2.2.4 Variation of ${\left( M/L \right)}$ with L (or M)

A physical interpretation of the FP should not only be able to account for the observed slope, but also for the small scatter of the FP, a scatter which is more or less constant along the FP (JFK96 found a scatter in ${\log{r_{\rm e}}}$ of 0.125 and 0.073 for galaxies with ${\log\sigma}$ below and above 2.0, respectively, and expected larger measurement errors for galaxies with ${\log\sigma}< 2.0$ to explain some of the difference). In this subsection we will consider the scenario of structural homology and a variation of ${\left( M/L \right)}$ with L (or M). In the next subsection we will consider the opposite scenario.

First we will summarize evidence that there actually is a variation of ${\left( M/L \right)}$ with L. van der Marel (1991) used a dynamical model to predict the kinematics of an elliptical galaxy on the basis of observed surface photometry. The predictions were compared with actual kinematical data along both the major and the minor axes of 37 bright E galaxies. From this, accurate mass-to-light ratios were derived that were corrected for the effects of rotation and radial anisotropy. Note, that these mass-to-light ratios were not based on the assumption of structural homology. van der Marel found, that the mass-to-light ratio correlated with luminosity as $(M/L_{\rm B}) \propto L_{\rm B}^{0.35 \pm 0.05}$.

Second we note, that part of the FP slope is due to a metallicity effect. Higher luminosity galaxies have higher metallicity than fainter galaxies, and because of the line-blanketing effect, brighter galaxies will emit more of their light at longer wavelengths than fainter galaxies. Therefore, the bolometric mass-to-light ratio could be constant with bolometric luminosity, while at the same time the blue mass-to-light ratio could increase with blue luminosity. Djorgovski & Santiago (1993) found the FP coefficient $\alpha$ to increase monotonically with the effective wavelength of the bandpass, from $\alpha \approx 0.95$ at U ( $\lambda_{\rm eff} \approx 0.35\mu {\rm m}$) to $\alpha \approx 1.5$ at K ( $\lambda_{\rm eff} \approx 2.2\mu {\rm m}$). $\beta$ remained constant at $\beta \approx -0.8$. The increase in $\alpha$ with wavelength is indeed a sign of line blanketing, since it implies a decrease in $\xi$, the coefficient in ${\left( M/L \right)}\propto L^\xi$. However, at $2.2\mu {\rm m}$ line blanketing should be negligible, and still Djorgovski & Santiago find ${\left( M/L_{\rm K} \right)}\propto L_{\rm K}^{0.17}$. Recillas-Cruz et al. (1990) found the same trend, namely $\alpha = 1.36\pm0.11$ at B, $\alpha = 1.48\pm0.13$ at V, and $\alpha = 1.69\pm0.11$ at K. At K this implies ${\left( M/L_{\rm K} \right)}\propto L_{\rm K}^{0.09\pm0.04}$. Dressler et al. (1987b) found that the metallicity effect only explains a minor part of the FP slope. They found ${\left( M/L_{\rm B} \right)}\propto L_{\rm B} ^{0.25}$, and after applying bolometric correction, the dependence on L was only reduced to ${\left( M/L_{\rm bol} \right)}\propto L_{\rm bol}^{0.18}$.

Renzini & Ciotti (1993) explored whether a systematic variation in either the IMF slope below $0.3M_\odot$ or the minimum stellar mass could produce the FP slope. They found that a major change of either of these parameters along the FP was required to reproduce the tilt. At the same time, an extremely small dispersion was required to reproduce the small constant thickness of the FP. In other words, fine tuning was needed, making this explanation unattractive.

Another possible explanation could be that the dark matter fraction $R \equiv M_{\rm dark}/M_{\rm luminous}$ increased with luminosity, while $(M_{\rm luminous}/L)$ remained constant. This was explored by Renzini & Ciotti (1993) and Ciotti, Lanzoni, & Renzini (1996), and also here it was found that a fine tuning was needed.

Part of the FP slope could also be due to a systematic variation of mean age along the FP (Faber et al. 1995), with the stellar populations of high luminosity galaxies having higher mean ages than for low luminosity galaxies.