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Astron. Astrophys. 351, 487-494 (1999)

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3. The relation between [FORMULA] and the central brightness profile

Fig. 1 plots the [FORMULA] relation for the galaxies in this sample, with different symbols for core and power law galaxies (full and open circles respectively). We note that:

1) The least optically luminous galaxies and the most optically luminous ones are respectively power law and core galaxies, as in the sample studied by F97.

2) The least X-ray luminous galaxies and the most X-ray luminous ones are again respectively power law and core galaxies, consistent with the known [FORMULA]-[FORMULA] correlation. For the whole Einstein sample of 148 early type galaxies [FORMULA], although with a large scatter of more than two orders of magnitude in [FORMULA] at any fixed [FORMULA] [FORMULA] (Eskridge et al. 1995). For a magnitude-limited sample of early type galaxies with X-ray emission measured by the ROSAT all-sky survey, Beuing et al. (1999) find [FORMULA] and a scatter at least as large as that found by Eskridge et al. (1995).

3) At intermediate [FORMULA], where they coexist, core galaxies span the whole range of [FORMULA] values (roughly two orders of magnitude in [FORMULA]), while power law ones are confined below log [FORMULA] (erg s[FORMULA] (hereafter [FORMULA]).

[FIGURE] Fig. 1. The [FORMULA]-[FORMULA] diagram for early type galaxies with inner surface brightness profile measured by HST. Power law galaxies are shown with open circles and core galaxies with full circles. Upper limits on X-ray luminosities are shown with a downward arrow. The data used are those in Table 1. The dashed line ([FORMULA] [FORMULA][FORMULA]) is an estimate of the stellar sources contribution to the X-ray emission (from Kim et al. 1992).

It looks as if power law galaxies cannot be more X-ray luminous than [FORMULA], while core galaxies show [FORMULA] values extending from the lowest to the highest [FORMULA] observed. In Fig. 2 this result is shown from the point of view of the relation between [FORMULA] and the central surface brightness slope [FORMULA] (defined in Sect. 2). A sharp transition in [FORMULA] as [FORMULA] drops below 0.3 (as for core galaxies) is clearly seen: the X-ray brightest galaxies are exclusively core galaxies, and power law galaxies are never X-ray brighter than [FORMULA], independently of the [FORMULA] value.

[FIGURE] Fig. 2. The relation between [FORMULA] and the slope of the central surface brightness profile [FORMULA] (left) or the [FORMULA] parameter (right) for the galaxies in Table 1. Power law galaxies are shown with open circles and core galaxies with full circles. Upper limits on X-ray luminosities are shown with a downward arrow. Those galaxies for which just the classification into core or power law is given are put respectively at [FORMULA], with an arrow pointing leftwards, and at the average [FORMULA] value for their family (i.e., 0.8, see Sect. 1). The vertical dashed line in the [FORMULA]-[FORMULA] plot separates boxy and disky galaxies.

3.1. Statistical analysis

How strong is the result presented above, from a statistical point of view? Is the confinement of power law galaxies to [FORMULA] [FORMULA][FORMULA] a result of the [FORMULA]-[FORMULA] correlation, or is it statistically significant in general, for all of them? To establish this, I made some statistical tests for the galaxies in the range of [FORMULA] values where power law and core galaxies coexist. This range has been chosen close to that indicated by F97, who find that core and power law profiles coexist for [FORMULA] (with [FORMULA] km s- 1 Mpc-1); by assuming a B-V=1 and rescaling to [FORMULA] km s-1 Mpc- 1, this range corresponds to [FORMULA] log [FORMULA] ([FORMULA]. Considering also the distribution of power law galaxies in Fig. 1, as overlap range for the present sample I have adopted [FORMULA] log [FORMULA] ([FORMULA]. In this range there are 15 core galaxies and 10 power law galaxies.

First I have checked if these two sets of galaxies are consistent with being drawn from the same distribution function of [FORMULA]. To check this I have used a variety of two-sample tests [discussed in Feigelson & Nelson (1985), and contained in the ASURV package] to verify the null hypothesis that the two sets are drawn from the same parent distribution. All these tests give a probability [FORMULA], from which one usually concludes that the two sets are consistent with coming from the same distribution 3. I obtain an even higher probability when using the Kolmogorov-Smirnov test ([FORMULA]).

Next, I have repeated the two-sample tests for the [FORMULA] values, in order to check whether power law and core galaxies (again for [FORMULA] log [FORMULA] ([FORMULA], where they coexist) are consistent with being drawn also from the same [FORMULA] distribution. I have obtained that the null hypothesis cannot be supported. The probabilities given by all the tests are around 0.003. So, the two sets come from different [FORMULA] distributions at the [FORMULA] level.

In conclusion, in the [FORMULA] range where they coexist, core and power law galaxies are similar from the point of view of their optical luminosity, while they clearly differ in their X-ray properties.

A linear regression analysis for data sets with censoring in one variable, with the expectation and maximization (EM) algorithm, and the Buckley-James algorithm (Isobe et al. 1986), gives a best fit slope for the [FORMULA]-[FORMULA] relation of [FORMULA] for core galaxies. This slope is close to the best fit slope obtained from the analysis of larger samples (Eskridge et al. 1995, Beuing et al. 1999; see Sect. 3). An estimate of the best fit slope for power law galaxies is not realistic because of the low number of detections (6 only); however, in this case a regression analysis gives a best fit slope consistent with unity. So, [FORMULA] [FORMULA][FORMULA] could be a good description of the [FORMULA]-[FORMULA] relation for power law galaxies, while the observed deviation of the [FORMULA]-[FORMULA] relation from a linear one could be produced by core galaxies.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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