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 Vc 100 km s-1 as a
condition for substantial gas removal. While this result has been widely
quoted, the agreement of Vc 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 Vc 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
Z2.7) 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 re
-
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 re -
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 M1.45 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 / LB
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 r1/4 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.