FIVE/2 Coordinate Systems


Our galaxy is a giant "pinwheel" of stars, with a close-to spherical core (the "bulge") and a flat disc containing the spiral arms. This disc is about 100,000 LY in diameter and 1000 LY thick, and rotates slowly. Outside the disc (above, below and beyond) is the galactic halo extending perhaps twice as far as the disc from the galactic core and in a sphere surrounding it. The bulge is something like 7000 LY across. For the Sun it takes about 250 million years to complete one galactic year, as we are about 28,000 LY from the center. The Sun lies about 50 LY above the plane of the galaxy, near the edge of a 15,000 LY short spur (a small spiral arm) often called the Orion spur. There are four big spiral arms: Sagittarius-Carina, Perseus, Cygnus and a fourth unnamed arm. The Orion Spur lies between S-C and Perseus, about 6000 LY away from either, and we are on the S-C side of the spur.


It is useful to be able to provide a defined location for star systems in the form of coordinates (X, Y, Z). The coordinates are often based upon Earth's sky, but as Earth's equator isn't the same as the galactic equator the resulting coordinates don't tell how stars are placed relative to the galactic plane and center. Thus, it may be wise to utilise galactic coordinates. The easiest way to generate galactic coordinates is to take the "normal" coordinates based upon right ascension, declination and parallax and convert them.


These three things (optionally, distance instead of parallax) are necessary to provide coordinates. Right ascension (ra) is the celestial version of longitude, declination (dec) the latitude equivalent and parallax the displacement angle the object shows due to the annual motion of the Earth. Parallax translates into distance as

distance = 1
63,115.2 x tan(parallax/206,264,806)

calculating in radians. Distance is in light years, parallax in milli-arc-seconds (mas). The coordinates of an object, when distance, right ascansion and declination are known, are as follows:

X = distance x cos(dec x hc) x cos(ra x hc) hc = /180
Y = distance x cos(dec x hc) x sin(ra x hc)
Z = distance x sin(dec x hc)

Use trigonometry in radians, distance in light years and dec/ra in decimal degrees.


However, the right ascension and declination of an object is subject to change. One reason is that the object actually moves fast enough to make a difference, but the main reason is that Earth itself undergo changes in orbital elements. The rotational axis slowly undergo precession and thus the pole do not face the exact same spot over time. Thus, the astronomical measurements are accompanied by an "epoch", a sort of time-tag. Some star data you may find (Gliese, for instance) are 1950, while Hipparcos (another big data source) is 1991. Depending on how painstakingly precise you decide to place stars (decimal fractions of light years?) this may be more or less important.


For 1991-data (Hipparcos), you can transform the X, Y and Z coordinates above to galactic coordinates (centered on Earth, but with X-Y-Z axis oriented according to the galaxy) by using

Xg = - (0.0571 x X) - (0.8733 x Y) - (0.4838 x Z)
Yg = (0.4938 x X) - (0.4458 x Y) + (0.7466 x Z)
Zg = - (0.8677 x X) - (0.1963 x Y) + (0.4567 x Z)


Stars move relative to each other. Over a short enough time, say 500 years, this will be of little importance (0.1 LY or so) unless you need very exact coordinates. (If you do, use the proper motion and radial velocity measurements many star lists provide.) But over longer time it will make a difference. 1 million years ago our skies looked different and our stellar neighborhood too. The Alpha Centauri system has not been our closest star for that long (and in only a few thousand years Barnard's Star will be closer).


To measure distances between star systems is simple.

distance = ( (X1 - X2)2 + (Y1 - Y2)2 + (Z1 - Z2)2)0.5

where the two sets (1 & 2) of coordinates are those of the two involved stars.