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Astron. Astrophys. 359, 447-456 (2000)

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6. Statistical trends

From Fig. 3 it is clear that, if most nuclei are truly central, we should expect to see, in the frequency distribution of measured nuclear offsets, a strong peak around [FORMULA] 0.5". This is indeed what is observed, as shown in Fig. 4. However, what is also quite evident from this distribution is the existence of a tail of large [FORMULA]-values; the distribution is strongly skewed towards large measured offsets, and this is clearly hinting at a population of dwarfs with true nuclear offsets. How can we quantify this? First, we note that the peak is centered on [FORMULA] = 0.4", rather than 0.5" (or 0.54", as is the precise value from Fig. 3). This means that we have apparently slightly overestimated the systematic errors with our Monte Carlo simulation. It should be noted that the observed peak, at [FORMULA] 0.5", must almost certainly stems from a population of dwarfs with central nuclei; at least we see no other explanation for it. Judged from the width of this peak towards smaller offsets (to the left), we estimate a Gaussian sigma of [FORMULA] 0.2" (as compared to 0.22" from our simulation; Fig. 3). Assuming, then, a Gaussian distribution with mean and sigma as 0.4" [FORMULA]0.2" for a population of perfectly central nuclei, any measured nuclear offset larger than 2[FORMULA] = 0.8" is very likely real (with a probability of 95%). There will be some true nuclear offsets among those with [FORMULA] 0.8", but for an individual case we cannot tell. Above 0.8" we should be on the safe side and be allowed to take the measured [FORMULA] for granted, albeit with a large random error of [FORMULA] 0.4". We therefore adopt a very rough division line at [FORMULA] = 0.8" (the arrow in Fig. 4) in the sense that most dwarf nuclei with [FORMULA] 0.8" are regarded as central, while those with [FORMULA] 0.8" are regarded as truly displaced from the galaxy center.

[FIGURE] Fig. 4. The frequency distribution of measured nuclear offsets [FORMULA]. There are three very large (however uncertain) nuclear offsets beyond the bounds of the figure. The arrow at [FORMULA] = 0.8" is indicating a critical line above which a nuclear offset can be regarded as real (see text).

In Fig. 5 we have plotted the nuclear offset [FORMULA] versus the total galaxy magnitude [FORMULA]. The nuclear offset distances are shown with their formal r.m.s. errors from averaging the five offsets determined at five isophotal levels. As discussed above, the true uncertainties are usually much larger, of order 0.4", without considering the systematic errors (see Fig. 3). However, it is evident that even these formal errors can be considerably large for dwarfs with [FORMULA] 0.8". Truly "uncertain" or inconsistent cases (cf. Sect. 3) - indicated by colons in Table 1 and flagged with open symbols in Figs. 5-7 - are also more common above the division line of [FORMULA] = 0.8".

[FIGURE] Fig. 5. Nuclear off-center distance [FORMULA] versus total galaxy magnitude [FORMULA]. Open symbols indicate uncertain offset distances, three of which are larger than 2" (arrows). The error bars are standard deviations of the mean [FORMULA]-values. The broken line at [FORMULA] is the fiducial division line between dwarfs with central and non-central nuclei.

[FIGURE] Fig. 6. Same as Fig. 5 but now plotting the dimensionless ratio [FORMULA] (nuclear offset divided by effective radius) versus total magnitude [FORMULA]. The broken line is the mean locus of [FORMULA] = 0.8" (see text).

[FIGURE] Fig. 7. The nuclear off-center distance [FORMULA] (in arcsecs) versus the effective mean surface brightness [FORMULA] (in B mag arcsec-2) of the parent galaxy. Otherwise as Fig. 5.

No systematic trend of [FORMULA] with [FORMULA] is obvious from Fig. 5. A few bright dwarfs with off-center nuclei are popping out, but this is caused by the large size of the parent galaxies (see below). The fraction of dwarfs above the chosen division line at [FORMULA] = 0.8" compared to the total, given as percentage of dwarfs with off-center nuclei, is 29% (23 out of 78) if uncertain cases are included, or 23% (15 out of 64) if uncertain cases are excluded. Overall, it seems fair (and conservative) to state that roughly 20% of all nucleated dwarf ellipticals have off-center nuclei .

So far we have given the nuclear off-center distances in absolute units of arcsecs. However, as the size of the dwarf galaxies varies, one might suspect that a relative measure of nuclear off-center distance, i.e. relative to some characteristic radius of the parent galaxy, is physically more meaningful. We have therefore plotted, in Fig. 6, the ratio [FORMULA] versus [FORMULA], to see whether we get a different picture.

One obvious change is that the large nuclear offsets in some bright dwarfs (see Fig. 5) are strongly reduced in relative units, i.e. the nuclear offsets in those dwarfs are large in distance but are relatively small as the dwarfs themselves are large in size. Less luminous dwarfs tend to have larger relative nuclear offsets. But where do we now have to draw the critical line between significant and non-significant nuclear offsets?

It turns out that this line does not strongly depend on [FORMULA], simply because the metric (effective) radius of dE galaxies does not. This is also expressed by the basic [FORMULA] relation of dEs, where the surface brightness is systematically decreasing with decreasing luminosity. To show this, we use the formula for the observed scaling law of Virgo cluster dEs given in BC91:

[EQUATION]

where [FORMULA] is in B mag arcsec-2, and the identity:

[EQUATION]

to derive the equation ([FORMULA] always in arcsecs):

[EQUATION]

Eq. (3) and [FORMULA] = 0.8" now define a critical relative nuclear offset [FORMULA], which is shown as broken line in Fig. 6. Obiously, using [FORMULA] instead of [FORMULA] does not make much difference, justifying a posteriori that we have worked with absolute nuclear offsets (in arcsecs or parsecs) from the beginning. Still, there is a hint that the offsets become more common in fainter dwarfs when they are measured in a relative way.

A most interesting and possibly real statistical trend of the nuclear offset distance with the mean effective surface brightness of the parent galaxy is shown in Fig. 7. There is a clear tendency of [FORMULA] to increase with decreasing surface brightness. Again, given the small sample and large errors, we cannot claim significance of the effect, but Fig. 7 is certainly very suggestive. What is most striking, of course, is the apparent lack of high-surface brightness dwarfs with large nuclear offsets (the void region in the upper left of Fig. 7). Remember that the relative offsets ([FORMULA]) for bright dwarfs, which on average also have higher surface brightness, would be even further reduced.

On the other hand, the apparent lack of very low-surface brightness dwarfs with small nuclear offsets (the void in the lower right in Fig. 7) is probably an observational bias (incompleteness of the sample), or an artifact from unaccounted-for errors which grossly increase at such faint surface brightness levels. In this case the trend might be better described as an increasing spread in the range of observed offsets with decreasing surface brightness .

This spread might result from the nuclear oscillations seen in the simulations of Miller & Smith (1992) and Taga & Iye (1998). Its increase towards fainter surface brightness, if real , would then simply mean an increase in the amplitude of oscillations. Although this was not explicitly explored by the simulations just mentioned, it appears quite plausible that a nucleus can oscillate more easily, i.e. with larger amplitude, in a galaxy of low surface brightness and hence shallow gravitational potential than in a deep, cuspy potential (high-surface brightness central part). Interestingly, the few normal elliptical galaxies with known nuclear displacements in the study of Lauer et al. (1995) all have cores , as noted by the authors themselves. Core ellipticals have lower central surface brightness than cuspy ellipticals, so this would be the same effect.

We have also looked for a trend with the "exponential" central surface brightness, [FORMULA], which is one of the parameters of the exponential law that was fitted to the dwarf profiles by BC91/93, but found this relation to be much weaker than that with the mean effective surface brightness. This is actually not too surprising, as [FORMULA] is nearly constant (although with very large scatter) in the magnitude range [FORMULA] = 14-17 mag (see Fig. 9a of BC91). The reason for this is that the exponential law was fitted to the outer part of the dE profiles. As there is often an extended brightness excess above the fitted exponential, [FORMULA] does not represent the "true" central surface brightness of the parent galaxy underlying the quasi-stellar nucleus (see BC91/93). A better measure for this would be the central surface brightness from the best-fitting King model (King 1966), but many dwarf profiles could not be fitted by a King model. Ironically, these are often those dwarfs that are found here to have an off-center nucleus. One reason for this could be that all dE,N profiles were automatically centered on their "central" nucleus. On the other hand, there are hints that a significant fraction of bright early-type dwarfs are rotating disk galaxies rather than King spheroids (e.g., Jerjen et al. 2000, and references therein). Rotation might, in fact, be the driving mechanism for the nuclear oscillations (see Taga & Iye 1998)!

We have looked for possible trends of [FORMULA] with many other parameters, among them are -: (1) the flattening of the parent galaxy (with a slight hint that apparently flatter dEs have higher nuclear offsets; but see the appendix for a significant nucleus-ellipticity relation); (2) the angular distance of the parent galaxy from M87 (as the nuclear displacement might be influenced from the environment within the Virgo cluster); (3) the isophotal shape [FORMULA] of the parent galaxy (diskyness [FORMULA] boxyness, as given by Ryden et al. 1999); (4) the strength of the nucleus itself (as given by BC91/93 from their King profile fitting, but unfortunately, as mentioned above, only available for a subsample).

No clear trends were found with these parameters. But this is no proof of their absence, as our qualitative analysis of a very limited data set is necessarily blind for any subtle effects, if present. The most interesting, or most expected trend would be a relation between nuclear displacement and nuclear strength. Such a relation, in the sense that stronger nuclei should be less displaced from the center (oscillate with smaller amplitude), is in fact predicted (Taga & Iye 1998).

Finally, we have looked for any systematic trend of the nuclear offset position angle with respect to the galaxy major axis, and found none.

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Online publication: July 7, 2000
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