Geometry/Coordinate Systems

Suppose you are an astronomer in America. You observe an exciting event (say, a supernova) in the sky and would like to tell your colleagues in Europe about it. Suppose the supernova appeared at your zenith. You can't tell astronomers in Europe to look at their zenith because their zenith points in a different direction. You might tell them which constellation to look in. This might not work, though, because it might be too hard to find the supernova by searching an entire constellation. The best solution would be to give them an exact position by using a coordinate system.

On Earth, you can specify a location using latitude and longitude. This system works by measuring the angles separating the location from two great circles on Earth (namely, the equator and the prime meridian). Coordinate systems in the sky work in the same way.

The equatorial coordinate system is the most commonly used. The equatorial system defines two coordinates: right ascension and declination, based on the axis of the Earth's rotation. The declination is the angle of an object north or south of the celestial equator. Declination on the celestial sphere corresponds to latitude on the Earth. The right ascension of an object is defined by the position of a point on the celestial sphere called the vernal equinox. The further an object is east of the vernal equinox, the greater its right ascension.

A coordinate system is a system designed to establish positions with respect to given reference points. The coordinate system consists of one or more reference points, the styles of measurement (linear measurement or angular measurement) from those reference points, and the directions (or axes) in which those measurements will be taken. In astronomy, various coordinate systems are used to precisely define the locations of astronomical objects.

Latitude and longitude are used to locate a certain position on the Earth's surface. The lines of latitude (horizontal) and the lines of longitude (vertical) make up an invisible grid over the earth. Lines of latitude are called parallels. Lines of longitude aren't completely straight (they run from the exact point of the north pole to the exact point of the south pole) so they are called meridians. 0 degrees latitude is the Earth's middle, called the equator. 0 degrees longitude was tricky because there really is no middle of the earth vertically. It was finally agreed that the observatory in Greenwich, U.K. would be 0 degrees longitude due to its significant role in scientific discoveries and creating latitude and longitude. 0 degrees longitude is called the prime meridian.

Latitude and longitude are measured in degrees. One degree is about 69 miles. There are 60 minutes (') in a degree and 60 seconds (") in a minute. These tiny units make GPS's (Global Positioning Systems) much more exact.

There are a few main lines of latitude:the Arctic Circle, the Antarctic Circle, the Tropic of Cancer, and the Tropic of Capricorn. The Antarctic Circle is 66.5 degrees south of the equator and it marks the temperate zone from the Antarctic zone. The Arctic Circle is an exact mirror in the north. The Tropic of Cancer separates the tropics from the temperate zone. It is 23.5 degrees north of the equator. It is mirrored in the south by the Tropic of Capricorn.

Horizontal coordinate system

One of the simplest ways of placing a star on the night sky is the coordinate system based on altitude and azimuth, thus called the Alt-Az or horizontal coordinate system. The reference circles for this system are the horizon and the celestial meridian, both of which may be most easily graphed for a given location using the celestial sphere.

In simplest terms, the altitude is the angle made from the position of the celestial object (e.g. star) to the point nearest it on the horizon. The azimuth is the angle from the northernmost point of the horizon (which is also its intersection with the celestial meridian) to the point on the horizon nearest the celestial object. Usually azimuth is measured eastwards from due north. So east has az=90°, south has az=180°, west has az=270° and north has az=360° (or 0°). An object's altitude and azimuth change as the earth rotates.

Equatorial coordinate system

The equatorial coordinate system is another system that uses two angles to place an object on the sky: right ascension and declination.

Ecliptic coordinate system

The ecliptic coordinate system is based on the ecliptic plane, i.e., the plane which contains our Sun and Earth's average orbit around it, which is tilted at 23°26' from the plane of Earth's equator. The great circle at which this plane intersects the celestial sphere is the ecliptic, and one of the coordinates used in the ecliptic coordinate system, the ecliptic latitude, describes how far an object is to ecliptic north or to ecliptic south of this circle. On this circle lies the point of the vernal equinox (also called the first point of Aries); ecliptic longitude is measured as the angle of an object relative to this point to ecliptic east. Ecliptic latitude is generally indicated by $\phi$ , whereas ecliptic longitude is usually indicated by $\lambda$ .

Galactic coordinate system

As a member of the Milky Way Galaxy, we have a clear view of the Milky Way from Earth. Since we are inside the Milky Way, we don't see the galaxy's spiral arms, central bulge and so forth directly as we do for other galaxies. Instead, the Milky Way completely encircles us. We see the Milky Way as a band of faint starlight forming a ring around us on the celestial sphere. The disk of the galaxy forms this ring, and the bulge forms a bright patch in the ring. You can easily see the Milky Way's faint band from a dark, rural location.

Our galaxy defines another useful coordinate system — the galactic coordinate system. This system works just like the others we've discussed. It also uses two coordinates to specify the position of an object on the celestial sphere. The galactic coordinate system first defines a galactic latitude, the angle an object makes with the galactic equator. The galactic equator has been selected to run through the center of the Milky Way's band. The second coordinate is galactic longitude, which is the angular separation of the object from the galaxy's "prime meridian," the great circle that passes through the Galactic center and the galactic poles. The galactic coordinate system is useful for describing an object's position with respect to the galaxy's center. For example, if an object has high galactic latitude, you might expect it to be less obstructed by interstellar dust.

Transformations between coordinate systems

A spherical triangle solved by the law of cosines.

One can use the principles of spherical trigonometry as applied to triangles on the celestial sphere to derive formulas for transforming coordinates in one system to those in another. These formulas generally rely on the spherical law of cosines, known also as the cosine rule for sides. By substituting various angles on the celestial sphere for the angles in the law of cosines and by thereafter applying basic trigonometric identities, most of the formulas necessary for coordinate transformations can be found. The law of cosines is stated thus:

$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)$

To transform from horizontal to equatorial coordinates, the relevant formulas are as follows:

$Dec=\arcsin {\Big (}\sin(Alt)\sin(Lat)+\cos(Alt)\cos(Lat)\cos(Az){\Big )}$

$RA=LST-\arccos \left({\frac {\sin(Alt)-\sin(Dec)\sin(Lat)}{\cos(Dec)\cos(Lat)}}\right)$

where $RA$ is the right ascension, $Dec$ is the declination, $LST$ is the local sidereal time, $Alt$ is the altitude, $Az$ is the azimuth, and $Lat$ is the observer's latitude. Using the same symbols and formulas, one can also derive formulas to transform from equatorial to horizontal coordinates:

$Alt=\arcsin {\Big (}\sin(Dec)\sin(Lat)+\cos(Dec)\cos(Lat)\cos(LST-RA){\Big )}$

$Az=\arccos \left({\frac {\sin(Dec)-\sin(Alt)\sin(Lat)}{\cos(Alt)\cos(Lat)}}\right)$

Transformation from equatorial to ecliptic coordinate systems can similarly be accomplished using the following formulae:

$\phi =\arcsin {\Big (}\sin(Dec)\cos(\epsilon )-\cos(Dec)\sin(\epsilon )\sin(RA){\Big )}$

$\lambda =\arcsin \left({\frac {\sin(Dec)-\sin(\phi )\cos(\epsilon )}{\cos(\phi )\sin(\epsilon )}}\right)=\arctan \left({\frac {\sin(RA)\cos(\epsilon )+\tan(Dec)\sin(\epsilon )}{\cos(RA)}}\right)$

where $RA$ is the right ascension, $Dec$ is the declination, $\phi$ is the ecliptic latitude, $\lambda$ is the ecliptic longitude, and $\epsilon$ is the tilt of Earth's axis relative to the ecliptic plane. Again, using the same formulas and symbols, new formulas for transforming ecliptic to equatorial coordinate systems can be found:

$Dec=\arcsin {\Big (}\sin(\phi )\cos(\epsilon )+\cos(\phi )\sin(\epsilon )\sin(\lambda ){\Big )}$

$RA=\arcsin \left({\frac {\sin(Dec)\cos(\epsilon )-\sin(\phi )}{\cos(Dec)\sin(\epsilon )}}\right)$