## 3. Galaxy evolutionThe formalism introduced in the previous section only deals with the formation of dark halos through the action of gravity. To infer the mass function of galaxies we need to precise when such a halo will form a galaxy, that is we have to express explicitly the physical processes which lead to the formation of a galaxy. ## 3.1. Galaxy formation: cooling constraintsFollowing the picture of the spherical model, an object with a
small overdensity at early times will gradually separate from the
general expansion, reach a maximum radius
at time
and finally collapse at time
. One usually assumes that the real
halo will virialize at the radius at
time , with
. We consider that once the halo of
dark matter is virialized, the gas can cool and fall into the
potential well as it is no longer thermally supported. However, at
time on average we assume that
another larger halo containing the previous one will eventually form
and will merge the initial overdensity with neighbouring ones. Indeed
in the usual hierarchical scenarios, which we consider in this
article, larger scales turn non-linear later so that small objects get
embedded within increasingly large mass condensations. If the gas
contained in the small halo has not cooled sufficiently, it will be
distributed over the newly formed overdensity. In other words,
Thus, we write this cooling constraint as
where
is the cooling time. Hence the
amount of gas which can cool and fall to the center of the halo, and
possibly form a disk and stars, is the mass of gas which is enclosed
by the cooling radius where
. However, for high redshifts or
small temperatures, this radius can be larger than the virialization
radius , where the averaged density
contrast is equal to
(given by where is the density of the
universe at the redshift we consider. The parameter where is the cooling function
(in ). We use the cooling function
given by Sutherland & Dopita (1993) for a gas with primordial
abundances. The expression (16) for
is the evaluation for the mean halo baryonic density. Cooling may be
much more efficient since the local density increases within the halo
and the baryon density gets higher during the collapse. This means
that the parameter It is convenient to express (15) as a relation virial temperature - density, so that the halos we consider can be completely described by their temperature. Hence we can write the curve corresponding to the first condition of the system (15) as: The curve corresponding to the second condition of (15), which characterizes just-virialized objects, is given by: where is the density contrast at
the time of virialization, at the redshift This relation temperature-density defines implicitely the relation
we introduced in Sect. 2 to
determine the mass function of galaxies. Note that there are only two
parameters:
## 3.2. Galaxies versus clusters and groupsThe curves and
are shown in Fig. 2. Objects below
either one of these curves violate one at least of the above
conditions. Objects above these curves satisfy the criteria, but it is
easy to see that slightly larger size objects (and whence lower
density objects) also satisfy the criteria. Hence we define galaxies
as being, for a given temperature (velocity dispersion), the largest
object (hence the one with the lowest density) that satisfies the
criteria (15). Thus the locus of galaxy formation, curve
, is the higher of the curves
and
in the
- The curve does not depend on
redshift while the curve does. At
fixed temperature This variation with redshift has important consequences. Galaxies,
that at a given epoch settle on the curve
because larger objects of lower
densities are not virialized, at a later epoch will lie Note that in our model all galaxies have roughly the same age (of
the order of the age of the universe) from the star formation
viewpoint, even if they did not exist as distinct objects along this
whole period. Indeed, present galaxies are the result of the merging
of many smaller and older sub-units where star formation was already
active at high ## 3.3. Star formationOur scenario describing the history of star formation is rather standard, and kept as simple as possible. Apart from a small component (10%) of dark baryonic matter (brown dwarfs, planets) that plays a negligible role in the stellar evolution history, we consider four components: -
short lived stars, of total mass , that will be recycled. -
long lived stars, of total mass , that will not be recycled. -
a central gaseous component, of total mass , at the sites of star formation, that is deplenished by star formation and ejection by supernova winds, replenished by infall from a diffuse gaseous component located in the dark halo potential well. -
a diffuse gaseous component, of total mass , deplenished by infall and replenished by the supernova winds.
The star formation rate is proportional to the mass of central gas with a time-scale proportional to the dynamical time-scale : The mass of gas heated and ejected by supernovae out of the central parts of the galaxy is proportional to the star formation rate and decreases for deep potential wells: where is a constant and Since we shall use: where and are parameters of order unity. We assumed , as it may describe gravitational instabilities within galaxies as well as the influence of neighbours. Moreover, the time-scale disappears for faint galaxies, as we shall see below, and our model would still be approximately valid even for bright galaxies if one has in fact , since it would simply correspond to a change of (then the system is governed by the longest time-scale among and ). This means that other forms for , for instance or (where is the gas density in the core) would give similar results, as we checked numerically. The details of this model of star formation are presented in Appendix B and Appendix C. We solve the corresponding equations (B1) numerically but we also give there approximate analytic solutions that allow one to follow the behaviour of these various components as a function of time and galactic mass. We can stress here that the fraction of non-luminous stars we adopt (10%) is consistent with observations (see discussion in Mera et al. 1998) while the values used in some other studies (50% in Kauffmann et al. 1993; 63% in Cole et al. 1994 for instance) are much too large. Using a smaller fraction of non-luminous stellar-like objects however would increase the luminosity of the galaxies obtained in the latter models and would lead to stronger discrepancies with the observed luminosity function. We also describe the evolution of the metallicity (E1) with three different values: for stars, for the central gaseous component and for the diffuse gaseous component. We summarize here the free parameters of our model: -
: proportionality factor in the cooling constraint (17). -
: proportionality factor for the star formation time-scale (23). -
: proportionality factor for the definition of the dynamical time-scale (23). -
: supernovae efficiency, (B3) and (21). -
: fraction of non-luminous objects (brown dwarfs) (D8). -
"yield" : mass of metals produced per mass of stars through SNII, see (E1).
The parameters where is the total gas mass. For massive galaxies the derivation leading to this relation is no longer valid but (24) still gives a reasonable description of the star formation rate because in our model so that for there is only one time-scale which is also given by since in this case . The value of the main parameters and is further discussed in Appendix F. © European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |