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Astron. Astrophys. 333, 795-802 (1998)

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9. Intrinsic scatter or spatial depth effects?

Even without any new datum, we still have one potentially decisive test that might be able to help us decide whether the large scatter observed in Binggeli & Jerjen's [FORMULA] - [FORMULA] relationship is due primarily to intrinsic scatter or depth effects. This test involves looking for departures from uni-modality and/or normality in the differential frequency distribution of scale-length residuals with respect to the best-fit curve for the data (not in this case a curve defined by galaxies from an external galaxy sample). Should the distribution not be consistent with a uni-modal Gaussian [measured in log(distance) space to be rigorous], we can say that the scatter is not consistent with the intrinsic scatter origin hypothesis.

In Fig. 9, we find some evidence for a tri or quad-modal distribution. Such a distribution could arise if early-type dwarfs exist in three or four discrete size ranges within the same cluster. However, not only does this seem most unlikely, but it would also be very hard to reconcile such a scenario with the theoretical work on the subject (Hjorth & Masden 1995, Gerbal et al. 1997, Prugniel & Simien 1997 & Ciotti & Lanzoni 1997). It is perhaps more likely that any multi-modality present is due to line-of-sight substructure in the galaxy sample's spatial distribution, albeit smoothed considerably due to the relative distance scales being different for different ranges in n.

[FIGURE] Fig. 9. The frequency of scale-length residuals with respect to Binggeli & Jerjen's best fit curve to their data as a function of the residuals in magnitude space. Note that the abscissa scale cannot be interpreted as a [negative] relative distance scale because no correction has been made for the differential Malmquist bias effects discussed in Sect. 8.

We have performed several non-parametric statistical tests as described by Lucey et al. (1986) and references therein. Comparing the Fig. 9 data against 100,000 Monte-Carlo realisations drawn from a Gaussian distribution having the standard deviation of the residuals, yielded probabilities that the Fig. 9 distribution could arise by chance. For the Kolmogorov-Smirnov (Lillefors) test and Geary's a-test (mean absolute deviation/standard deviation), the probabilities were 0.255 and 0.222 respectively; which were suggestive but not by any means conclusive. However, for the skewness and kurtosis tests and the u-test (data range/standard deviation) the probabilities were 0.041, 0.011 and 0.005 respectively. We take these results as significant evidence against Binggeli & Jerjen's intrinsic scatter interpretation.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998