to topocentric (always relative systems), geocentric and heliocentric, barycentric and galactic coordinate systems are also used.

The galactic coordinate system has its origin in the galactic center, its reference plane is the Milky Way disk .

A barycentric coordinate system , for example, has its origin in the barycentre (common center of gravity), for example of the earth and moon, or in the solar system.

These systems rotate around the observer. In astronomy, however, it is common to view the fixed star sky as stationary, which is why one speaks of "absolute", while the observer-related positions are referred to as " apparent ".

## Angle specifications in hours instead of degrees

With the hour angle (stationary equatorial coordinate system) and the right ascension (rotating equatorial coordinate system), the information in hours, minutes and seconds (hourly or time ) is preferred to those in degrees. The reason for the hour angle is that the change in the hour angle of the sun determines the change in the time of day . 15 ° change is one hour , that is its original definition.

The cause of this custom for right ascension is the influence of the earth's rotation, of which it is in principle independent, on the measurement. Two stars with a 15 ° difference in right ascension run through the meridian circle of an observatory with one hour difference in sidereal time . A sidereal time hour is about 10 seconds shorter than an hour. The observation plan in an observatory is based on the sidereal time, which is known for each star and can be read on a corresponding clock. This shows 0 o'clock sidereal time when the vernal equinox, the reference point for equatorial celestial coordinates, passes the meridian circle. The time of day lags sidereal time by one day a year, just as the sun (apparently) wanders backwards through the starry sky once a year.

## Overview table

Coordinate system Origin of coordinates Fundamental level Pole Coordinates Reference direction
vertical horizontal
horizontal observer horizon Zenith / Nadir Elevation angle h Azimuth a North or south point of the horizon
equatorial
"dormant"
Observer or center of the earth Celestial equator Celestial poles declination angle δ Hour angle τ Observer's meridian
equatorial
"rotating"
Center of the earth Right ascension α Spring equinox
ecliptical Center of the sun or

Center of the earth

Ecliptic Ecliptic poles ecliptical latitude β ecliptical length λ
galactic Center of the sun galactic plane galactic poles galactic latitude b galactic longitude l galactic center

## Conversions

The conversions are carried out using the representations in Cartesian coordinates of both systems. The transformation - a rotation around the y-axis - takes place between the Cartesian forms of the systems (the y-coordinates are the same in both systems): rotation through the angle 90 ° -φ (φ = geographical latitude ) in the first, around the Angle ε ( skew of the ecliptic ) in the second case.

To convert: Horizontal system ↔ equatorial system (resting) The zenith is perpendicular above the observer and the nadir perpendicular below the observer in the center of the representation. If the observer is on the north or south pole ( ), then the horizon and equatorial planes are identical, and the zenith and nadir are on the polar axis (blue). ${\ displaystyle \ phi = 0}$

The meridian is the great circle through the north celestial and south celestial poles as well as the north (N) and south (S) directions as seen from the observer.

In the horizontal system (gray disk) the observer sees a point (violet) in the sky below the azimuth a (black), which is measured from the meridian in the horizontal plane, and the elevation angle h (green), which is perpendicular to the horizontal plane on the great circle between Zenith and nadir (green) is measured, which goes through the observed point. These angles can be converted into the Cartesian coordinates x, y and z in the horizontal system.

In the equatorial system (turquoise disc) the hourly angle τ (cyan) is determined from the meridian in the equatorial plane and the declination angle δ (red) perpendicular to the equatorial plane on the great circle that goes through the celestial poles and the observed point.

East point (O) and west point (W) are identical in both systems, and the inclination of the two planes is given by the polar height φ (blue), which corresponds to the degree of latitude on which the observer is located. The observed point in the sky (purple) moves within half a day apparently on a semicircle from east to west, which runs parallel to the equatorial plane with a constant declination angle δ.

In the following tables, in addition to the final results of the conversions, the Cartesian coordinates x , y and z of the unit sphere in the target system are given as intermediate results. It should be noted that the first two systems (horizontal and equatorial at rest ) are defined as left systems , the other two (rotating equatorial and geocentric-ecliptical) as right systems .

### Resting equatorial (δ, τ) into rotating equatorial coordinates (δ, α) and vice versa

 ${\ displaystyle \ theta}$ = Sidereal time at the point of observation ${\ displaystyle \ tau}$ = Hour angle ${\ displaystyle \ alpha}$ = Right ascension ${\ displaystyle \ delta}$ = Declination

The declination δ remains unchanged.

${\ displaystyle \ alpha = \ theta - \ tau}$
${\ displaystyle \ tau = \ theta - \ alpha}$

### Horizontal (a, h) → Cartesian coordinates → equatorial coordinates at rest (τ, δ)

 ${\ displaystyle \ phi}$ = latitude ${\ displaystyle a}$ = Azimuth ${\ displaystyle h}$ = Elevation angle ${\ displaystyle \ tau}$ = Hour angle ${\ displaystyle \ delta}$ = Declination
Cartesian coordinates in the target system ( , ):${\ displaystyle \ tau}$${\ displaystyle \ delta}$
${\ displaystyle x = \ cos \ delta \ cdot \ cos \ tau = \ cos \ phi \ cdot \ sin h + \ sin \ phi \ cdot \ cos h \ cdot \ cos a}$
${\ displaystyle y = \ cos \ delta \ cdot \ sin \ tau = \ cos h \ cdot \ sin a}$
${\ displaystyle z = \ sin \ delta = \ sin \ phi \ cdot \ sin h- \ cos \ phi \ cdot \ cos h \ cdot \ cos a}$
Angular coordinates in the target system:
${\ displaystyle \ delta = \ arcsin \ left (\ sin \ phi \ cdot \ sin h- \ cos \ phi \ cdot \ cos h \ cdot \ cos a \ right)}$
${\ displaystyle \ tau = \ arctan \ left ({\ frac {\ sin a} {\ sin \ phi \ cdot \ cos a + \ cos \ phi \ cdot \ tan h}} \ right)}$
(Here the determination of the quadrant applies according to the conversion from Cartesian to polar coordinates)

### Resting equatorial (τ, δ) → Cartesian coordinates → horizontal coordinates (a, h)

 ${\ displaystyle \ phi}$ = latitude ${\ displaystyle a}$ = Azimuth ${\ displaystyle h}$ = Elevation angle ${\ displaystyle \ tau}$ = Hour angle ${\ displaystyle \ delta}$ = Declination
Cartesian coordinates in the target system ( , ):${\ displaystyle a}$${\ displaystyle h}$
${\ displaystyle x = \ cos h \ cdot \ cos a = - \ cos \ phi \ cdot \ sin \ delta + \ sin \ phi \ cdot \ cos \ delta \ cdot \ cos \ tau}$
${\ displaystyle y = \ cos h \ cdot \ sin a = \ cos \ delta \ cdot \ sin \ tau}$
${\ displaystyle z = \ sin h = \ sin \ phi \ cdot \ sin \ delta + \ cos \ phi \ cdot \ cos \ delta \ cdot \ cos \ tau}$
Angular coordinates in the target system:
${\ displaystyle h = \ arcsin \ left (\ sin \ phi \ cdot \ sin \ delta + \ cos \ phi \ cdot \ cos \ delta \ cdot \ cos \ tau \ right)}$
${\ displaystyle a = \ arctan \ left ({\ frac {\ sin \ tau} {\ sin \ phi \ cdot \ cos \ tau - \ cos \ phi \ cdot \ tan \ delta}} \ right)}$
(Here the determination of the quadrant applies according to the conversion from Cartesian to polar coordinates)

### Rotating equatorial (α, δ) → Cartesian coordinates → horizontal coordinates (a, h)

 ${\ displaystyle \ phi}$ = latitude ${\ displaystyle \ theta}$ = Sidereal time at the point of observation ${\ displaystyle \ alpha}$ = Right ascension ${\ displaystyle \ delta}$ = Declination ${\ displaystyle a}$ = Azimuth ${\ displaystyle h}$ = Elevation angle

Cartesian coordinates in the target system (a, h)

${\ displaystyle x = \ cos h \ cdot \ cos a = - \ cos \ phi \ cdot \ sin \ delta + \ sin \ phi \ cdot \ cos \ delta \ cdot \ cos (\ theta - \ alpha)}$
${\ displaystyle y = \ cos h \ cdot \ sin a = \ cos \ delta \ cdot \ sin (\ theta - \ alpha)}$
${\ displaystyle z = \ sin h = \ sin \ phi \ cdot \ sin \ delta + \ cos \ phi \ cdot \ cos \ delta \ cdot \ cos (\ theta - \ alpha)}$

Angular coordinates in the target system

${\ displaystyle a = \ arctan {\ frac {\ sin (\ theta - \ alpha)} {\ sin \ phi \ cdot \ cos (\ theta - \ alpha) - \ cos \ phi \ cdot \ tan \ delta}} }$
(Here the determination of the quadrant applies according to the conversion from Cartesian to polar coordinates)
${\ displaystyle h = \ arcsin \ left (\ sin \ phi \ cdot \ sin \ delta + \ cos \ phi \ cdot \ cos \ delta \ cdot \ cos (\ theta - \ alpha) \ right)}$

### Rotating equatorial (α, δ) → ecliptical coordinates (λ, β, geocentric)

 ${\ displaystyle \ epsilon}$ = 23.44 ° = inclination of the ecliptic ${\ displaystyle \ alpha}$ = Right ascension ${\ displaystyle \ delta}$ = Declination ${\ displaystyle \ lambda}$ = ecliptical length ${\ displaystyle \ beta}$ = ecliptical latitude
Cartesian coordinates in the target system ( , ):${\ displaystyle \ lambda}$${\ displaystyle \ beta}$
${\ displaystyle x = \ sin \ epsilon \ cdot \ sin \ delta + \ cos \ epsilon \ cdot \ cos \ delta \ cdot \ sin \ alpha}$
${\ displaystyle y = \ cos \ delta \ cdot \ cos \ alpha}$
${\ displaystyle z = \ cos \ epsilon \ cdot \ sin \ delta - \ sin \ epsilon \ cdot \ cos \ delta \ cdot \ sin \ alpha}$
Angular coordinates in the target system:
${\ displaystyle \ beta = \ arcsin z}$
${\ displaystyle \ lambda = \ arccos \ left ({\ frac {y} {\ cos \ beta}} \ right) = \ arccos \ left ({\ frac {y} {\ sqrt {1-z ^ {2} }}} \ right)}$
or:
${\ displaystyle \ lambda = \ arcsin \ left ({\ frac {x} {\ cos \ beta}} \ right) = \ arcsin \ left ({\ frac {x} {\ sqrt {1-z ^ {2} }}} \ right)}$

### Ecliptical (λ, β, geocentric) → rotating equatorial (α, δ) coordinates

 ${\ displaystyle \ epsilon}$ = 23.44 ° = inclination of the ecliptic ${\ displaystyle \ alpha}$ = Right ascension ${\ displaystyle \ delta}$ = Declination ${\ displaystyle \ lambda}$ = ecliptical length ${\ displaystyle \ beta}$ = ecliptical latitude
Cartesian coordinates in the target system ( , ):${\ displaystyle \ alpha}$${\ displaystyle \ delta}$
${\ displaystyle x = \ sin \ alpha \ cdot \ cos \ delta = - \ sin \ epsilon \ cdot \ sin \ beta + \ cos \ epsilon \ cdot \ cos \ beta \ cdot \ sin \ lambda}$
${\ displaystyle y = \ cos \ alpha \ cdot \ cos \ delta = \ cos \ beta \ cdot \ cos \ lambda}$
${\ displaystyle z = \ sin \ delta = \ cos \ epsilon \ cdot \ sin \ beta + \ sin \ epsilon \ cdot \ cos \ beta \ cdot \ sin \ lambda}$
Angular coordinates in the target system:
${\ displaystyle \ delta = \ arcsin \ left (\ cos \ epsilon \ cdot \ sin \ beta + \ sin \ epsilon \ cdot \ cos \ beta \ cdot \ sin \ lambda \ right)}$
${\ displaystyle \ alpha = \ arctan \ left ({\ frac {\ cos \ epsilon \ cdot \ sin \ lambda - \ sin \ epsilon \ cdot \ tan \ beta} {\ cos \ lambda}} \ right)}$