7.3 Physical Basis

The fact that luminous disk galaxies have flat rotation curves over the entire extent of their optical region and beyond (e.g., Rubin et al. 1980; Bosma 1981) implies the existence of large amounts of non-luminous matter. If spherical symmetry is imposed, then the integrated mass (~ V²R/G) grows linearly with radius, with the implication that the ``dark matter'' density scales as ~ R^-2. This also is the expectation for an isothermal halo (e.g., Binney and Tremaine 1987), along with a flat rotation curve at large R. If we also assume that galaxies have roughly constant surface brightness (L ~ R²) and that the mass-to-light ratio (M/L) is constant, then we have R ~ V_max², or L ~ V_max⁴. This is roughly the slope of the TF relations at the longer wavelengths (M_H ~ 10 log W_Rⁱ). A similar argument was originally presented by Schechter (1980). More elaborate physical justifications have been presented by Aaronson et al. (1979) in an attempt to explain the origin of the slope of the H-band TF relation.

Unfortunately, the situation is considerably more complicated than these derivations would suggest. It is now clear that the basic photometric scaling properties of low luminosity systems differ from those of luminous systems (e.g., Binggeli et al. 1984; Pierce 1988, 1991). For example, the I-band M/L begins to increase rather abruptly for galaxies less luminous than about M_B ~ -19.5 (Pierce 1988; 1991). These results suggest that while the visible matter in luminous galaxies is self-gravitating, the low luminosity systems are dark-matter dominated (e.g., Persic and Salucci 1988; 1990). Consequently, the existence of a tight TF relation over this broad range in luminosity becomes difficult to understand. Yet, searches for second parameters in the TF relations have thus far been unsuccessful (e.g., Cornell 1989; Bivano et al. 1990).

Central to this problem is the fact that disk galaxies have flat rotation curves over a broad range of luminosity and mass, with dark matter dominating at moderately large radii. At the same time, it is clear that the TF relation implies a ``disk-halo conspiracy'', such that the mass within some characteristic optical radius is coupled with the mass (Sancisi & van Albada 1985). One of the more successful attempts at understanding this phenomena is given by Blumenthal et al. (1984). They find that gravitational interplay between the luminous and non-luminous matter in luminous disk galaxies is sufficient to redistribute the total mass in such a way as to produce the flat rotation curves which are observed. Dekel and Silk (1986) have developed a model for the formation of dwarf galaxies in an attempt to explain some of the properties of these systems, including their distinctive photometric scaling properties. They suggest that low mass galaxies are more susceptible to significant sweeping of gas via supernovae-driven winds during protogalactic collapse than are more massive systems due to a lower binding energy. In such a situation, star formation can be radically slowed and, in some cases, essentially quenched. As a result, these systems today would have a larger dark-to-luminous matter ratio than more massive systems since the dark matter (assumed to be non-baryonic) would be unaffected by radiation and gas dynamics.

While this model is capable of reproducing the general trend of M/L with L, including the characteristic onset mass (luminosity), it cannot explaining the tight TF relations found for low luminosity systems (Pierce 1991). Evidently, some important physics is still missing and a complete understanding of the TF relations will have to wait for a more highly developed model of the formation of galaxies. Nevertheless, the TF relations remain strong empirical relations between the photometric and kinematic scaling properties of spiral and irregular galaxies.