The inverse transformation, to take our derived (X,Y) for each object to RA and DEC, is ( Note that the HAIP has a sign error in the second equation above, and an incorrect expression for the declination equation below)Īs we mentioned earlier, commonly available astronomical software packages include routines to perform all these transformations for you and to do all the least-squared fitting. With these relations we can compute the (X,Y) position for each of our reference stars, and then use those in our least squares routine. Here the Greek alpha ( ) and delta ( ) denote RA and DEC, respectively, with the subscript zero ( ) meaning it is the RA and DEC of the tangent point of our image. The transformation is derived from spherical trigonometry. Of course, we have not said how to get the (X,Y) coordinates of our reference stars in the first place. See the HAIP or a reference on linear least squares fitting methods to learn how this is done. In fact, we may use many dozens of reference stars spread across our image, and then determine the “best fit” set of coefficients via a least squares fitting method. In practice we typically use more than the minimum three reference stars. Recall that for each star we have two constraints, (X,Y), so three reference stars gives us six constraints on our six unknowns. In principle this can be done using three reference stars of known position to determine the six unknown quantities. The constant coefficients a, b, c, d, e and f are to be determined for each image. The symbols (X,Y) and (x,y) have the meanings given above. ![]() The transformation equations are as follows: ![]() The first step is to determine the transformation between image and standard coordinates. Transformation From Image to Celestial Coordinates The transformation equations are given below. Once that has been done, the standard coordinates for any object can be converted to RA and DEC using straightforward spherical trigonometry. The usual way this is done is to assume a simple linear transformation between the standard coordinates and the image coordinates, and to determine the coefficients for that transformation. The trick of astrometry is to turn your image coordinates into celestial coordinates. Ideally the (x,y) and (X,Y) coordinates would be the same. These differ from standard coordinates because of things like tilt or rotation of the CCD, and also because of possible optical distortions.
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