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 and , 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 and data input to or output from ASTROM are converted to and from this internal standard as required. The conversion from to plate coordinates consists of the following steps:
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 and by the factor , the coefficient depending on the telescope type specified. The values for each telescope type are given in the following table:
|AAT2||AAT PF doublet||+147.1|
|AAT3||AAT PF triplet||+178.6|
For the 4- and 6-coefficient linear models, the fitting process consists of finding a set of coefficients which transform the measured reference star data into plate coordinates which approximate those calculated from the data. For the 7-9 coefficient solutions, revised estimates of the plate centre and/or radial distortion coefficient are made as well.
The models relate the following three types of coordinate:
Two varieties of 4-coefficient linear model are tried, one the mirror-image of the other. The standard model is:
The laterally inverted model is:
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:
Instead of the coefficients being found directly, the fits are, in fact, implemented in terms of corrections to assumed approximate values of . For example, the 6-coefficient model is fitted as:
When determining the plate centre, the following extra non-linear terms are added to the basic 6-coefficient linear model:
The coefficients and estimate the offset between the pole of projection and the current origin. This offset is used to improve the plate centre (and to correct the zero point ) prior to recomputing for each reference star.
When determining the radial distortion coefficient, the following extra terms are added:
The obtained from the fit is added to the current 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 . Each reference star thus produces two rows of design matrix - one for and one for . Internally, the measured coordinates 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 Basic astrometry program