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Calculating Stellar Positions (near the Sun)

If you've done a lot of astronomical observing or calculations, you're familiar with the equatorial coordinate system of right ascension and declination (if not, go to the Introduction to Astronomical Terms page for an explanation). Though this is probably the most widely used coordinate system in astronomy, it has its limitations. Since it's closely related to Earth-based latitude and longitude, it works best from the Earth. 3D star mappers and astrometry buffs often work from vantage points many light years away, so equatorial coordinates don't make much sense as a general system. For example, suppose you want to calculate the positions of a bunch of stars as seen from a very distant location (say, the Pleiades). There is no simple way to do this using only equatorial coordinates. Some other coordinate system is needed.

For our purposes, the most flexible coordinate system is Cartesian coordinates, which are simply the components of the star's position along three perpendicular axes (x,y, and z). Cartesian coordinates have two big advantages over most other systems:

  1. It's very easy to translate Cartesian coordinates from one place to another. For example, if you want to figure out what the sky looks like from, for example, Sirius, all you have to do is subtract the coordinates of Sirius from the coordinates of all the other stars. It's also very easy to calculate the distances between stars after such a translation. (More on this when we discuss calculating the night sky from other stars.)
  2. Converting from Cartesian coordinates to other coordinate systems, such as equatorial coordinates, is generally fairly straightforward. If you have to convert between two non-Cartesian coordinate systems, it's often easiest to convert the first one to Cartesian coordinates, then convert those to the second coordinate system. Thus, Cartesian coordinates are a lingua franca
    for handling lots of coordinate conversions.

With this in mind, let's define the three axes needed for our coordinate system:

  1. +x: towards \delta = 0 degrees, \alpha = 0.0 hours (the vernal equinox)
  2. +y: towards \delta = 0 degrees, \alpha = 6.0 hours
  3. +z: towards \delta = +90.0 degrees (north celestial pole)

The origin -- the zero point -- of this coordinate system is at the Sun.

Though the locations of these axes are defined from certain values of \alpha and \delta, they point to specific, fixed regions in space that are independent of the Earth. With this in mind, we don't need to worry about the Earth anymore, and can locate stars in this coordinate system anywhere in space.

Converting equatorial coordinates to Cartesian coordinates

If you know the equatorial coordinates (\alpha, \delta, d) of a star, you can get its Cartesian coordinates (x,y,z) by three simple formulas.

Remember, since right ascension is normally given in hours, you must multiply it by 15 to get degrees before using these formulas.

  1. x = d cos \delta cos \alpha;
  2. y = d cos \delta sin \alpha;
  3. z = d sin \delta.

The x,y,z coordinates are distances, with respect to the origin (i.e., Earth) and have the same units as the original distance, d.

A Full Example

The closest star to the Sun is Proxima Centauri, a faint red star only about 0.01% as bright as the Sun. Its coordinates are:

  1. \alpha = 14.4966 hours
  2. \delta = -62.681 degrees
  3. d = 1.29 parsecs

First, convert \alpha to degrees: 14.4966 * 15 = 217.45 degrees. Then:

  1. x = 1.29 cos (-62.681) cos (217.45);
  2. y = 1.29 cos (-62.681) sin (217.45);
  3. z = 1.29 sin (-62.681).

which results in:

  1. x = -0.472 parsecs
  2. y = -0.361 parsecs
  3. z= -1.151 parsecs

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