4 SBF distances of the dE galaxies

The results in Tables 6 and 7 show that for most galaxies the fluctuation magnitude $\overline{m}_R^0$ varies from field to field together with the local (B-R)₀ colour. This dependency was investigated by Jerjen et al. (2000, 2001) using Worthey's (1994) on-line model interpolation engine combined with the evolutionary isochrones from the Padova library (Bertelli et al. 1994). The locus of theoretical (B-R)₀ and $\overline{m}_R^0$ values for mainly old stellar populations consists of two branches (Fig. 8). A parabolic branch stretching from 0.85< (B-R)₀< 1.35 is solely defined by single-burst, old (>12 Gyr), metal-poor ([Fe/H] <-0.5) stellar populations. Slightly younger (8-12 Gyr), more metal-rich populations or populations with a second burst of star formation fall onto a linear branch from 1.0 < (B-R)₀ < 1.5 (see also Fig. 8 of Jerjen et al. 2000). In their sample of Cen A group dwarfs, Jerjen et al. (2000) identified the dwarf ESO384-016 as "single-burst, old, metal-poor'' type with data on the parabolic branch, whereas the three dEs ESO269-066, AM1339-445, and AM1343-452 produced data on the linear branch implying a younger and/or more complex underlying stellar population. These findings are consistent with the diversity of star formation histories revealed in Local Group (LG) dwarf ellipticals (Da Costa 1998; Grebel 1998). Moreover, the SBF data for Cen A group dwarfs provided first observational support for the predicted relative offset between the two theoretical branches.

Figure 8: Worthey+Padova model predictions of a (B-R)0 colour-R-band fluctuation luminosity relation for a grid of stellar populations (open circles and symbols): a set of single burst populations that covers the {age = 8, 12, 17 Gyr} $\times$ {[Fe/H] =-1.7, -1.6, ..., -1.0, -0.5, -0.25, 0} parameter space (with [Fe/H] $\geq -1.3$ in the case of 17 Gyr due to model limitations) and a set of composite populations where the previously defined populations were mixed at the 10, 20 and 30% level (in mass) with a second generation of 5 Gyr old stars with solar metallicity. The solid lines are the best least squares fits to the two branches exhibit by the 116 data points. The parabolic branch stretching from 0.85< (B-R)< 1.35 is solely defined by single-burst, old (>12 Gyr), metal-poor ([Fe/H] <-0.5) stellar populations. Slightly younger (8-12 Gyr), more metal-rich populations or populations with a second burst of star formation fall onto a linear branch from 1.0 < (B-R) < 1.5. A colour independent offset of 0.13 mag was applied to the model data to match the empirical zero point (see text).

An empirical zero point for the (B-R)₀- $\overline{m}_R^0$ relation was established from a comparison of SBF data for three M81 dwarf galaxies (Jerjen et al. 2001) with independent distances measured by Karachentsev and collaborators (1999, 2000) by means of the I-band magnitude of the red giant branch tip (TRGB). Lee et al. (1993) demonstrated that this method is a reliable distance indicator as relatively independent of age and metallicity. According to Da Costa & Armandroff (1990), the TRGB is located at $M_I = -4.05\pm0.1$ mag for metal-poor systems, a calibration value that was recently confirmed by Ferrarese et al. (2000) from galaxies with Cepheid distances and by Bellazzini et al. (2001) based on photometry and a distance estimate from a detached eclipsing binary in the Galactic globular cluster Centauri.

The SBF and TRGB observations suggested a systematic underestimation of $\overline{M}_R$ by 0.13 $\pm$ 0.03 mag. The origin of this discrepancy is not understood. But we refer the reader to Tonry et al. (2001) for a discussion of similar problems of the stellar evolution theory and population synthesis to reproduce the I-band SBF zero point. At the moment, the empirical R-band SBF calibration resting on the TRGB zero point appears the most reliable. Based on this value two analytical expressions were defined (Jerjen et al. 2001), one for each branch of the colour-fluctuation luminosity relation. They are currently used to calibrate R-band SBF data:

$$\overline{M}_R 6.09 (B-R)_0-8.94 \quad \mbox{for}~~1.00<(B-R)_0<1.35$$ = (1)

$$\overline{M}_R 1.89 [(B-R)_0-0.77]^2-1.39\quad \mbox{for}~~0.80<(B-R)_0<1.35.$$ = (2)

Plotting $\overline{m}_R^0$ as a function of (B-R)₀ colour in Fig. 7 revealed a strong linear correlation between the two quantities for all galaxies but FCC116 which will be discussed separately. The linear trend demanded the use of Eq. (1) for the calibration of a data set by shifting the points simultaneously along the ordinate to find the best least squares fit. These best fits are shown in the panels of Fig. 7. The given error bars reflect the $1\sigma$ uncertainty for a single measurement due to the sources discussed in Sect. 3. The corresponding distance moduli for the galaxies are compiled in Table 8. The quoted overall distance errors there (Col. 3) also include the systematic error of $\pm$0.1 mag from the TRGB zero point uncertainty.

FCC116 is a good example for the problem that occurs if no significant colour variation is detected among the analysed SBF fields in a galaxy. The lack of a clear trend in the [(B-R)₀, $\overline{m}_R^0$] data makes it generally impossible to decide on which of the two branches to use for the calibration and thus prevents a secure distance measurement. However, we can utilize other arguments in the case of FCC116. The distance modulus of $30.95\pm0.19$ inferred from the parabolic branch is quite small moving this dwarf into the foreground of the Fornax cluster and $\sim$4.5 Mpc away from its center. Such a short distance also appears incompatible with the observed velocity of $v_\odot=1635$ km s^-1 for FCC116 that is only 150 km s^-1 larger than the mean heliocentric velocity of the cluster (see below). Because of these reasons we feel confident that $(m-M)=31.30\pm0.19$ derived from the linear branch and formally consistent with the cluster center distance is the true distance modulus of FCC116.

 

 

Table 8: Distances of the dwarf ellipticals and the Fornax cluster.

		(m-M)0	Random error	Method
Galaxy	Type	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)
FCC043	dS0,N	$31.66\pm0.11$	0.05	SBF(R)
FCC050	dE0,N	$31.55\pm0.28$	0.26	SBF(R)
FCC082	dE1,N	$31.82\pm0.36$	0.35	SBF(R)
FCC085	dE0,N	$31.37\pm0.18$	0.15	SBF(R)
FCC100	dE4,N	$31.84\pm0.16$	0.12	SBF(R)
FCC116	dE1,N	$31.30\pm0.19$	0.16	SBF(R)
FCC136	dE2,N	$31.55\pm0.14$	0.10	SBF(R)
FCC150	dE4,N	$31.31\pm0.16$	0.13	SBF(R)
mean		$31.54\pm0.07$