2.2.5 Deviations from Homology

Observational evidence of systematic deviations from homology includes the following. Burkert (1993) found that E galaxies in general were well fitted by the r^1/4 profile, but that they nevertheless showed a small change of slope at some point $x_{\rm cut}$ in the profile (with $x_{\rm cut} \approx$ 0.8-0.9 ${r_{\rm e}}$). This change of slope was quantified by the parameter $\delta b$. It was found, that $\delta b$ was correlated with luminosity. Caon et al. (1993) fitted r^1/n profiles and found n to correlate with ${r_{\rm e}}$ or L. The values of n spanned a large range, from n=0.5 at ${r_{\rm e}}$ = 0.3 kpc to n=16 at ${r_{\rm e}}$ = 25 kpc. Note, that none of these two studies explicitly take into account whether the galaxy has a disk. This is important since many E and S0 galaxies have disks (e.g. JF94).

Hjorth & Madsen (1995) used a model based on statistical mechanics of violent relaxation to give a possible explanation of the result from Burkert (1993). The one free parameter in their model is the dimensionless central potential $\psi$. They found $\delta b$ to vary with $\psi$. They also found that the FP tilt could be explained by a variation of $\psi$ with L, while keeping ${\left( M/L \right)}$ constant, and that the result from Burkert (i.e. that $\delta b$ varied with L) supported this. They did not address the question whether a fine tuning of the $\psi$-L relation was needed to reproduce the small and constant thickness of the FP. Note, that their model does not include any disk component.

The effect of a trend in the shape of the surface brightness profile was also studied by Ciotti et al. (1996). It was found that the FP slope could be explained in this way, but that a fine tuning was needed.

Renzini & Ciotti (1993) and Ciotti et al. (1996) also explored the effect of a trend in the relative distribution within the galaxy of luminous and dark matter. They parametrized this by the parameter $\beta' \equiv r_{\rm dark}/r_{\rm luminous}$, with r being the half mass radius of the given component. They found, that a decrease in $\beta'$ could produce the FP tilt, but that still fine tuning was needed. Note, that a variation in $\beta'$ implies a variation in ${k_{\rm R}}$ (Eq. ), and thereby non-homology.

Djorgovski (1995) found that an FP also existed for globular clusters. When using the core parameters, the FP coefficients $\alpha=2.2\pm0.15$ and $\beta=-1.1\pm0.1$ were found, indicating structural homology and a constant (M/L) ratio. When using half-light parameters, the FP coefficients $\alpha=1.45\pm0.2$ and $\beta=-0.85\pm0.1$ were found, similar to what is found for elliptical galaxies. Also Nieto et al. (1990) and Burstein et al. (1997) found globular clusters to have similar FP coefficients to those of giant elliptical galaxies when using half-light parameters. Djorgovski argued, that for globular clusters this almost certainly implies non-homology, and that this suggests that a similar explanation may be at work for the elliptical galaxies. The argumentation for the latter statement seems somewhat dubious. Along those lines, Djorgovski later warns against assuming that the similar half-light FPs for globular clusters and elliptical galaxies reflect entirely the same physics.

Finally we mention, that Prugniel & Simien (1996) found that about half of the FP tilt was due to a trend in the global stellar population (age and/or metallicity), and that the other half could be accounted for by partly a trend in the amount of rotational support and partly a trend in spatial structure (as seen by Caon et al. 1993).