7.5. Suppression of star formation: internal agents

Feedback of energy into the interstellar medium from the first generation of stars in a protogalaxy may regulate subsequent star formation. It seems clear that some feedback is needed. The luminosity-metallicity relation suggests that massive galaxies are able to retain their ISM longer than low-mass galaxies, or had different initial mass functions (although the existence of such a relation remains to be demonstrated for cluster dE's). Feedback from star formation is the simplest mechanism for producing such a relation. However, this feedback can take several forms, and it is not yet clear which one dominates.

7.5.1. Supernovae

By far the most widely studied mechanism for regulating star formation in dwarf galaxies (or galaxies in general, for that matter) is feedback from supernovae. The simplest criterion for supernova regulation is the hypothesis that the ISM is spontaneously ejected from a galaxy when a fraction f of the cumulative energy of all supernovae ever formed exceeds the binding energy of the remaining gas (Larson 1974). The details of the transfer of kinetic energy from the supernova remnants to the gas are hidden in the parameter f. Vader (1986) points out that storage of energy in such models is a major problem; the accumulation of large amounts of hot gas prior to ejection inevitably leads to gas densities so high that rapid cooling ensues. Dekel and Silk (1986) solve the problem by requiring that the SNR's must cover a significant fraction of the volume (i.e. they must overlap) before gas can be ejected. Combining this with the assumption that the star formation rate scales with the free-fall time and the assumption that most of the energy is deposited into the ISM during the Sedov (adiabatic expansion) phase, Dekel and Silk (1986) derive a critical halo velocity dispersion V_c 100 km s^-1 as a condition for substantial gas removal. While this result has been widely quoted, the agreement of V_c with the rough dividing line between giant and dwarf ellipticals could be a coincidence, given the uncertainties in the input physics.

Perhaps more important than the existence of a critical velocity dispersion V_c is the conclusion, reached by both Vader (1986) and Dekel and Silk (1986), that mass loss in the form of a chemically homogeneous wind in a self-gravitating (as opposed to dark-matter dominated) galaxy cannot simultaneously reproduce both the metallicity-luminosity and surface-brightness-luminosity relation. Dekel and Silk (1986) solve the problem by postulating that the mass loss takes place in a halo made up of such a large fraction of dark matter that the gas loss would have no dynamical effect on the stellar system that is left behind. This, combined with an instantaneous recycling approximation for the evolution of metals, leads to a one-parameter model which governs the relation between L, R, [Fe/H], and . Dekel and Silk (1986) choose the free parameter by requiring that the model follow the observed radius-luminosity relation L R⁴ (equivalent to the surface-brightness-luminosity relation). The success of the model is that, having fixed this one parameter, it can match the observed luminosity-metallicity relation (which Dekel and Silk 1986 quote as L Z^2.7) and the zero-point of the surface-brightness-luminosity relation. The largest discrepancy with observations is the large M / L predicted for dE's in the mass range of Fornax or Sculptor. One of the biggest departures of the model from standard lore is the assumption that stars can form in a gas cloud that is not self-gravitating.

Vader (1986) adopts a different approach. Assuming constant M / L and gas removal that is slow compared to the crossing time of the system in a self-gravitating halo, she arrives at a r_e - relation for dE's that is inconsistent with the observations. Dekel and Silk went through the same argument and arrived at the same conclusion. However, rather than assuming that galaxies are not self-gravitating, Vader assumes that the efficiency with which SN energy is converted into gas escape energy is a strong function of the velocity dispersion of the galaxy, with dwarf galaxies having lower efficiencies than giants or globular clusters. The basis for this conclusion is the assumption of a constant initial mass-density relation for globular clusters, dE's and giant E's. Vader (1986; 1987) goes on to consider the different conditions for wind removal in different environments and the conditions under which the metallicity of the wind will be enhanced relative to the overall ISM, driving out more metals per unit mass than might be expected in models with uniform winds. The model is able to explain why dE's apparently have lower metallicities for their velocity dispersions than giant ellipticals (but see Bender et al. 1993), but can only qualitatively reproduce the position of dE's in the r_e - plane.

Yoshii and Arimoto (1987) argue that both the relations of metallicity and surface brightness with luminosity can be reproduced with a chemically homogeneous wind model with no dark matter. They point out that models generally predict a mass-weighted metallicity, while the observations measure a luminosity-weighted metallicity. The latter can be lower than the former by up to 1 dex. Yoshii and Arimoto (1987) construct a model based on the assumption that the binding energy of elliptical galaxies scales as M^1.45 and the star-formation timescale in the mass range of dwarf galaxies scales as _SF M^-1/3 (from consideration of free-fall time and the collision time of fragmentary clumps). The different structural properties of E's, dE's and globular clusters are determined by the balance between the star-formation timescale and the timescale for the formation of a supernova-driven wind. Winds in proto-dE galaxies set in at ~ 10⁷ yr, but are slow enough that galaxies can respond to the mass loss, and hence lose more than half their total mass without becoming unbound (however, see Angeletti and Giannone 1990) for a comment on the specific numerical estimate of the time of occurrence of a global wind). Expansion as a result of mass loss accounts for the low dE surface brightnesses. Mass-to-light ratios in this model are expected to be M / L_B 5-6 for dE galaxies. The high M / L values for the Draco and UMi dwarfs are thus problematical.

In an ambitious attempt to explain the excess faint blue galaxies, Lacey et al. (1993) propose a model of tidally triggered galaxy formation. Their model for dwarf galaxy formation is similar to that of Dekel and Silk, in that they assume star formation ceases when the gas is ejected by supernova-driven winds. However, they assume that stars form in the baryon-dominated cores of the dark-matter halos, and therefore that the baryonic cores expand homologously when the mass is ejected. They are able to reproduce the surface-brightness-luminosity relation extremely well, but do not consider chemical evolution, and therefore cannot reproduce the metallicity-luminosity relation. Their prediction for B-V vs. mass is completely contrary to observations (more massive galaxies are bluer - although the colors in this model are due to star-formation histories rather than metallicity).

Other efforts to evolve galaxies in hierarchical models adopt a quite different prescription for how supernovae regulate star formation (White and Frenk 1991; Kauffmann et al. 1994; Cole et al. 1994). In these models, heating of the gas by supernovae and subsequent cooling to form more stars reaches equilibrium, thus governing the rate of star formation in the sense that low-mass galaxies have lower star formation rates. The assumption that SN heat the gas rather than eject it is based on the argument that the time between star formation and energy injection is short compared to the sound crossing time or the cooling time. Hence once star formation starts it quickly becomes self-regulating. The star formation timescales _SF for dwarf galaxies in these models are long. For example, a typical galaxy with velocity dispersion = 50 km s^-1 in the Cole et al. (1994) model has _SF = 29 Gyr. Clearly the dwarfs in these models are dwarf irregulars, rather than dE's. The models do have the advantage that they can schematically reproduce the luminosity-metallicity relation. No attempt has yet been made to match the surface-brightness-luminosity relation. Perhaps the most problematical observation for such models is the apparent lack of rotational support in dE's (Sect. 3.2). It is not clear how a self-regulated, mostly gaseous galaxy could remain anisotropic for several billion years.

Athanassoula (1994) describes the results of hydrodynamical simulations of dE formation that include feedback from supernovae. The simulations reproduce the µ - L relation and the range of dE surface-brightness profiles reasonably well. The models match the observations much better when dE's are assumed not to have dark-matter halos. For the self-gravitating models, the model locus in the µ - L plane is actually somewhat narrower than the observational data. However it is important to note that each model point at fixed magnitude represents a different set of physical assumptions (i.e. initial density profile, coefficients governing cooling, star formation, and supernova feedback). What is remarkable is that varying these parameters does not fill in the µ - L plane. Evidently the details of cooling, star formation, and feedback are unimportant for producing the surface-brightness-luminosity relation. No attempt has yet been made to match the metallicity-luminosity relation or the variation of profile shape with luminosity for these models.

de Young and Heckman (1994) consider the effect of geometry on the ability of supernovae to clear a galaxy of its ISM. If the galaxy is non-spherical, the wind may clear holes along the minor axis and then be very inefficient at clearing out the rest of the galaxy. de Young and Heckman (1994) find that galaxies of 10⁷ M can easily remove their entire ISM, while galaxies of 10¹¹ M are very resistant to disruption. Galaxies with masses ~ 10⁹ M show the widest range of behavior. Such an enhanced sensitivity to geometry might help account for the scatter (if real) in the metallicity-luminosity relation of dE galaxies in this mass range.

7.5.2. OB star winds

If the initial mass function (IMF) extends to high masses ( 60 M), the energy and momentum deposited into the ISM from O and B stars may be comparable to that deposited (at later times) by supernovae. Leitherer et al. (1992) have constructed detailed models of the effect of OB star winds for a variety of assumptions for metallicity and IMF. In a typical case, OB stars dominate both the energy and momentum flux until ages t 5 x 10⁶ yr. Thereafter, SN dominate. The total contribution from OB star winds depends critically on the IMF and the metallicity. For constant IMF, lowering the upper mass cutoff of the IMF from 120 M to 30 M decreases the energy output of winds by nearly 2 orders of magnitude. For constant IMF, the energy input increases by 1.5 orders of magnitude from Z = 0.1 Z to Z = 2 Z. In contrast, the contribution from supernovae is very insensitive to the upper mass cutoff and metallicity. OB star winds, therefore, offer a possible mechanism for introducing variations in the star-formation feedback efficiency as a function of galaxy metallicity, and hence luminosity.

7.5.3. OB star photoionization

Most of the energy output by OB stars during their main-sequence lifetime comes in the form of UV photons. This energy (~ 10⁵³ erg over the life of a 30 M star) exceeds that of the final supernova by roughly two orders of magnitude. As with winds, the energy input from this radiation comes well before supernovae become important. Lin and Murray (1992) studied the effect of OB star photoionization on a collapsing protogalaxy and concluded that it could regulate star formation, holding the gas at a temperature T 10⁴ K, with gas cooling and stars forming at a rate sufficient to keep the gas marginally ionized. Until supernovae become a significant source of energy (at ages ~ 10⁷ yr), this self-regulation will cause stars to form over timescales longer than a free-fall time, and with different rates as a function of radius in the galaxy. With this self-regulation Lin and Murray (1992) are able to reproduce the r^1/4 and exponential surface brightness profiles of ellipticals and disk galaxies, respectively, and the Faber-Jackson and Tully-Fisher scaling relations between luminosity and kinematics. The model does not naturally produce exponential surface-brightness profiles for non-rotating galaxies, and therefore may not provide the best explanation for the formation of faint dE galaxies. Nevertheless, photoionization from OB stars may still be important in governing their early star-formation history.

7.5.4. Chemodynamical Models

Burkert et al. (1994) emphasize that cooling and feedback from star formation are highly sensitive to metallicity. Winds from OB stars, photoionization, dust and the rate of H₂ formation, the shape of the IMF, and the minimum mass of type-II supernovae all to some degree depend on metallicity. Hence, processes that were unimportant in the balance between heating and cooling during the early collapse phase of a galaxy may gain in importance as the metallicity of the system increases. Burkert and collaborators attempt to incorporate metallicity dependence into a hydrodynamical model of galaxy formation. For galaxies less than 10¹⁰ M in these models the effect of feedback is to slow the star-formation timescales such that _SF is much longer than a free-fall time. However, in contrast to the models of White and Frenk (1991) and Cole et al. (1994), the variation in feedback efficiency with metallicity allows a galaxy that was not initially able to eject its ISM to do so after a few Gyr. Gas later shed from evolved stars cools and sinks to the center, allowing subsequent generations of stars. In contrast to models involving rapid expulsion of gas (e.g. Dekel and Silk 1986), dE galaxies in these models have small surface densities because they never experience a dissipation and collapse phase. The gas which is lost in the wind has a small fraction of the total mass, but contains most of the metals (as in Vader 1987). Although this has yet to be explicitly demonstrated, such models may be able to match simultaneously both the surface-brightness-luminosity and metallicity-luminosity relations. The initial extended period of star formation that is a robust feature of such models may be tested with deep HST color-magnitude diagrams of the local dE's.

While there is clearly much work to be done to bring the models and observations to the point where they can be quantitatively compared, we conclude this section by suggesting that the correlations of surface brightness and metallicity with luminosity tend to favor internal over external agents for suppressing star formation, and slow as opposed to rapid mechanisms for gas removal. If gas removal is indeed slow, then external mechanisms for triggered star formation and/or gas removal, acting well after the time of halo collapse, may greatly influence dwarf galaxy evolution as a function of environment.