9. The adjustment
The set of coefficients that define the model were found through least squares adjustment (Arley 1950, Bevington 1968). Each of the data points provides us with two equations of condition; namely, the and components of the spherical vector equation (15). The design matrix can be constructed straightforwardly, although its huge size (Mb for and simple precision) made necessary a careful programming of the least squares routine.
The existence of large systematic errors in the SAO catalogue for polar stars suggested the introduction of a weighting scheme. For each difference in each observation i an effective standard deviation was introduced in the form:
To simplify the construction of the weighted normal equations, each of the conditional equations was multiplied by the square root of the effective weight (19).
and to the Birge ratio
used in the least squares process as a scale factor of the uncertainties (Cohen & Taylor 1987).
For each of the groups of stars, several least squares adjustments were carried out varying the degree of the Fourier series from 1 to 20. It is found that the RMS stabilizes around mas for in the case of positional parameters. On the other hand, was enough to find stable values of the proper motion parameters.
As a final process, outliers were eliminated using the r-statistic of Arley (1950). These were defined as stars with reduced residuals (as defined in reference Arley (1950)) with . Several fits were carried with this process with and 35, eliminating as many outliers as found. The number of remaining stars in each group, as well as the final RMS and Birge ratio are shown in Table 2.
Table 2. Final statistics of the adjustment. The columns show the number of retained stars , the RMS of the adjustment and the squared Birge ratio . For the adjustment, the stars were eliminated with a previous adjustment
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000