3 The t system

This system was first adopted by Young (1994) and Young & Currie (1998) in the compilation of their Virgo Photometry Catalogue (hereunder VPC); as well as by Young & Currie (1995), Drinkwater et al. (1996) and Young (1997); who all quote t-system total magnitude values from the VPC. The t system is a system for extrapolating surface-brightness profiles of sufficiently low resolution, to $r=\infty$where r is reduced radial distance ($\sqrt{r_{\rm major}r_{\rm minor}}$). This means that high-resolution images (those of non-nucleated dwarf and intermediate ellipticals excepted) must be smoothed sufficiently prior to any surface-brightness profile being parameterised, but has the important advantage that low-resolution images can be measured even if their profiles are significantly distorted by e.g. seeing effects or poor sampling. Although alternative smoothing functions can be used, we recommend a (radially-symmetric two-dimensional) Gaussian function.

In order to generate an extrapolated total t-system magnitude from a low-resolution surface-brightness profile, Sérsic's (1968) law is adopted:

$$\sigma(r)= \mbox{ } \sigma_{0} \mbox{ } \exp \mbox{ } \left[ -\left( \frac{r}{r_{0}}\right) ^{n} \right],$$ (5)

in which $\sigma(r)$ is the surface brightness in linear units of luminous flux density at r, $\sigma_0$ is the central surface brightness and r₀ is the angular scalelength. The extrapolated central surface brightness is therefore: $\mu_{0}=-2.5 \log_{10}\sigma_{0}$ in mag arcsec^-2, whence the equivalent expression in logarithmic surface-brightness units is:

$$\mu(r)= \mbox{ } \mu_{0} \mbox{ } + \mbox{ } 1.086 \left( \frac{r}{r_{0}} \right)^{n},$$ (6)

enabling values of $\mu_0$ and r₀ to be obtained by linear regression when the optimum value of n has been derived. The analytical solution:

$$2\pi\int_{0}^{\infty} \sigma_{0} r {\rm e}^{-(\frac{r}{r0})^... ...} \pi \sigma_{0} \Gamma \left( \frac{2}{n} \right) {r_{0}}^{2},$$ (7)

then yields an estimate of the total luminous flux within the pass-band concerned.

Clearly, this generalisation incorporates not only the $r^{\frac{1}{4}}$ law ($n=\frac{1}{4}$) but also exponentials (n=1) and Gaussians (n=2) as well as both intermediate and more extreme cases. Although it is a one-component model, galaxies exhibiting two-component (or yet more complicated) structure can be comfortably accommodated by this scheme after their images have been smoothed sufficiently.

For the benefit of readers wishing to use this extrapolation system, a listing of the relevant FORTRAN code can be found in Appendix A. The subroutine EXTRAPOL.FOR: (1) increments the profile-curvature parameter n, and attempts (for each n) to fit Sérsic's law to a surface-brightness profile defined by elliptical isophotes (r_i, $\mu_i$(r), $\sigma_{\mu_{i}(r)}$); (2) quantifies the quality of the fit obtained for each value of n; and (3) integrates the volume beneath the surface defined by the best-fitting profile form (after rotation through 2$\pi$ radians about r=0) to $r=\infty$, thereby yielding a total-magnitude estimate. It calls the subroutine FIT and the function GAMMLN, both from Press et al. (1986). Note that FIT actually fits a straight line of the form Y=A+BX, not one of the form Y=AX+B. Also, in order to reduce the dependence on other subroutines and functions, we removed the lines:
Q=1. and Q=GAMMQ(0.5$\ast$(NDATA-2),0.5$\ast$CHI2),
from this subroutine and removed the parameter Q from Line 1.

Although t-system total magnitudes cannot generally be derived directly from high resolution surface photometry of bright galaxies in the literature, they can be derived from surface photometry of virtually all galaxies fainter than $B_{25}\sim15$ mag provided that the resolution of the photometry (after smoothing) is coarser than about $4\hbox{$.\!\!^{\prime\prime}$}5$ (FWHM). In the compilation of their VPC, Young & Currie (1998) found that a reasonable amount of smoothing of their plate-scan data was necessary simply in order to prevent the fragmentation of images during the image segmentation process. They found that the minimum degree of smoothing required by the segmentation software was actually sufficient for the derivation of t-sytem total magnitudes from the surface-brightness profiles of all unsaturated galaxies with enough isophotes above the limiting one for the fits to be performed.