Re: Mixed Coordinate conversion - ra from long and dec

Dave Blake wrote:
Can any one help me with a bit of math.
I would like to get right ascension given the celestial longitude and
declination of an object. Yes, that's right I do mean mixed co-ordinate
systems. I know the equations that convert ecliptical to equatorial
co-ordinates e.g. celestial long and lat to ra and dec (and vica versa)
but I'm darned if I can get the trig re-arranged to give ra as a
function of celestial longitude and declination. Not so worried about
celestial lat, but it could be part of it.

It might sound odd, but I really do need to solve this. Hope this is
right place to try.

Yup. In what follows, I'll use (a,d) for (RA,Dec) and (l,b) for
ecliptic (lon,lat) and e for the obliquity, just to save keystrokes.
(Note that l and b are really lambda and beta, NOT galactic coordinates!)

Use rectangular coordinates. The conversion from (RA,Dec) to (lon,lat):

cos l cos b = cos a cos d
sin l cos b = cos e sin a cos d + sin e sin d
sin b = -sin e sin a cos d + cos e sin d

Divide the second equation by the first:

sin l cos e sin a cos d + sin e sin d
----- = -------------------------------
cos l cos a cos d

This equation contains RA, longitude, declination, and obliquity -- the
latitude has been removed. We need to solve it for RA in terms of the
others. Cross multiply and gather everything on one side:

sin a (cos e cos d cos l) - cos a (cos d sin l) + sin e sin d cos l = 0

Now -- here's the tricky part -- make the substitution

cos e cos d cos l = m cos M
cos d sin l = m sin M
sin e sin d cos l = C

so the previous equation reduces to

sin a (m cos M) - cos a (m sin M) + C = 0

or

m sin (a-M) = -C

whose solution is

a = M + arcsin (C/m)

where

M = atan2 (sin l, cos e cos l)
m = cos d sqrt (sin^2 l +cos^2 e cos^2 l)

Use the principal value for arcsin (C/m) and the 4-quadrant arctan
to get M.

As one would expect, this blows up at the poles. :-)

I'll leave it as an exercise for the reader to prove that |C/m| <=1 for
all physically possible combinations of declination and longitude.

-- Bill Owen
.

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