Method

For each input sequence, up to three astrometric solutions are reported. The first is a four coefficient linear model (zero points, scale and orientation), requiring at least two reference stars. The second, computed in addition to the 4-coefficient model if there are at least three reference stars, is a six coefficient linear model (zero points, scales in x and y, orientation and nonperpendicularity). The third solution, which is performed on request and providing at least 10 reference stars have been supplied, has 7-9 coefficients and includes in the model the radial distortion coefficient and/or the plate centre, along with the six linear terms.

The 4-coefficient model is useful (1) for rough and ready astrometry, e.g. from a print using a ruler or graph paper, and (2) for identifying an erroneous reference star, the higher order fits tending to disguise the error. On most occasions, the 6-coefficient solution will be the most useful.

Internally, the modelling is done in idealized ``plate coordinates'', and the various $[\alpha,\delta\,]$ and $[x,y\,]$ data input to or output from ASTROM are converted to and from this internal standard as required. The conversion from $[\alpha,\delta\,]$ to plate coordinates consists of the following steps:

1.

Appropriate operations to transform the supplied $[\alpha,\delta\,]$ into either observed coordinates (if the optional observation data have been provided) or mean coordinates at the plate epoch (if not).

2.

Conventional gnomonic projection, using the given plate centre $[\alpha,\delta\,]$, to obtain tangential coordinates $[\xi,\eta\,]$.

3.

A small adjustment to allow for departures from tangent-plane geometry.

The distortion model in step 3 is the usual ``cubic'' one, where the vector from the plate centre to the star image is lengthened by an amount proportional to the cube of the length of this vector. The adjustment is carried out by multiplying each of $\xi$ and $\eta$ by the factor $(1 + q (\xi^{2}+\eta^{2}))$, the coefficient q depending on the telescope type specified. The values for each telescope type are given in the following table:

telescope type	description	q
ASTR	astrograph	zero
SCHM	Schmidt	-1/3
AAT2	AAT PF doublet	+147.1
AAT3	AAT PF triplet	+178.6
AAT8	AAT f/8	+21.2
JKT8	JKT f/8	+14.7
GENE	general	specified

Notes:

• Positive q values correspond to pincushion distortion, negative to barrel distortion.

• In the case of telescope type GENE (generalized pincushion/barrel distortion), q is specified directly as a numeric parameter, and therefore can be used for any telescope or camera which is adequately described by the distortion model.

• The difference between the Schmidt and tangent-plane projections is conventionally assumed to be that between r and $\tan r$; ASTROM's q = -1/3 is equivalent to making the approximation $\tan r \simeq r + r^3/3$.

• The coefficient q can, optionally, be determined automatically.

For the 4- and 6-coefficient linear models, the fitting process consists of finding a set of coefficients which transform the measured reference star $[x,y\,]$ data into plate coordinates which approximate those calculated from the $[\alpha,\delta\,]$ data. For the 7-9 coefficient solutions, revised estimates of the plate centre $[\alpha,\delta\,]$ and/or radial distortion coefficient q are made as well.

The models relate the following three types of coordinate:

• The estimated coordinates [x_e,y_e], derived from the reference star $[\alpha,\delta\,]$ by gnomonic projection and the application of radial distortion, using the current estimates of the plate centre and radial distortion coefficient.

• The measured coordinates [x_m,y_m], as supplied.

• The predicted coordinates [x_p,y_p], derived by the application of the current linear model to the measured coordinates.

Two varieties of 4-coefficient linear model are tried, one the mirror-image of the other. The standard model is:

$$\begin{eqnarray*} x_{e} & \simeq & a_{1} + a_{2} x_{m} + a_{3} y_{m} \\ y_{e} & \simeq & b_{1} - a_{3} x_{m} + a_{2} y_{m} \end{eqnarray*}$$

The laterally inverted model is:

$$\begin{eqnarray*} x_{e} & \simeq & a_{1} + a_{2} x_{m} + a_{3} y_{m} \\ y_{e} & \simeq & b_{1} + a_{3} x_{m} - a_{2} y_{m} \end{eqnarray*}$$

The one delivering the smallest RMS error is selected. If only two reference stars have been supplied, the standard model is used.

The 6-coefficient linear model is as follows:

$$\begin{eqnarray*} x_{e} & \simeq & a_{1} + a_{2} x_{m} + a_{3} y_{m} \\ y_{e} & \simeq & b_{1} + b_{2} x_{m} + b_{3} y_{m} \end{eqnarray*}$$

Instead of the coefficients a_n,b_n being found directly, the fits are, in fact, implemented in terms of corrections $\Delta a_{n},\Delta b_{n}$ to assumed approximate values of a_n,b_n. For example, the 6-coefficient model is fitted as:

$$\begin{eqnarray*} x_{e} - x_{p} & \simeq & \Delta a_{1} + \Delta a_{2} x_{m} + ... ...\simeq & \Delta b_{1} + \Delta b_{2} x_{m} + \Delta b_{3} y_{m} \end{eqnarray*}$$

When determining the plate centre, the following extra non-linear terms are added to the basic 6-coefficient linear model:

$$\begin{eqnarray*} x_{e} - x_{p} & \simeq & \cdots + p_{1} (x_{p}^{2} + q (3 x_{p... ... x_{p}y_{p})) + p_{2} (y_{p}^{2} + q (x_{p}^{2} + 3 y_{p}^{2})) \end{eqnarray*}$$

The coefficients p₁ and p₂ estimate the offset between the pole of projection and the current [x_p,y_p] origin. This offset is used to improve the plate centre $[\alpha,\delta\,]$ (and to correct the zero point [a₁,b₁]) prior to recomputing [x_p,y_p] for each reference star.

When determining the radial distortion coefficient, the following extra terms are added:

$$\begin{eqnarray*} x_{e} - x_{p} & \simeq & \cdots - \Delta q (x_{p}^{2} + y_{p}^... ...y_{p} & \simeq & \cdots - \Delta q (x_{p}^{2} + y_{p}^{2}) y_{p} \end{eqnarray*}$$

The $\Delta q$ obtained from the fit is added to the current q to provide a better estimate.

The above expressions are similar to those derived by Murray in sections 8.3.1ff of Vectorial Astrometry (Adam Hilger, 1983). The main difference is that in ASTROM the centres of the gnomonic projection and cubic distortion are assumed to be coincident.

All three types of solution are found by the iterative application of a least-squares algorithm based on singular value decomposition of the design matrix. (See sections 2.9 and 14.3 of Numerical Recipes, Press et al., Cambridge University Press, 1986.) This algorithm gives identical results to the traditional normal equations approach, but copes better with the ill-conditioned character of the 7-9 coefficient model. The fit minimizes $\Sigma ((x_{e}-x_{p})^{2}+(y_{e}-y_{p})^{2})$. Each reference star thus produces two rows of design matrix - one for x and one for y. Internally, the measured coordinates [x_m,y_m] are scaled to unit RMS to reduce the risk of numerical problems during the fitting process.

In the case of the 4- and 6-coefficient linear models, a single iteration is, in principle, all that is needed, whatever the starting values for the coefficients. However, a second iteration is performed in order to minimize rounding errors.

The 7-9 coefficient models are highly nonlinear, with adjustments of plate centre and - especially - radial distortion producing large changes in the scales and zero points which depend on the distribution of reference stars. To ensure convergence, given reasonable starting values for the plate centre and radial distortion coefficient, the following strategy is used:

• ASTROM insists that at least 10 reference stars be available if a non-linear fit is to be attempted.

• A fixed and ample number of iterations is used - currently 20.

• Iterations which fit the plate centre and/or the distortion alternate with ones containing the six linear terms alone. The final iteration is the linear model.

• Where fitting of both the plate centre and the distortion has been requested, for the first few iterations only one or other of these is included in the fit, in alternation. Once reliable estimates of each have been obtained the full model is fitted at once.