How do you compute the physical distance between two stars with known coordinates and distances?

If r1 is the distance to the first star, and r2 is the distance to the second star from the Sun, the distance, D, between these stars is then:

 2        2       2
D    =  r1   +  r2   - 2r1 r2 cos(Theta)

Example, if the two stars are on opposite sides of the sky from each other so that Theta = 180 degrees, then cos(Theta) = -1 and you get D = r1 + r2. If they are 90 degrees from each other, then cos(90) = 0 and

          
 2      2      2
D   = r1   + r2     

The calculation of Theta can be found in my answer to a previous question, as follows:

Let ra1 and d1 be the right ascension and declinations of star 1 in degrees Let ra2 and d2 be the right ascension and declination of star 2 in degrees, then the angular separation A, in degrees, between them is simply:

cos(A) = sin(d1)sin(d2) + cos(d1)cos(d2)cos(ra1-ra2)

Example. Sirius is at 6h 41m and -16d 35' so ra1 = 6.68h = 100.2 and dec = -16.58 so d1 = -16.58. Betelgeuse is at 5h 50m and +7d 23' so ra2 = 87.5d and d2 = 7.38. Then

cos(A) =  -0.285 x 0.128 + 0.958 x 0.9917 x cos(100.2 - 87.5)
       = -0.0364 + 0.9268
       =  0.890

so A = 27.1 degrees. You can check this by putting both stars at the same RA and getting their difference in declination as cos(a) = .9136 so a = 23.9 degrees which equals d2-d1= 16.58 + 7.38 = 23.9 degrees.

In this formula A = Theta.

Copyright 1997 Dr. Sten Odenwald

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