The coordinate systems considered here are all based at one reference point in space with respect to which the positions are measured, the origin of the reference frame (typically, the location of the observer, or the center of Earth, the Sun, or the Milky Way Galaxy). Any location in space is then described by the "radius vector" or "arrow" between the origin and the location, namely by the distance (length of the vector) and its direction. The direction is given by the straight half line from the origin through the location (to infinity). In the spherical coordinate systems used here, the direction is fixed by two angles, which are given as follows:
A reference plane containing the origin is fixed, or equivalently the axis through the origin and perpendicular to it (typically, an "equatorial" plane and a "polar" axis); elementarily, each of these uniquely determines the other. One can assign an orientation to the polar axis from "negative" to "positive", or "south" to "north", and simultaneously to the equatorial plane by assigning a positive sense of rotation to the equatorial plane; these orientations are, by convention, usually combined by the right hand rule: If the thumb of the right hand point to the positive (north) polar axis, the fingers show in the positive direction of rotation (and vice versa, so that a physical rotation defines a north direction).
The reference plane or the reference axis define the set of planes which contain the origin and are perpendicular to the "equatorial" reference plane (or equivalently, contain the "polar" reference axis); each direction in space then lies precisely in one of these "meridional" planes (or half planes, if the reference axis is taken to divide each plane into halfs), with the exception of the (positive and negative) polar axis which lies in all of them by definition.
The first angle used to characterize a direction, typically the "latitude", is taken between the direction and the reference plane, within the "meridional" plane. For the second angle, it is required to select and fix one of the "meridional" half planes as zero, from which the angle (of "longitude") is measured to the "meridional" half plane containing our direction.
Note that this selection of angles to characterize a direction in a given reference frame is chosen by convention, which is especially common in astronomy and geography, and which is used in the following here, as well as in most astronomical databases. Other, equivalent, conventions are possible, e.g. physicists often use instead of the "latitude" angle to the reference plane, the angle between the direction and the "positive" or "north" polar axis (called "co-latitude"; co-latitde = 90 deg - latitude). It depends on taste at last what the reader likes to use, but here we will stay as close to standard astronomical convention as possible. In order to minimize the requirement of case-to-case enumeration of conventions, we also recommend the reader to do the same.
The natural reference plane here is that of the Earth's equator, and the natural reference axis is the rotational, polar axis which cuts the Earth's surface at the planet's North and South pole. The circles along Earth's surface which are parallel to the equator are the latitude circles, where the angle at the planet's center is constant for all points on these circles. Half circles from pole to pole, which are all perpendicular to the equatorialplane, are called meridians. One of the meridians, in practice that through the Greenwich Observatory near London, England, is taken as reference meridian, or Null meridian. Geographical Longitude is measured as the angle between this and the meridian under consideration (or more precisely, between the half planes containing them); it is of course the same for all points of the meridian.
Because Earth is not exactly circular, but slightly flattened, its surface (defined by the ocean surface, or the corresponding gravitational potential) forms a specific figure, the so-called Geoid, which is very similar to a slightly oblate spheroid (the reference ellipsoid). This is the reason why there are two common but different definitions of latitude on Earth:
Taking any of the meridional planes, the meridian has the approximate shape of a half ellipse. The major half axis represents the equatorial radius of the planet, while the minor axis is the polar (and thus the rotational) half axis, which is about 1/298 shorter than the equatorial radius. More precisely:
tan B' = (b/a)^2 * tan BThe maximal difference occurs for B = 45 deg, and amounts to 11.5 arc minutes.
In the following, we always deal with geographic latitude unless otherwise mentioned.
Thus each observer can look at the skies as being manifested on the interior of a big sphere, the so-called celestial sphere. Then each direction away from the observer will intersect the celestial sphere in one unique point, and positions of stars and other celestial objects can be measured in angular coordinates (similar to longitude and latitude on Earth) on this virtual sphere. This can be done without knowing the actual distances of the stars. Moreover, any plane through the origin cuts the sphere in a great circle. Examples for celestial coordinate systems are treated below.
In times up to Copernicus, people believed that there is actually a
solid sphere to which the stars beyond the solar system are fixed: This idea
was overcome when it was realized that stars are sunlike bodies, in the time
of Newton and Halley. Today, the celestial sphere is only a virtual
construct to make our understanding of positional astronomy easier.
Through any direction, or point on the celestial sphere, e.g. the position of a star, a unique [half] plane (or great [half] circle) perpendicular to the horizon can be found; this is called vertical circle; all vertical [half] circles contain (and intersect in) both the zenith and the nadir. Within the plane of its verticle circle, the position under consideration can be characterized by the angle to the horizon, called altitude a. Alternatively and equivalently, one could take the angle between the direction and the zenith, the zenith distance z, which is related to the altitude by the relation: z = 90 deg - a. All objects above the horizon have positive altitudes (or zenith distances smaller than 90 deg). The horizon itself can be defined, or recovered, as the set of all points for which a = 0 deg (or z = 90 deg).
In contrast to the apparent horizon which defines coordinates of objects as the observer perceives them, the true horizon is defined by the plane parallel to the apparent horizon, but through the center of Earth. The angle between the position of an object and the true horizon is referred to as true altitude. For nearby objects such as the Moon, the measured position can vary notably between these two reference systems (up to 1 deg for the Moon). Also, the apparent altitudes are subject to the effect of refraction by Earth's atmosphere.
The second coordinate of a position in the horizon system is defined by the point where the verticle circle of the position cuts the horizon. It is called azimuth A and, in astronomy and on the Northern hemisphere (the present author does not know the southern standards for this thread), is the angle from the south point (or direction) taken to the west, north, and east to the foot point of the vertical circle on the horizon, thus running from 0 to 360 deg. In geodesy, the north direction is often taken as zero point (this angle is sometimes called bearing and is given by A +/- 180 deg). Note that these conventions are not always uniquely used so that it may be advisable to clear up which conventions are used (e.g., by saying A is taken to the West).
Taking the astronomical standard, the south, west, north, and east points on the horizon are defined by A = 0 deg, 90 deg, 180 deg, and 270 deg, respectively. The vertical circle passing through the south and north point (as well as zenith and nadir) is called local meridian; the one perpendicular to it through west point, zenith, east point and nadir is called prime vertical. The local meridian coincides with the projection of the geographical meridian of the observer's location to the sky (celestial sphere) from Earth's center.
The terms introduced here are helpful in understanding the effects of Earth's rotation.
In principle, the celestial coordinate system can be introduced in the simplest way by projecting Earth's geocentric coordinates to the sky at a certain moment of time (actually, each time when star time is O:00 at Greenwich or anywhere on the Zero meridian on Earth, which occurs once each siderial day); the reader will hopefully understand this statement after reading this section. These coordinates are then left fixed at the celestial sphere, while Earth will rotate away below them.
Practically, projecting Earth's equator and poles to the celestial sphere by imagining straight half lines from the Earth's center produces the celestial equator as well as the north and the south celestial pole. Great circles through the celestial poles are always perpendicular to the celestial equator and called hour circles for reasons explained below.
The first coordinate in the equatorial system, corresponding to the latitude, is called Declination (Dec), and is the angle between the position of an object and the celestial equator (measured along the hour circle). Alternatively, sometimes the polar distance (PD) is used, which is given by PD = 90 deg - Dec; the most prominent reference known to the present author using PD instead of Dec is John Herschel's General Catalogue of Non-stellar Objects (GC) of 1864, but this (equivalent) alternative has come more and more out of use since, so that virtually all current astronomical databases use Dec.
It remains to fix the zero point of the longitudinal coordinate, called Right Ascension (RA). For this, the intersection points of the equatorial plane with Earth's orbital plane, the ecliptic, are taken, more precisely the so-called vernal equinox or "First Point of Aries". During the year, as Earth moves around the Sun, the Sun appears to move through this point each year around March 21 when spring begins on the Northern hemisphere, and crosses the celestial equator from south to north (Southerners are asked to forgive a certain amount of "hemispherism" in the official nomenclature). The opposite point is called the "autumnal equinox", and the Sun passes it around September 23 when it returns to the Southern celestial hemisphere. As a longitudinal coordinate, RA can take values between 0 and 360 deg. However, this coordinate is more often given in time units hours (h), minutes (m), and seconds (s), where 24 hours correspond to 360 degrees (so that RA takes values between 0 and 24 h); the correspondence of units is as follows:
24 h = 360 deg 1 h = 15 deg, 1 m = 15', 1 s = 15" 1 deg = 4 m, 1' = 4 sSo the vernal equinox, where the Sun appears to be when Northern spring begins around March 21, is at RA = 0 h = 0 deg, the summer solstice where the Sun is when Northern summer begins around June 21, is at RA = 6 h = 90 deg, the autumnal equinox is at RA = 12 h = 180 deg, and the winter solstice is at RA = 18 h = 270 deg. Thus RA is measured from west to east in the celestial sphere.
Because of small periodic and secular changes of the rotation axis of Earth, especially precession, the vernal equinox is not constant but varies slowly, so that the whole equatorial coordinate system is slowly changing with time. Therefore, it is necessary to give an epoch (a moment of time) for which the equatorial system is taken; currently, most sources use epoch 2000.0, the beginning of the year 2000 AD.
To go over from equatorial coordinates fixed to the stars to the horizon system, the concept of the hour angle (HA) is useful. In principle, this means introducing a new, second equatorial coordinate system which co-rotates with Earth. This system has again the celestial equator and poles as reference quantities, and declination as latitudinal coordinate, but a co-rotating longitudinal coordinate called hour angle. In this system, a star or other celestial object moves contrary to Earth's rotation along a circle of constant declination during the course of the day; various effects of this diurnal motion are discussed below. This rotation leaves the celestial poles in the same invariant position for all time: They always stay on the local meridian of the observer (which goes through south and north point also), and the altitude of the north celestial pole is equal to the geographic latitude of the observer (thus negative for southerners, who cannot see it for this reason, but the south celestial pole instead). This meridian always coincides with in hour circle for this reason. Thus, as may be suggestive, the local meridian is taken as the hour circle for HA=0.
Celestial objects are at constant RA, but change their hour angle as time proceeds. If measured in units of hours, minutes and seconds, HA will change for the same amount as the elapsed time interval is, as measured in star time (ST), which is defined so that a siderial rotation of Earth takes 24 hours star time, which corresponds to 23 h 56 m 4.091 s standard (mean solar) time; see our article on Astronomical Time Keeping for more details. This is actually the reason why RA and HA are measured in time units. The standard convention is that HA is measured from east to west so that it increases with time, and this is opposite to the convention for RA !
Star time is ST = 0 h by definition whenever the vernal equinox, RA = 0 h, crosses the local meridian, HA = 0. As time proceeds, RA stays constant, and both HA and ST grow by the amount of time elapsed, thus star time is always equal to the hour angle of the vernal equinox. Moreover, objects with "later" RA come into the meridian HA = 0, more precisely with RA which is later by the amount of elapsed star time, so that also star time is equal to the current Right Ascension of the local meridian.
More generally, for any object in the sky, the following relation between right ascension, hour angle, and star time always holds:
HA = ST - RA(here given to determine the current HA from known RA and ST).
cos Dec * sin HA = cos a * sin A sin Dec = sin B * sin a + cos B * cos a * cos A cos Dec * cos HA = cos B * sin a + sin B * cos a * cos AThe inverse transformation formulae from given HA, Dec to A, a read:
cos a * sin A = cos Dec * sin HA sin a = sin B * sin Dec + cos B * cos Dec * cos HA cos a * cos A = - cos B * sin Dec + sin B * cos Dec * cos HAFor practical calculation in either case, evaluate e.g. the second formula first to obtain Dec or a, and then use the result in the first formula to get HA or A, respectively. (Get HA from or transform it to Right Ascension according to the relation given at the end of the last section if star time is known)
By doing so, stars will cross the local meridian (defined e.g. by zero hour angle HA) twice a day; these events are called transits or culminations, i.e., the upper and the lower transit, or the upper and the lower culmination. These events also mark the maximal and minimal altitude a the objects can reach in the observer's sky, and may both take place above or below the horizon of the observer, depending on the declination Dec of the object and the geographic latitude B of the observer.
The altitudes for upper transits are as follows:
a = 90 deg - |B - Dec|where the transit takes place north of the zenith if Dec > B and south otherwise. If |B - Dec| > 90 deg, the upper transit will take place at negative altitude, i.e. below the horizon, so that the object will never come above the horizon and thus never be visible; for the Northern hemisphere, this is true for all objects with
Dec < B - 90 deg (< 0),and for the Southern hemisphere for
Dec > B + 90 deg (> 0).The altitudes for the lower transit are given by
a = (B + Dec) - 90 deg B > 0 (North) a = - (B + Dec) - 90 deg B < 0 (South)For an observer on the Northern hemisphere, stars with Dec > 90 deg - B (> 0), and for southern hemisphere observers, stars with Dec < - 90 deg - B (< 0) will have their lower transit at positive altitudes, i.e., above horizon, and will never set; such stars are called circumpolar.
All stars which are neither circumpolar nor never visible will have their upper transit above and their lower transit below horizon, and thus rise and set during a siderial day. Disregarding refraction effects, the hour angle of the rise and set of a celestial object, the semidiurnal arc H0, is given by
cos H0 = - tan Dec * tan Bwhile the azimuth of the rising and setting points, the evening and morning elongation A0 is
cos A0 = - sin Dec / cos Bwhere A0 > 90 dec if Dec and B have same sign (i.e., are on the same hemisphere). Rising and setting times differ from transit time by the amount of the diurnal arc H0, given in time units (hours), taken as hours of star time.
If Dec and B have same sign (i.e., are on the same hemisphere), one of the following situations occurs:
sin a = sin Dec / sin B cos HA = tan Dec * cot B
sin a = sin B / sin Dec cos HA = cot Dec * tan B
The ecliptic latitude (be) is defined as the angle between a position and the ecliptic and takes values between -90 and +90 deg, while the ecliptic longitude (le) is again starting from the vernal equinox and runs from 0 to 360 deg in the same eastward sense as Right Ascension.
The obliquity, or inclination of Earth's equator against the ecliptic, amounts eps[ilon] = 23deg 26' 21.448" (2000.0) and changes very slightly with time, due to gravitational perturbations of Earth's motion. Knowing this quantity, the transformation formulae from equatorial to ecliptical coordinates are quite simply given (mathematically, by a rotation around the "X" axis pointing to the vernal equinox by angle eps):
cos be * cos le = cos Dec * cos RA cos be * sin le = cos Dec * sin RA * cos eps + sin Dec * sin eps sin be = - cos Dec * sin RA * sin eps + sin Dec * cos epsand the reverse transformation:
cos Dec * cos RA = cos be * cos le cos Dec * sin RA = cos be * sin le * cos eps - sin be * sin eps sin Dec = cos be * sin le * sin eps + sin be * cos eps
Ecliptical coordinates are most frequently used for solar system calculations such as planetary and cometary orbits and appearances. For this purpose, two ecliptical systems are used: The heliocentric coordinate system with the Sun in its center, and the geocentric one with the Earth in its origin, which can be transferred into each other by a coordinate translation.
Here, the galactic plane, or galactic equator, is used as reference plane. This is the great circle of the celestial sphere which best approximates the visible Milky Way. For historical reasons, the direction from us to the Galactic Center has been selected as zero point for galactic longitude l, and this was counted toward the direction of our Sun's rotational motion which is therefore at l = 90 deg. This sense of rotation, however, is opposite to the sense of rotation of our Galaxy, as can be easily checked ! Therefore, the galactic north pole, defined by the galactic coordinate system, coincides with the rotational south pole of our Galaxy, and vice versa.
Galactic latitude b is the angle between a position and the galactic equator and runs from -90 to +90 deg. Glalactic longitude runs of course from 0 to 360 deg.
The galactic north pole is at RA = 12:51.4, Dec = +27:07 (2000.0), the galactic center at RA = 17:45.6, Dec = -28:56 (2000.0). The inclination of the galactic equator to Earth's equator is thus 62.9 deg. The intersection, or node line of the two equators is at RA = 18:51.4, Dec = 0:00 (2000.0), and at l = 33 deg, b=0.
The transformation formulae for this frame get more complicated, as the transformation is consisted of (1.) a rotation around the celestial polar axis by 18:51.4 hours, so that the reference zero longitude matches the node, (2.) a rotation around the node by 62.9 deg, followed by (3.) a rotation around the galactic polar axis by 33 deg so that the zero longitude meridian matches the galactic center. This complicated transformation will not be given here formally.
Before 1959, the intersection line had been taken as zero galactic longitude, so that the old differred from the new latitude by 33.0 deg (the longitude of the node just discussed, but for the celestial equator of the epoch 1950.0):
l(old) = l(new) - 33.0 degFor a transition time, the old coordinate had been assigned a superscript "I", the new longitude a superscript "II", which can be found in some literature.
For some considerations, besides the geo- or heliocentric galactic coordinates described above, galactocentric galactic coordinates are useful, which have the galactic center in their origin; these can be obtained from the helio/geocentric ones by a parallel translation.
Angles are most often measured in degrees (deg), arc minutes (arc min, ') and arc seconds (arc sec, "), where
1 deg = 60' = 3,600"and the full circle, or revolution, is 360 deg. Mathematicians and physicists often use units of arc instead, where the full circle (i.e., 360 deg) is given by 2 pi, so that
pi = 180 deg = 10,800' = 648,000" 1 deg = pi/180 = 1/57.2958 = 0.0174533 1' = pi/10,800 = 1/3,437.75 = 0.000290888 1" = pi/648,000 = 1/206,265 = 0.00000484813