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Scaling laws are new constraints on the nature of DM and on galaxy formation and evolution. Most of these remain to be explored. Simple implications include:

1 - The surprisingly high DM densities in dwarf spheroidals are normal for galaxies of such low luminosity. This implies that dSphs are real galaxies and not tidal fragments. Tides almost certainly pull bound fragments out of more luminous galaxy progenitors, but they cannot retain even the relatively low DM densities in those progenitors (Barnes & Hernquist 1992), much less increase the DM density to the high values characteristic of dwarf spheroidal galaxies.

2 - Dwarf spheroidal galaxies are not included in the least-squares fits in Figures 2 and 4 because only rho0 can be derived for their halos. However, these rho0 values lie on the extrapolation to low luminosity of the correlations for spiral and irregular galaxies. That is, the DM halos of dSph and Sc - Im galaxies appear to form a single physical sequence as a function of DM core mass.

3 - Since virialized density depends on collapse redshift zcoll, rho0 propto (1 + zcoll)3, the smallest dwarfs formed at least Delta zcoll appeq 7 earlier than the biggest spirals. Correction for baryonic DM compression will make rho0 smaller for giant galaxies. This will slightly increase Delta zcoll.

4 - The visible matter parameters rc and sigma of dSphs are a factor of about 2 smaller than their extrapolated DM parameters. This is reasonably consistent with the hypothesis that extreme dSphs have low visible matter densities (M / LB ~ 102) because they lost most of their baryons early. Possible reasons include galactic winds (e.g., Dekel & Silk 1986) or the difficulty of holding onto baryons in shallow DM potential wells when the Universe was ionized (e.g., Klypin et al. 1999). In the absence of a dark halo, the loss of most baryons would unbind the few stars that had already formed. But since these galaxies contain DM halos, we expect instead that the distribution of stars has expanded to fill the halo's core. Unlike the situation in giant galaxies, visible matter and DM would then have similar scale parameters.

5 - The fact that, as luminosity decreases, dwarf galaxies become much more numerous and also more nearly dominated by DM raises the possibility that there exists a large population of objects that are completely dark (Freeman 1987; Kormendy 1990; see also Tully 2004). Undiscovered DM dwarfs would help to solve the well known problem that the spectrum of initial density fluctuations predicted by CDM theory predicts far too many dwarf satellites of giant galaxies (Moore et al. 1999; Klypin et al. 1999). The favored explanation for why these dwarfs are not seen is that they virialized early, before or during the reionization of the Universe, and therefore lost or never captured the canonical fraction of baryons because those baryons were too hot to be confined in the puny potential wells of the dark dwarfs. Our observations suggest that empty halos - if they exist - are likely to be small and dense and to have small total masses. They would be darker versions of Draco and UMi.

6 - Djorgovski (1992) has compared an earlier version of the DM parameter correlations to the scaling laws predicted by hierarchical clustering (Peebles 1974; Gott & Rees 1975). For a power spectrum of initial density fluctuations that is a power law in wavenumber k, |deltak|2 propto kn, the size R, density rho, and velocity dispersion sigma of a bound object are related approximately by

Equation 33-35 (33)



Here we have used the relation rho propto sigma2 R-2 for an isothermal sphere. Djorgovski pointed out that the DM parameter correlations in Kormendy (1990) imply that n appeq - 2.45, close to the value n appeq - 2 expected for giant galaxies in CDM theory. With the more accurate fits in equations 20 - 22,

Equation 36-38 (36)



we get n = - 1.80 ± 0.10, n = - 2.12 ± 0.10, and n = - 1.81 ± 0.10, respectively. Note that these values are not independent. Their average is n = - 1.91. If we use the fits (equations 5 - 7) determined from decompositions using isothermal DM, then the average is n = - 2.1 ± 0.2. Both values are remarkably close to the value n appeq - 2.1 expected in CDM theory at a halo mass of 1012 Modot (Shapiro & Iliev 2002). We need to correct the slopes for baryonic DM compression; this will be done in Kormendy & Freeman (2003). The above comparison provides a measure of the slope of the fluctuation power spectrum on mass scales that are smaller than those accessible to most other methods.

Shapiro & Iliev (2002) have made a more detailed comparison of the DM parameter correlations published by Kormendy & Freeman (1996) with their predictions based on COBE-normalized CDM fluctuation spectra. They found that the agreement between predictions and observations was best for CDM.

It is interesting to note a consequence of the theoretical prediction that the slope n gets steeper at smaller mass scales. If n appeq - 2.6 for the smallest dwarfs (Shapiro & Iliev 2002; Ricotti 2002), then the straight lines in the left panels of Figure 4 should curve downward toward the visible matter parameters of dSph galaxies. This would strengthen the inference that the visible and dark matter in these galaxies is distributed similarly. It will be important to look for curvature in the correlations as more data become available for dwarf galaxies.

Finally, we note that the scatter in Figures 2 - 4 has surely been increased by problems with the data. (1) Distance errors are not negligible. For our calibrating galaxies, we can compare accurate distances to those given by our Virgocentric flow field machinery. This implies errors in logD of ± 0.11. Since rho0 propto D-2, distance errors are a significant - although not the dominant - source of scatter in equations 5 - 25. (2) If some disks are submaximal, then this affects the scatter in the correlations. If the degree to which they are submaximal depends on MB (Kranz, Slyz, & Rix 2003), this affects the correlation slopes, too. (3) The assumption that DM halos have isothermal cores is challenged by CDM theory, although it is suppported by many observations. It will be important to see how the correlations are affected if NFW halos are used. (4) The correlations in Figures 2 - 4 require correction for DM compression by the baryons before a definitive comparison with theory can be made. We will address these issues in future papers.


JK is grateful to the staff of Mt. Stromlo Observatory for their hospitality during three visits when part of this work was done. We thank S. Djorgovski, S. M. Fall, and P. Shapiro for helpful discussions on the comparison of predicted and observed DM scaling laws. This work used the NASA/IPAC Extragalactic Database, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.

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