4 The $\vec{L-N}$ relationship

Each of the three "outliers'' that depart from the L-N relation in the NGC5044 sample studied by C99 (namely N42, N49, and N50), displays a different kind of peculiarity. As shown in Sect. 2, N42 has a normal colour (B-V=0.75), but its morphology reveals the presence of a central bulge that should be accounted for in a multi-component fitting model.

On the other hand, N49 is a very blue (B-V=0.49) irregular galaxy (C99), and cannot therefore be included in our analysis of the dwarf elliptical population.

N50 takes only apparently the look of a standard dE object. In the previous section we showed that in spite of its quite normal B-V, this galaxy shows the signature of relatively fresh star formation in its centre.

Given a so wide range of morphological features it is difficult to firmly assess their systematic influence on the dE L-N relationship. We should certainly agree with YC00 that any simple colour argument such as that relying on the integrated B-V, is not sufficient alone to secure a fair sample selection.

Attempting a very summary analysis in this sense, one could expect bulge-enhanced systems to affect the L-N relation in a systematic way, especially at the bright tail (M_V < -15) of dE luminosity function, by forcing a lower value of N (see Fig. 3 in C99). While in some cases this could even (artificially) improve the match with the standard L-Nrelationship, it does not help much for calibrating galaxy distance since in any case the Sérsic shape parameter poorly tracks galaxy luminosity at brighter magnitudes.

On the other side, intervening star formation, especially in case of clumpy features like those in N50, could dramatically affect galaxy surface brightness enhancing the spread in the profile fitting procedure. Again, this problem would more severly affect brighter dEs for which both "convex'' and "concave'' Sérsic profiles could result.

A fair estimate of the frequency of peculiar objects like N42 or N50 is in this regard the real concern to ultimately assess the reliability of the L-N relation as a distance indicator. M32-like galaxies seem rather rare objects (Ferguson & Binggeli 1994; Ziegler & Bender 1998) but, as a matter of fact, even a rather coarse sample like that of C99 resulted affected by over 20% of such "deviating'' objects.

In any case, it is clear from our results that any useful application of the L-N relation as a distance indicator should forcedly be pursued on a statistical basis mainly relying on non-nucleated (faint) dEs. Obviously, such a tight sampling constraint might be the most stricking drawback of this approach for extragalactic studies.

   
4.1 The $\mathsf{\alpha}$ vs. N coupling

It is well known that reliable extragalactic distance indicators are either based on the luminosity of a given standard candle (e.g., SNIa, globular clusters luminosity function, etc.), or on the relation between two independently measured parameters of galaxies, one depending on distance and the other distance-independent (e.g., the Tully-Fisher and $D_n-\sigma$ relations) (see for example Jacoby et al. 1992 or Trimble 1997 and references therein). For the L-Nrelation to fulfill the condition of independence between both parameters, the total apparent magnitude should be obtained independently from N by means of aperture or growth-curve photometry, instead of calculating it from the integration of Eq. (1), i.e.,

$$V_{\rm T} = S_0-2.5\,\log(2\pi\alpha^2) -2.5\,\log\biggl[{\Gamma\bigl(\frac{2}{N}\bigr)\over N}\biggr]\cdot$$ (3)

However, as long as there is a good correspondence between total magnitudes obtained from Eq. (3) and from aperture or growth curve photometry (excluding nuclei, when necessary), integrated magnitudes should also work. Although the physical ground behind the L-N relation is far from being understood, certainly Young & Currie (1994, 1995) have made some interesting hypotheses in that direction, and first attempts towards a self-consistent theoretical scenario are due to Lima Neto et al. (1999).

The situation with the $\alpha-N$ relation is substantially different, since it involves two free parameters obtained from the same fitting formula. Worse, it is evident from Eq. (1) that the coupling between $\alpha$ and N must be strong, i.e., any error in the measurement of N will propagate (non linearly) to $\alpha$. This is important because there is a sizable scatter among values of N measured for the same galaxies by different researchers (Ryden et al. 1999).

As an illustrative example, we have re-fitted from the C99 data the two high-S/N profiles of dwarfs N29 (N = 0.54) and N83 (N = 0.93) after varying the adopted sky level by a $\pm 20\,\%$ of $\sigma_{\rm SKY}$ (this is a relative fluctuation $\Delta {\rm sky/sky} \sim 0.1 \%$). In addition, based on the C99 observational setup, we also generated an artificial profile for a faint N > 1 galaxy and explored sky-fluctuation uncertainty likewise.

The results are displayed in Fig. 7 where solid bars represent the ranges spanned by the fitted Sérsic parameters. It is evident that variations in the adopted sky level cause the galaxies to move along the $\alpha-N$ relation (the dotted line is the polynomial fit for Fornax Cluster dwarfs given by YC00), showing that observational errors at least partially contribute to the $\alpha-N$ relation. This feature is to some extent a straightforward geometrical consequence of the fit: a more concave profile will generally predict a more "spiked'' nucleus, that is a sharper pseudo scalelength.

Figure 7: Shape parameter (N) vs. pseudo scalelength ($\alpha$) for two galaxies in the C99 NGC5044 Group sample and a simulated N>1 model. Bars represent the spanned range of the ($N,\alpha$) parameters after varying the sky level by $\pm 20\%$ of the sky rms. The dotted line is the YC00 polynomial fit to Fornax Cluster dwarfs. Dashed lines are galaxy loci for different constant luminosity (see text for details)

Also shown in Fig. 7 (dashed lines) are the loci for galaxies with constant luminosity and $V_{\rm T} \propto S_0$, according to BJ98. The three dashed lines correspond to $V_{\rm T}=14$, 16, and 18 mag, respectively, from upper-left to lower-right. Even before Sérsic law was first used to quantify the shape of dE profiles, it was qualitatively known that brighter dwarfs tend to be larger (and, at the same time, of higher surface brightness, and with more "concave'' profiles; e.g., Ferguson & Binggeli 1994, and references therein). Figure 7 shows, instead, that brighter galaxies following the $\alpha-N$ relation tend to have smaller $\alpha$(and N) values (this can also be seen from the data in YC95; see also Jerjen & Binggeli 1997). This is actually why the parameter $\alpha$ in the Sérsic law (Eq. (1)) can no longer be taken as a physical scalelength. Our suggestion is therefore that special caution should be deserved in using the $\alpha-N$ relationship to infer galaxies distances.