Astrometric Position Calculator |
In order to do so, you must know how the celestial sphere is projected on your flat film-/sensor plane. The underlying geometry and equations are, again, fairly simple and straightforward and can be expressed by analytically exact formulae:
Let RA0 and Dec0 be the co-ordinates of the centre point of your image (i.e.: the co-ordinates at which your telescope / lens was pointing during exposure). Then, a star with co-ordinates RA1 and Dec1 will be projected on your focal plane in an x-y co-ordinate system with origo (x=0, y=0) at the centre of your image as follows:
x1 = FAC * (sin(RA1-RA0) * cos(Dec1)
y1 = FAC * (cos(RA1-RA0) * cos(Dec1 * sin (Dec0) - sin(Dec1) * cos(Dec0))
where FAC is a constant depending upon the focal length of your optical system and the magnification of your image, (if applicable)
And thus, from the known characteristics of your system/image and measured values of x1 and y1 on your image, you may calculate the celestial co-ordinates of your subject as follows:
RA1 = RA0 + arctan( x'1 / (cos(dec0) - y'1 * sin(Dec0)) )
Dec1 = arcsin( (sin(Dec0) + y'1 * cos(Dec0)) / sqrt(1 + (x'1^{)2} +( y'1^{)2}))
where x'1 = x1/FAC and y'1 = y1/FAC
We would normally have a good idea - but not an exact knowledge - of RA0 and Dec0 as well as the manufacturer's information on the system's focal length (individual, real focal lengths may well differ some 10% from the manufacturers' specifications) , and in theory we could find exact values for these unknowns if we had three stars with known co-ordinates in our image to set up three equations with three unknowns. However, we also have an issue with the orientation of our x-y system as our camera/sensor's axis were probably not totally totally parallel with and perpendicular to the celestial equator.
To make a long story short, for astrometric purposes you will normally need more stars with known co-ordinates in order to "calibrate" your image using least square fitting. Thus, programming becomes rather long-haired, but fortunately there are people out there, who has already done the job for us. The calculator below is a slightly simplified version of that published in ref. 10 on the reference page to this section. There, you will find the full script including help files and references. This astrometric position calculator is based on a BASIC program written by Jordan D. Marche and explained by him in the July 1990 issue of Sky & Telescope, page 71.
You should note the following:
1. Your image must be the full, original frame - i.e.: do not crop !
2. Stellar celestial co-ordinates can be obtained from a number of sources. There are databases publicly available with millions of stars. My personal preference (as an amateur) is the Cartes du Ciel, (ref. 11).
3. Be careful if you compare your image with star maps (for example by trying to overlap image and map). A flat image obtained with a rectilinear optical system will be a tangential projection of the celestial sphere, while maps may be produced (as in Cartes du Ciel) using various projection methods ("ARC", "CARTESIAN", "SIN", TAN"....). Be sure to have your settings right. Also remember that comparison by trying to overlap a map and a photo is only meaningful if the map's centre of projection and the point at which your imaging system's optical axis was pointing are identical.
Astrometric Position Calculator |
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Use this page to compute accurate equatorial coordinates (Right
Ascension and Declination) for an object in your image -- an asteroid
or comet, for example. A
set of example entries is inserted and you must note their format. Clicking on the
"Calculate" button will compute the constants required to convert a given set of
x,y co-ordinates to RA and Dec. Note: For x,y you would normally use pixel co-ordinates as you may read those directly on-screen with almost any decent digital imaging software. However, the calculator will also work if you use millimetres as measured on a print or an analogue negative. |
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Copyright © 2010 - Steen G. Bruun