Separation Equations

Spherical triangles are different from planar triangles. The sides of a spherical triangle are also angles themselves. In the diagram above, the side 'c' on the surface is the same as the angle 'c' shown near the center of the sphere. 'C' is the angle opposite 'c'.

If 'A' is the angle between the sides 'b' and 'c', and, if 'a' is the separation angle between point 'B' and 'C', then:

cos(A) = ( cos(a) - cos(b)*cos(c) ) / sin(b)*sin(c)

The angle A is found by taking the inverse cosine - a function found on most calculators.

For our triangle:

cos( DRA ) = ( cos( CODEC1 ) - cos( CODEC2 ) * cos( SEPARATION ) ) / sin( CODEC1 ) * sin( CODEC2 ))

and the difference in the right ascensions of the two stars is found by taking the inverse cosine of cos( DRA).

I might note that, in the late 1500s, the Sine and Cosine functions had not yet been fully developed so Tycho, and the ancients, used chord lengths across the unit circle (laboriously calculated ahead of time) to find the equivalent of Sin and Cos.

If we collect the DRAs of stars in a belt around the sky, their sum should be 360 degrees.

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