SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 359, 1195-1200 (2000)

Previous Section Next Section Title Page Table of Contents

9. The adjustment

The set of coefficients [FORMULA] that define the model were found through least squares adjustment (Arley 1950, Bevington 1968). Each of the [FORMULA] data points provides us with two equations of condition; namely, the [FORMULA] and [FORMULA] components of the spherical vector equation (15). The design matrix can be constructed straightforwardly, although its huge size ([FORMULA]Mb for [FORMULA] 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 [FORMULA] in each observation i an effective standard deviation was introduced in the form:

[EQUATION]

and the weight factors were introduced as:

[EQUATION]

where the normalization factor [FORMULA] is:

[EQUATION]

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).

Another important parameter to adjust is the maximum degree of the Fourier series [FORMULA]. This was done by a simple examination of the RMS of the adjustment:

[EQUATION]

which is related to the [FORMULA] statistic through the equation:

[EQUATION]

and to the Birge ratio

[EQUATION]

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 [FORMULA] from 1 to 20. It is found that the RMS stabilizes around [FORMULA] mas for [FORMULA] in the case of positional parameters. On the other hand, [FORMULA] 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 [FORMULA]. Several fits were carried with this process with [FORMULA] 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 [FORMULA] are shown in Table 2.


[TABLE]

Table 2. Final statistics of the adjustment. The columns show the number of retained stars [FORMULA], the RMS of the adjustment and the squared Birge ratio [FORMULA]. For the [FORMULA] adjustment, the stars were eliminated with a previous [FORMULA] adjustment


Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
helpdesk.link@springer.de