3. The relation between and the central brightness profile
Fig. 1 plots the 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 - correlation. For the whole Einstein sample of 148 early type galaxies , although with a large scatter of more than two orders of magnitude in at any fixed (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 and a scatter at least as large as that found by Eskridge et al. (1995).
3) At intermediate , where they coexist, core galaxies span the whole range of values (roughly two orders of magnitude in ), while power law ones are confined below log (erg s (hereafter ).
It looks as if power law galaxies cannot be more X-ray luminous than , while core galaxies show values extending from the lowest to the highest observed. In Fig. 2 this result is shown from the point of view of the relation between and the central surface brightness slope (defined in Sect. 2). A sharp transition in as 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 , independently of the value.
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 a result of the - 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 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 (with km s- 1 Mpc-1); by assuming a B-V=1 and rescaling to km s-1 Mpc- 1, this range corresponds to log (. Considering also the distribution of power law galaxies in Fig. 1, as overlap range for the present sample I have adopted log (. 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 . 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 , 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 ().
Next, I have repeated the two-sample tests for the values, in order to check whether power law and core galaxies (again for log (, where they coexist) are consistent with being drawn also from the same 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 distributions at the level.
In conclusion, in the 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 - relation of 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, could be a good description of the - relation for power law galaxies, while the observed deviation of the - relation from a linear one could be produced by core galaxies.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999