2 Existing extrapolation methods

In his pioneering study, Hubble (1930) found that the surface-brightness profiles of elliptical galaxies (which he measured along either the major or minor axes) seemed to be well fitted by the law:

$$\sigma(r) = \frac{\sigma_{\rm H}}{ \left( 1 + \frac{r}{r\rm _H} \right) ^{2}},$$ (1)

where an elliptical-galaxy profile can be uniquely described by two terms: $\sigma_{\rm H}$ (a central surface brightness) and $r_{\rm H}$ (a scale length). The main problem with this law is that its integrand with respect to r diverges as r increases, and it is therefore unsuitable for extrapolation to large r.

An alternative representation was proposed by de Vaucouleurs (1948):

$$\sigma(r) = \sigma_{\rm e} \; \mbox{dex} \; \left\{ -3.33 \l... ... \frac{r}{r_{\rm e}} \right) ^{\frac{1}{4}}-1 \right] \right\},$$ (2)

where an elliptical-galaxy profile can be uniquely specified by $r_{\rm e}$(the effective radius which contains half the galaxy's light). $\sigma_{\rm e}$ is the surface brightness at $r=r_{\rm e}$. This representation has the advantage of a convergent integrand:

$$2 \pi \int_{o}^{\infty} \sigma(r) r \mbox{ } {\rm d}r = 22.4 \sigma_{\rm e} {r_{\rm e}}^{2}.$$ (3)

However, it has the disadvantage that strictly, the total luminosity needs to be known before the effective parameters can be evaluated.

A move away from wholly empirical functions was made by King (1966). King's models were constructed with tidally truncated globular star clusters in mind but have also been applied to elliptical galaxies. They assume that at all points within a stellar system, the frequency distribution of stars as a function of position and velocity can be described by an isothermal Gaussian minus an offset. At any position within the system, the offset is chosen so as to ensure that the frequency distribution is zero for stars with velocities equal to and in excess of the local escape velocity. This succeeds in reducing an otherwise infinite isothermal space distribution of stars to one with a finite radius. Nevertheless, the empirical law of de Vaucouleurs actually offers a better description of tidally unperturbed galaxies than does King's model.

Despite the intricate structure exhibited by many spiral galaxies, it has long been realised, e.g. by de Vaucouleurs (1958) that their smoothed light profiles could be separated into two components: a central component approximately obeying an $r^{\frac{1}{4}}$ law (corresponding to the spheroidal bulge, denoted s) and a more extended component approximately obeying an exponential law (corresponding to the disc, denoted d):

$$\begin{eqnarray} \lefteqn{\sigma(r)=} \nonumber \\ & &\! \sigma_{\rm e,s} \mbox... ...! - \! \left( \! \frac{r}{r_{\rm 0,d}}\! \right) \! \right] \! ,\!\end{eqnarray}$$ (4)

where $\sigma_{\rm 0,d}$ (the central surface-brightness due to the disc alone) and $r_{\rm 0,d}$ (the scale length of the disc component) uniquely describe the disc component. The contribution due to this exponential component decreases as one progresses from late-type spirals to earlier types, but is not completely absent from lenticular objects or even classical ellipticals.

This trend in galaxy-profile characteristics from early through late types was exploited by de Vaucouleurs et al. (1976, 1991) during the compilation of their Second and Third Reference Catalogs of Bright Galaxies (hereunder RC2 and RC3 respectively). Their elaborate and widely-used scheme for extrapolating both aperture and surface-photometry measurements of galaxies, in order to estimate total magnitudes, is known as the T system. Although the T system has been used successfully to extrapolate the profiles of well resolved images of classical galaxies, its applicability to dwarf galaxies (as well as ellipticals intermediate between true dwarfs and true classicals) and to low-resolution images (of all galaxy types) is questionable. There appear to be at least three major limitations to the T system, the main consequences of which are summarised by Young (1997) and will be dealt with in more detail by Young et al. (in preparation).
(1) The scheme does not take into account the possibility that galaxies with profiles steeper than exponentials exist. As is evident from Young & Currie (1994, 1995) many dwarfs have profiles that are steeper than exponentials and some even have profiles as steep as Gaussians. These objects are therefore beyond the scope of the T system which necessarily over-estimates their luminosities.
(2) A morphological classification must be attempted first in order to be able to select the most appropriate extrapolation model for each galaxy concerned. This is normally done by eye. Many non-classical elliptical galaxies (including some with some characteristics of irregulars) e.g. IC 3475, 3349, 3457 and 3461, were classified as $r^{\frac{1}{4}}$-law objects by de Vaucouleurs et al. (1976, 1991) and their profiles extrapolated accordingly. In fact, IC 3475 has a very exponential profile as demonstrated by Vigroux et al. (1976), and the other three objects listed have profiles slightly steeper than an exponential as demonstrated by Young & Currie (1998). This means that whilst these objects could be accommodated by the T system if they were treated as objects with exponential profiles, their luminosities were in fact severely over-estimated by of the order of 100% in both the RC2 and the RC3 (Young 1997; Young et al., in preparation) because of limitations in the morphological typing procedure. The need to estimate the profile shape by eye before an extrapolation can be performed is therefore a serious short-coming of the T system.
(3) No account is taken of atmospheric or instrumental effects that degrade the resolution of a galaxy image and thereby modify its measurable surface-brightness profile. Clearly a low-resolution image of a particular galaxy is likely to have a more Gaussian-like profile than is a high-resolution image of the same galaxy. This makes the T system difficult to apply consistently to images of different resolution and even to images of almost identical galaxies at widely differing distances.

A worthy alternative to the T system has long been the Kron (1980) system, in which the rate of growth of the signal with respect to the signal itself is considered as a function of radial distance from the centre of each galaxy image. The light is then measured within a circular region of a radius corresponding to the point at which the logarithmic derivative of the light growth curve becomes smaller than the same upper limit for all target galaxies. In practice there are strong constraints on the suitable values for this limit, and the range of suitable values generally require an aperture of radius equal to about twice the effective (half-light) radius. The fraction of the total light within this aperture (which in practice is typically of the order of 95%) is then assumed to be constant for all galaxies, and total magnitudes can be obtained by extrapolating the measured aperture magnitude values by the same amount (which in practice simply means adding typically about -0.05 mag to each aperture magnitude).

The Kron system has two major advantages over the T system, namely that the magnitude measurement process is independent of galaxy morphological type and that atmospheric effects that degrade image resolution and distort galaxy luminosity profile shapes are taken into account. However, it does have at least three major drawbacks.
(1) Large random errors are present due to having to measure luminosity growth curves and [even harder] their derivatives, out to very large radial distances where the signal due to the galaxy is a very small fraction of the noise. Note that the Kron system does not take the signal-to-noise ratio as a function of radial distance into account at all. This can be a particularly serious problem when dealing with low-surface-brightness galaxies such as dwarfs.
(2) The need for very large apertures restricts which objects can be measured in crowded fields when undesirable galaxy or stellar images lie adjacent to the target objects. For this reason the Kron system has mainly been applied to field galaxies rather than cluster ones.
(3) Unlike the T system, which is based on aperture and surface photometry measurements that can be extracted from the literature in their published form, the Kron system does not allow one to compute total magnitudes without access to the original digital-image data.

Despite the existence of the elaborate T and Kron systems, most large machine surveys of galaxies have, understandably, adopted simpler algorithms for extrapolating large numbers of isophotal magnitudes to totals. In the APM Southern-sky Survey of Maddox et al. (1990) the majority of galaxy images were assumed to be seeing dominated, and therefore to have approximately Gaussian profiles (on average at least). This was a particularly efficient method, as knowledge of an isophotal magnitude and the angular area of that isophote was sufficient to specify the parameters of a Gaussian profile uniquely; as described in detail in Maddox et al. (1990). However, such an approach is only applicable to unresolved galaxy images from sufficiently small and/or sufficiently distant objects.

In addition to the extrapolation systems discussed in this section, there are of course others. However, such systems have generally only been applied to limited galaxy samples of a particular morphological type.