9.9 Astrometry

Astrometry was done on all 68 focused standard star images (cf. Table , p.  further on), using tasks in the package finder. The main purpose was to determine the CCD pixel scale, since that had not been determined before with sufficient accuracy.

First, the task tfinder was run on the images. It is given the epoch and the center coordinates of the image, the intended orientation of the CCD (in our case north at the bottom, and east to the right), and a good guess on the CCD (``plate'') scale (in our case 0.51 ''/pixel). It then uses the Hubble Guide Star Catalog (GSC; on CD-ROM) to find the GSC stars present in the given CCD image. The GSC stars are drawn as rings on top of the image, which is displayed in SAOimage. The user does the coarse centering, after which the task does the real centering, and outputs a catalog image.tab, in which the following columns are of particular interest: RA_DEG, DEC_DEG, X_CENTER, and Y_CENTER.

A note should be made about the two coordinate systems involved. (x,y) are the ideal plate coordinates, in radians. They are given in the columns RA_DEG and DEC_DEG, with a conversion from degrees to radians. The x-axis points towards east, and the y-axis points towards north, making it a left-handed system. (x',y') are the CCD coordinates, in pixels. They are given in the columns X_CENTER and Y_CENTER. In our case, the x'-axis points approximately towards east, and the y'-axis points approximately towards south, making it a usual right-handed system. See Figure .

 

The task tastrom takes the catalog image.tab produced by tfinder and computes a 6 coefficient co-called plate solution of the form

$$\left( \begin{array}{c} x \\ y \end{array}\right) = \left( ... ...ht) + \left( \begin{array}{c} b_1 \\ b_2 \end{array}\right).$$ (9.27)

These 6 coefficients, a_ij and b_i, are written to the ASCII file image.ast, along with a number of other informations. An example of the lines giving the 6 coefficients from d1318.ast is:

1PLATE SOLUTION - EMPIRICAL, 6-COEFF
 --------------   ------------------

     X,Y = IDEAL PLATE COORDINATES (RADIANS)

     X =   -0.001269                                Y =    0.001301
       +   0.2458474E-05 * XMEAS                      +  -0.2458925E-05 * YMEAS
       +  -0.1837236E-07 * YMEAS                      +  -0.1840916E-07 * XMEAS

We identify a_ij and b_i as

$$\begin{array}{lclllcl} x & = & b_1 & ~ & y & = & b_2 \\ ~ & ... ...& + & a_{12} \cdot y' & ~ & ~ & + & a_{21} \cdot x' \end{array}$$ (9.28)

The transformation matrix ${\bf A}$ can be written in terms of the following geometrical quantities

1.

$\alpha$: The angle from the x-axis (east) to the x'-axis (CCD abscissa), measured counterclockwise

2.

$\beta$: The angle from the -y-axis (south) to the y'-axis (CCD ordinate), measured counterclockwise

3.

s_x': The CCD scale in the x' direction

4.

s_y': The CCD scale in the y' direction

as

$${\bf A} = \left( \begin{array}{rr} s_{x'} \cos\alpha & -s_{y... ...\\ -s_{x'} \sin\alpha & -s_{y'} \cos\beta \end{array}\right),$$ (9.29)

where the minus signs on a₂₁ and a₂₂ are due to the fact that we have a usual right-handed coordinate system being rotated with respect to a left-handed coordinate system, cf. Figure .