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4. Total magnitudes

4.1. The t system

A reasonably discerning extrapolation procedure was required for the VPC galaxy sample as (1) most VPC galaxies are too bright and/or extended for their images to be seeing dominated, and (2) a wide variety of morphological types (including of course a very large number of low-surface-brightness galaxies) are represented. The total-magnitude system we adopted is the t system of Young (1994) and and Young et al. (submitted); as outlined by Young (in press). This system involves the smoothing of galaxy images prior to modeling them by means of Sérsic's (1968) law. Note that in the specific case of un-nucleated ellipticals, prior smoothing is not actually necessary as demonstrated by Young & Currie (1994) who fitted Sérsic's law to high-resolution images of dwarf and intermediate ellipticals.

4.2. Total magnitudes and colours for unsaturated images

In the case of the low-resolution VPC data, Sérsic's law was found to offer an excellent fit to virtually every unsaturated surface-brightness profile, irrespective of morphological type. This can be seen from the thirty example plots presented in Appendix A (click here)gif. No total magnitude estimates were attempted for saturated galaxies. Instead, Young et al. (in preparation) will present t-system total magnitudes for these objects based on even lower-resolution CCD photometry.

During the image-segmentation stage of the reductions, profile information was generated for each galaxy image in the form of ellipticities, position angles and reduced radii, for all unsaturated isophotes in tex2html_wrap_inline3647 intervals brightward of the limiting one. In the case of the tex2html_wrap_inline3309 plates, this corresponded to a maximum of 12 levels (a maximum of 16 in the case of plate U9362 and a maximum of 12 in the case of R2936).

The profile parameterizations were achieved by incrementing n from 0.2 through 3.0 in steps of 0.01, and attempting to fit a straight line to tex2html_wrap_inline3653 as a function of rn. Points corresponding to isophotes that overlapped with those of other images (as represented by five-pointed stars in Appendix A (click here)) were assigned weights 50% lower than those assigned to the remaining points (as represented by asterisks in Appendix A (click here)). tex2html_wrap_inline3657 was evaluated for each fit, and the parameters (n, r0 and tex2html_wrap_inline3663) that provided the best fit for a particular galaxy, were used to derive a total-magnitude estimate for that galaxy by means of Young & Currie's (1994) Eq. (3).

In the case of the tex2html_wrap_inline3309 band, although we had only parametrised the profiles from Plate J9229, as we had already computed two usable tex2html_wrap_inline3571 values for those galaxies lying within Fields A and B (one value from each tex2html_wrap_inline3309 plate), we were still able to average the extrapolated total magnitude values. This was achieved by subtracting tex2html_wrap_inline3671 from the total magnitudes derived from the parametrizations of the Plate J9229 images.

The transformation of total magnitudes from the tex2html_wrap_inline3309 system to the B system was achieved by means of Eq. (1), and the approximate relationship:


 equation566

  figure571
Figure 8: Unsaturated VPC galaxies for which (B-V) measurements have already been published (see text for references). This two-colour plot was used to derive a transformation equation between (tex2html_wrap_inline3609) equal-area colours and (B-V) by means of the best-fitting least-squares (on (tex2html_wrap_inline3609) because the (B-V) values are expected to be more accurate as they are photoelectric measurements rather than photographic ones) straight line

which was derived from Fig. 8 (click here); a two-colour plot for unsaturated VPC galaxies with (B-V) colours previously measured by Caldwell (1983) Gallagher & Hunter (1986) Bothun et al. (1986) and/or Drinkwater & Hardy (1991). Note that although the (tex2html_wrap_inline3609) and (B-V) colour measurements were not based on identical regions of each galaxy concerned, errors due to colour-gradients are expected to be considerably smaller than those due to the inherent scatter in such a colour-colour plot.

4.3. VPC magnitudes already published

The total magnitudes listed in Table 4 of Young & Currie (1995) as well as those VPC magnitude values listed in Table 9 of Drinkwater et al. (1996), were extracted from an earlier version of the VPC; that of Young (1994). The magnitudes were computed by a variant on the method described in Sect. 4.2. This method involved adding the integral of the same function between the isophotal radius r=r25 and tex2html_wrap_inline3697 to the linear-unit equivalent of the tex2html_wrap_inline3571 value tabulated in the VPC for each galaxy. The integral was evaluated numerically using the Compound Form of Simpson's Rule, with 2000 intervals spanning the range r=r25 to the point at which tex2html_wrap_inline3653 had dropped to tex2html_wrap_inline3705 tex2html_wrap_inline3441:


eqnarray597

where tex2html_wrap_inline3709 and tex2html_wrap_inline3711. It was adopted at a time when the catalogue listed tex2html_wrap_inline3713 and tex2html_wrap_inline3715 values for Field A and B objects; but did not list the senses of the differences between the tex2html_wrap_inline3717 and tex2html_wrap_inline3719 values. With this method, it was therefore still possible to obtain averaged total magnitudes based on the isophotal magnitudes from two plates [together with extrapolations from one plate] without knowing the senses of the differences between the plates.

In a small minority of cases there was a significant difference between the sizes of the extrapolations obtained by the different methods, but on average the choice of method made little difference. The mean extrapolation [for the VPC as a whole] beyond tex2html_wrap_inline3721 due to the old method was 0.348 mag, compared to a mean of 0.352 for the more rigorous method used in this paper; though these mean extrapolations were 0.670 and 0.696 respectively for the dwarf elliptical subsample of 64 objects presented in Young & Currie (1995). The systematic difference between the two sets of magnitude values is due to the minority of objects for which the light contribution from regions exterior to tex2html_wrap_inline3723 would be significant should the profiles be extrapolated to tex2html_wrap_inline3725.


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