The profiles of surface brightness (
), ellipticity
(
), position angle (
) and coefficient of the
Fourier analysis of the residuals (c4), as a funtion of the
semi-major axis a in arcseconds (a1/4 scale), are available
on the online version of the journal (Fig. 1), together with the proper error bars
(see
previus section). In the same figures we report for comparison the
PSF profile as a dotted line.
Figure 1:
Luminosity and geometrical profiles of elliptical galaxies in the HDF
We divided our galaxy sample in 3 different classes, according to the luminosity profiles:
In Col. (6) of Table 1 (click here) the luminosity profile class of each galaxy is reported and the three classes are indicated with 1, 2 and 3, respectively. The same convention is used in Table 2 (click here).
Concerning the "Flat'' class, it is worth stressing that the
observed inward flattening of the luminosity profiles with respect to
the de Vaucouleurs law cannot be interpreted as due to the presence of
some core-like structure. Actually, in all galaxies of this
class for which the redshift has been measured (6 objects), the linear
size of the involved regions turns out to be much greater than the
typical core size (
). For instance, at 
,
correspond to 
, with small
differences in the range 
. On the other hand, galaxies
with luminosity profile flattening confined inside 
have been included by default in the "
Normal'' class. Moreover, by comparing the fraction of HDF
ellipticals belonging to the "Flat'' class with that obtained in
a similar way from the sample of local ellipticals provided by
Djorgovski (1985), we found a relevant difference in favour of the HDF
sample (
vs. 
). We will see in a forthcoming
paper (Fasano et al. 1998) that the different classes of luminosity
profiles are often associated with different physical properties of
the galaxies (i.e. colors and sizes).
Finally, we mention that the luminosity profile of the galaxy
strongly suggests the presence of a nuclear point
source. This galaxy also belongs to the lists of radio and ISO sources
in the HDF (Fomalont et al. 1997; Mann et al. 1997).
The shape of the ellipticity profiles appears to be correlated with the above mentioned luminosity profile classes. Actually, the galaxies belonging to the "Normal'' class show increasing or almost constant ellipticity profiles (see Fig. 1). This is the most common behaviour in nearby early-type galaxies (see Bettoni et al. 1996). On the other hand, 25 out of the 44 galaxies belonging to the "Flat'' class or to the "Merger'' class show strongly decreasing ellipticity profiles (see Fig. 1). This is an unusual behaviour in the local samples of early-type galaxy (Bettoni et al. 1996). Table 2 (click here) reports some statistical data on the shapes of the ellipticity profiles.
Figure 2: a) effective ellipticity distribution of the "Normal'' galaxies (full
line histogram) campared with that of "Flat+Merger'' (dotted line
histogram). The shaded histogram refer to the "Merger'' class alone.
b) effective ellipticity distribution of the total sample (full line
histogram) compared with that of "local'' sample from the literature
(Fasano & Vio 1991)
This peculiarity of the ellipticity profiles is likely to reflect on
the ellipticity distribution (computed at the effective radius, see
Col. 9 of Table 1 (click here)) of galaxies in our sample. Figure 2 (click here)a shows that
the ellipticity distribution of the "Normal'' class looks
remarkably different from that relative to the other two classes, the
last ones being shifted towards flatter configurations. The
Kolmogorov-Smirnov test confirms this difference at high significance
level (
). Nevertheless, the ellipticity distribution of the
whole sample (Fig. 2b) seems to be in fair agreement with that
relative to nearby galaxy samples (see Fasano & Vio 1991). It is
worth stressing that the comparison with the local samples is not
invalidated by the fact that the selection criteria of our sample
extend to the limit of recognition between stars and galaxies. In fact
the error bars shown in the ellipticity profiles take into account the
influence of the PSF (see Sect. 3.2).
In Table 2 (click here) we report some statistics on the shape of the isophotes ("disky'' for c4 > 0; "boxy'' for c4 < 0) in our galaxy sample for the different luminosity profile classes. There is a weak indication that the fraction of boxy galaxies increases from the "Normal'' to the "Flat'' and "Merger'' classes.
Figure 3:
Maximum ellipticity vs. isophotal twisting (open circles: "Normal''
class; full circles: "Flat'' class; crosses: "Merger'' class)
Concerning the position angle profiles, although the uncertainties involved in measuring the position angle of outer isophotes are relevant for HDF ellipticals, we estimate that the amount of isophotal twisting in our sample is larger (on average) than that found in local galaxy samples (see Fasano & Bonoli 1989). This fact may be explained as a consequence of the high fraction of morphologically perturbed objects in the HDF.
Figure 3 (click here) shows the distribution of HDF early-type
galaxies in the "maximum ellipticity - twisting'' plane, which,
even being qualitatively similar to that of local samples (see
Galletta 1980), shows an higher fraction of significantly twisted
objects. The maximum ellipticity 
and the total
isophotal twisting (in degrees) are reported in Cols. (10) and (11)
of Table 1 (click here), respectively.
| Class | 1 | 2 | 3 | All |
| average ellipticities | ||||
 | 0.17 | 0.28 | 0.38 | 0.23 |
 | 0.21 | 0.30 | 0.40 | 0.26 |
ellipticity profiles ( ) | ||||
 | 36 | 14 | 0 | 25 |
 | 58 | 33 | 22 | 46 |
 | 6 | 53 | 75 | 29 |
| isophotal shape () | ||||
| disky | 33 | 25 | 11 | 28 |
| elliptical | 14 | 11 | 0 | 12 |
| boxy | 22 | 33 | 50 | 29 |
| irregular | 31 | 31 | 33 | 31 |
| #gal. | 55 | 36 | 8 | 99 |
In order to derive total magnitudes and half-light radii of the
galaxies, it is convenient to use some analytical representation
of the luminosity profiles. This allows a suitable smoothing
of each profile and provides an easy way to extrapolate it.
For obvious reasons it is also convenient to operate on the
"equivalent" luminosity profiles, that is to multiply the
semi-major axis (a) by the factor 
.
The most common technique to get suitably smooth representations of any observed function is the bicubic-spline interpolation. In the case of luminosity profiles of elliptical galaxies, a nice representation is also given by the Sersic function (1968, see also Ciotti 1991). However, we preferred to represent the equivalent luminosity profiles by means of sums of gaussian functions whose peak intensities regularly decrease at increasing the standard deviations (multi-gaussian expansion technique). In a forthcoming paper (Fasano et al. 1998) we will see that this representation is useful to perform the deconvolution of the luminosity profiles (Bendinelli 1991; Emsellem et al. 1994). Here we wish only to mention that, in our particular case, this method gives usually a better representation with respect to the bicubic-spline method, especially in the inner part of luminosity profiles.
The multi-gaussian representation was also used to extrapolate the
profiles. In general, it was forced to follow the de Vaucouleurs law
down to very faint values of . However, if the galaxy size is
comparable with the PSF size (very steep profiles), it is necessary
to impose that the multi-gaussian extrapolation of the outer galaxy
profile does not fall below the very extended wings of the
HST-PSF itself. To this end we
forced the extrapolation of the luminosity profiles of very small
galaxies to converge smoothly towards the PSF profile at large
radii. Luminosity profiles with an effective radius larger than three
times the FWHM were extrapolated simply by a de Vaucouleurs' law.
A special warning is needed when computing the total magnitude of
galaxies belonging to the above defined "Merger'' class, since
their complex inner structures make undefined (or unreliable) the
inner part of luminosity profiles. In these cases the flux inside the
innermost reliable ellipse was directly mesured on the frame and was
considered as an additional contribution to the integral of the
luminosity profile, computed from that ellipse
and extrapolated by a de Vaucouleurs law.
The total magnitudes in the V606 band (STMAG system) and the
corresponding equivalent effective (half-light) radii 
are
reported in the Cols. (7) and (8) of Table 1 (click here), respectively.
The quantities in brackets close to the Col. (7) represent
the surplus magnitudes 
due to the extrapolation procedure.
They give an indication of the quality of the total magnitude estimates.
Adding these quantities to 
one obtain, for
each galaxy, the magnitude before extrapolation.
Figure 4:
Difference between our total magnitudes and the magnitudes given
by Williams et al. (1996) as a function of the effective radius
a) and of the average surface brightness b).
The meaning of the symbols is the same as in Fig. 3
In Fig. 4 (click here) we compare the V606-STMAG total magnitudes from our
detailed surface photometry with the automated FOCAS magnitudes
given by Williams et al. (1996), corrected for the average offset of
between the ABMAG and the STMAG systems. There are two
galaxies (
and 
) for which the flux has
been probably overestimated by FOCAS, due to the presence of very
close companions. Apart from these cases, the total fluxes computed
with our procedure tend to be systematically greater than those
obtained with the automated photometry. Moreover, the difference
increases at increasing both the average surface brightness of the
galaxies and (weakly) their angular size. This fact is not surprising
and it is likely to indicate that the automated photometry tends to
underestimate the halos of the ellipticals. To this concern, it is also
worth noticing that the differences 
in Fig. 4 turn out
to be roughly proportional to the previously mentioned quantities
.
