## Appendix A: characteristics of galactic halosAny dark matter halo of radius which leads to The energy equipartition , where is the mean molecular weight of the gas and the proton mass, gives for the gas temperature: which also leads to the hydrostatic equilibrium for the gas, using , where and are the baryon number density and mass density, if . Hence we shall assume that the gas initially follows the spatial distribution of the dark matter in the virialized halo, before it starts cooling, which is consistent with numerical simulations (Evrard 1990). More precisely we assume: where is the present ratio of the baryon density to the critical density. We shall take in the case , and in the case . Both values are consistent with the bounds given by primordial nucleosynthesis (Walker et al. 1991). Finally, we note: which correspond to halos of primordial composition (with an helium
mass fraction ), where
is the electron number density. We
note We also define a luminous radius , and the circular velocity at , by: We shall use to compare our model to the observed Tully-Fisher relation, which relates the luminosity to the circular velocity at a radius . Since we have (the rotation velocity is roughly constant throughout the halo). The time introduced in Sect. 3.1 corresponding to the turn-around epoch of a given halo (or to the time when it became non-linear) is given by the spherical collapse model (Peebles 1980): The time of turn-around is simply , and the radius of maximum expansion is . When the overdensity virializes, at , its radius is , and the averaged density within is . Thus we obtain: which gives the time of turn-around of the halo as a function of its density. The slope of the dark matter
halos mainly enters our calculations as a numerical factor of order
unity in (A2). Hence we do not need a precise description of the
detailed density profile of the objects we deal with. Moreover, we
explicitly consider that very large halos defined by
which correspond to clusters of
galaxies (at low ## Appendix B: star formation model for an isolated haloWith the prescription introduced in Sect. 3.1 we are able to obtain the mass function of galaxies, or the temperature, radius functions, but we do not have the luminosity function yet. Indeed, the luminosity of galaxies depends on the processes which govern star formation, which we need to take into account explicitly. We shall adopt a very simple model for the formation of stars within the halos we previously considered. ## B.1. ModelFor each galaxy we divide stars in two populations: a first class
of massive stars, with a life-time
shorter than the age where is the mass fraction of the IMF corresponding to short-lived stars, for of gas which is processed into stars, and is the fraction of mass which is locked within white dwarfs or neutron stars after the death of these massive stars, and is not recycled in the galaxy to form other generations of stars. We have because in the case of usual stellar initial mass functions (IMF) most of the mass is within low mass long-lived stars. In a fashion similar to Kauffmann et al. (1993) we write the mass of gas heated and ejected by supernovae as: where is the fraction of the energy delivered by supernovae transmitted to the gas, and we defined: using and erg. This value of is consistent with the IMF we shall use, and it is constrained by the observed supernovae rates. The effect of stellar winds can also be incorporated into this model through . Using the fact that by definition of short-lived stars we can make the approximation that evolves in a quasi-static way: . Hence we obtain: Then we solve numerically (B1) to get the evolution of , and all other quantities. ## B.2. Analytic approximationsFor faint galaxies characterized by a small virial temperature the system (B1) leads to a simple approximation, which also provides usefull insight into the behaviour of more massive galaxies, as can be checked numerically. More precisely, this approximation is valid provided . In this case, a quasi-stationary regime sets in very quickly, where the galactic evolution is regulated by supernovae (and stellar winds) and the dynamical time-scale: the sink term for the star-forming gas , corresponding to matter ejected by supernovae (which involves the star formation rate and the supernovae efficiency parameterized through ), balances the source term due to the infall of gas from the extended halo (which involves the dynamical time-scale). Hence the global star formation rate is governed by the interplay of the supernovae efficiency and the dynamical time-scale. Then, evolves in a quasi-static way and follows closely the mass of the reservoir : Note that the condition of validity of this approximation implies that . Then, since we obtain where is the age of the galaxy and we defined: Thus, as we explained above, the galactic evolution is governed by and , the time-scale does not appear in the global gas mass, or stellar content. The instantaneous star formation rate is since , hence (we neglect the mass in the form of short-lived stars). The mass within massive short-lived stars is: while the mass in the form of small long-lived stars is: The mass locked in star remnants, white dwarfs or neutron stars, is ## Appendix C: redshift evolution and merging of galaxies## C.1. ModelIn the previous section we considered non-evolving halos, in the
sense that the total mass of the galaxy, and its time-scales
and
, remained constant with time.
However, as we explained in Sect. 3.2, the characteristics of the
galactic halos we consider evolve with time, following the curve
(see Fig. 2). Thus, we now have to
model the evolution of the matter located on
through its merging history.
Similarly to White & Frenk (1991), we can write for the comoving
stellar content of the universe at time where is the probability (which
we do not know) that matter which was in a halo of parameter
at time
will be part of a halo of parameter
In fact, when one expects that
the correlation disappears, or at
least gets weaker, but this is not a problem if the stellar content of
a given object at In fact, this approach simply means that we can still use the sytem
(B1) to get the proportion of the mass of gas which is converted into
stars in a galaxy, but the time-scales
and
, and the virial temperature
## C.2. Analytical solutionsAlthough in practice we compute numerically the solution of the
system (B1), we present now the case of the simplified
quasi-stationary regime, corresponding to small galaxies, to give a
clear illustration of the effects of this additional time-dependence.
Moreover, since the temperature Since , and in this case, we obtain: Thus we have an equation similar to the one describing halos located on , but now the global star formation time-scale depends on . The density of the halos located on , which virialize at the time , is according to the spherical model: Hence , whatever the value of . If clustering is stable, for , the quantity is constant, hence . Thus, for a constant parameter we obtain for halos on : As a consequence, since for halos such that , see (B7), we get: Hence, if we note the age of the universe at the redshift we consider, and the star formation time-scale at this date, we can write: using . We can note that this
integral converges for , and
In the case we have for the gas mass fraction at time , when we calculate , Thus, we obtain a relation similar to (B6), with an "effective age"
for the galaxy given by
. Hence the time-evolution of
reduced this age by a factor
, as compared to (B6), since for
the time-scale
increases at high redshifts, which
leads to a less efficient star formation, because the temperature
Note that halos on satisfy
by definition. Thus, the gas mass
fraction within halos at time is
given by a relation of the form ,
whatever the precise dependence on We can see that the time evolution of these virialized halos does
not depend on . This is due to the
fact that the internal dynamics of overdensities given by the
spherical model does not depend on the background universe. However,
the number of such halos will naturally vary with the cosmological
parameters. Besides, the redshift evolution depends on
, in parallel to the relation
time-redshift. For instance, if we
have while
if
and
. Thus, in the case
we have for halos on
defined by a fixed parameter
where is the present age of the universe. In the case of a low-density universe the redshift evolution is slower, because of the faster expansion which implies that the same multiplying factor in redshift corresponds to a smaller factor in time. We can note that for halos on , the time-scale which governs their merging history is . Hence, if star formation is also enhanced by gravitational interactions with surroundings and merging with other halos, the natural time-scale is again which gives another justification for our star formation time-scale . The gas of halos which undergo this succession of mergings can be reheated to the virial temperature by the energy released during these violent encounters. However, for halos located on we have by definition , hence we can neglect these successive phases of reheating. The evolution equation (C3) we obtained is interesting as it
readily shows the influence of the parameterization adopted for the
star formation time-scale on the
history of galaxies. Indeed, since ,
we can see that the integral on the right-hand side diverges for
if
is a strong power-law of the
density: with
. In fact, there is a cutoff at high
redshifts because for a fixed parameter ## C.3. Total mass in stars and global star formation rateWe can also consider the evolution with redshift of the total mass in stars per comoving . The mass of baryons within galaxies is . Using we obtain: where is the stellar density
parameter and is the mass fraction
in galaxies of mass between We obtain the comoving star formation rate in the same way. We have , for long-lived stars, hence: For the quasi-stationary regime we get for the derivative of the stellar comoving mass density: It first increases with redshift because the gas content of bright galaxies gets higher (depletion term ) and the star formation time-scale decreases. Indeed, since , see (B7), and for most galaxies which are close to , we obtain as long as . However, at high redshifts the comoving star formation rate gets smaller because the mass contained in deep potential wells starts to decrease (influence of the cutoff at K), and more importantly because as the virial temperature declines the star formation time-scale starts to increase (in the case ), see (B7). Naturally, if we neglect the very small apparent mass loss at K, we have: by construction, as implied by (C2) (and checked numerically). We can also look at the evolution with redshift of the star
formation rate of individual halos
defined by a fixed temperature ## Appendix D: stellar properties of galactic halosWe can note that the mass in the form of short-lived stars is always much smaller than the mass contained in long-lived stars, and increasingly so for well evolved galaxies (when ) as could be expected. Indeed these galaxies have already consumed most of their gas content, hence their present star formation rate is relatively small (as compared to their past) and their stellar population is dominated by all stars which were formed all along the galaxy history. Thus, for the quasi-stationary approximation we obtain: where we used , since in the case of usual stellar initial mass functions (IMF) most of the mass is within small long-lived stars, and we have . Thus, luminous galaxies, which correspond to large circular velocities (as is observed through the Tully-Fisher relation) and high densities (because of the cooling constraint, see the curve in Fig. 2), hence to a small global star formation time-scale , have consumed most of their gas and will be redder than faint galaxies. This is an important success of our model as this trend is actually observed (Lilly et al. 1991, Metcalfe et al. 1991), but usually difficult to get by common models. We can also notice that the mass in the form of white dwarfs or neutron stars is always very small as compared to the total stellar mass from (B11). Indeed we get (with and ): as is the case in any reasonable stellar evolution model. We can check that mass is conserved with time in the sense that: In fact we have a slight excess of mass , which is negligible since we noticed above that for instance. This is due to our quasi-static approximation . Finally, we also consider that for of gas converted into stars a fraction goes into "invisible" compact objects, such as brown dwarfs, which we included among the long-lived stars. Since observations seem to show that this mass fraction is rather small we choose in the numerical calculations (in fact we could as well use or , since this parameter has almost no influence as long as it remains small). Then, the mass of luminous stars is in our model , since we have to remove the part of formed by dark objects, but this fraction is negligible. Besides, , and we have already noticed that . Thus, we obtain: At small times when the gas content of the galaxy is still important we have: It is clear on this expression, which is valid for any model of star formation and does not rely on the quasi-stationary approximation, that the observed ratio of the Milky Way and its age give directly its star formation time-scale (whence the parameter ). Thus we now have attached a peculiar stellar content to each halo,
or galaxy. To get the luminosity of such a galaxy, we only need to
precise the luminosity of its stars. We note
and
the global luminosity of
short-lived and long-lived stars per unit mass, so that the luminosity
Now we have to precise the values of the quantities and . We shall derive them from the initial mass function (IMF) of stars and mass luminosity and mass life-time relations. For of matter converted into stars, the number of stars formed in the mass range is where which defines the normalisation of . We note the mass which separates the two classes of stars we introduced above. Then, we obtain: and The mass is formed of luminous stars, with , and of dark objects. Next we use the mass luminosity and the mass life-time relations: with , which is consistent with the mean observed mass - B band luminosity relation for stars on the main sequence. Hence we have: and Similarly, we get: Using (D7) and the previous mass-luminosity relation we can see that the luminosity of the galaxy is dominated by the contribution of stars of mass close to . Finally we use , and the mass which separates the two classes of stars we introduced above is chosen to be given by: so that is smaller than the age of the galaxies, which is close to the age of the universe at the time of interest. For instance, in the case a galaxy like the Milky Way, that is with a circular velocity km/s, corresponds in the present universe to years, and . Thus we can attach a luminosity to each galaxy, which enables us to get the luminosity function of galaxies from the mass function: We can also use this model to obtain the supernovae rate in galaxies. Thus, we assume that stars more massive than will explode as supernovae after they leave the main sequence. Naturally, they must also be part of our class of short-lived stars to explode during the life-time of the galaxy. The mass in the form of these massive stars is: Thus, in the case of the quasi-stationary regime we obtain for the number of such stars at any time: and the supernova rate is: with In the case , we obtain for a galaxy similar to the Milky Way a supernovae rate of 2.5 explosions per century, which is close to the observed value for type II supernovae. ## Appendix E: metallicityFinally, we can derive the metallicity from our model (B1). The metallicity here is understood as being the abundance of Oxygen, or any other element that is not significantly produced in SNI which are not included in our model. We define and as the fractions of metals within the diffuse gas and within the gas in the core of the galaxy . We note the mean stellar metallicity. Thus, we obtain: where while we get for the dense central phase: where is the age of the galaxy. Since the halo is not enriched directly, but through the mass loss of the galactic core, its metallicity remains for a long time much smaller than the one attached to this central component which receives the stellar ejecta. The second term of , which is constant in time, corresponds to the fact that very quickly a "quasi equilibrium" regime sets in where the gain of metals within this central phase from stellar ejecta balances the loss due to the exchange of matter with the diffuse phase, which replaces some gas with the metallicity by some gas falling from the halo with the metallicity . Since this exchange of gas between both components and is driven by supernovae, or stellar winds, whose importance is parameterized by , see (B2), the equilibrium metallicity is smaller for small temperatures (weak potential wells) where these flows of matter are more important. Then increases linearly with time as receives some metals from at a constant rate (the term in corresponds to the second term in , simply multiplied by a temperature-dependent factor and time). Naturally, when reaches the "stationary" value described above for , this "equilibrium" regime stops, both phases have the same metallicity which increases linearly with time as the enrichment process goes on. The stellar metallicity is: The second term in , which is
constant in time, corresponds to the "stationary" regime we described
for . Since stars form from the
dense phase , their metallicity
quickly reaches the surrounding gas metallicity
which is constant with time. Later,
when most baryons are incorporated into stars
(), the stellar metallicity
increases to reach ## Appendix F: role of parameters and scalingsThe model we constructed in the previous sections has only two main specific parameters: and , which determine respectively the size (and mass) of galaxies and their star formation rate. For massive galaxies, with , there is also a dependence on , while for faint galaxies there is a dependence on , but this is not critical. Naturally, it also depends on the usual cosmological parameters , and on the power-spectrum for the multiplicity functions. Now we shall see how we can get the value of and , for a given set of cosmological parameters and a fixed . Let us consider a galaxy like the Milky Way located on the curve
(see Fig. 2), defined by a fixed
temperature and Besides, since we have , and increasingly so for massive galaxies like the Milky Way as we noticed earlier (in fact, for faint galaxies which have not exhausted their gas content while for bright galaxies which have already consumed most of their gas content ). Hence, a larger leads to a higher gas/star mass ratio and a larger luminosity, which is quite natural since it means a less constraining cooling constraint, whence a broader, more massive and lower density halo. Similarly, a larger leads to a higher gas/star mass ratio, since it means a longer star formation time-scale, whence fewer stars. It does not influence the luminosity because the mass in the form of long-lived stars is of the order of the initial mass of gas for these dense galaxies. As a consequence, the observed luminosity and gas/star mass ratio of the Milky Way, which corresponds to a circular velocity of 220 km/s and a temperature K, give the value of and , as a function of the cosmological parameters. Thus we obtain: Now, we can use these values of and to get the variations of all physical characteristics of the galaxies we considered in our model with the cosmological parameters. Some of these scalings are shown in Table F1. However, these relations are quite general. For instance, the fact that for the Milky Way gives if the initial ratio of baryons in the halo is representative of the Universe, there has been no loss of baryons since the formation of the galaxy and most of the gas is in the disk. Then, the luminosity of the Galaxy, and its ratio , lead to . In this way we obtain the mass and the radius of the halo from which the baryons which constitute the Milky Way came, whence the position of the curve and the value of the product . For , we get . In Table F1 we consider two types of galaxies: i) massive galaxies like the Milky Way located on the cooling curve , which have a high luminosity and have already consumed most of their initial gas, and ii) small galaxies located on (hence characterized by the density contrast ), which are faint and have a small stellar content.
Thus, the values of the cosmological parameters imply well-defined
galaxy characteristics. For instance, for a given
, a larger
(that is a higher baryon fraction)
leads to a smaller radius ## Appendix G: approximate power-law regimesFrom the locus of galaxy formation implied by virialization and cooling we may distinguish using the quasi-stationary approximation three regimes characterized by a specific power- law behaviour: 1) Very faint galaxies, located on
and
K, with a 2) Faint galaxies, located on :
kpc and
K, with a nearly 3) Bright galaxies located on
kpc and K, which ## G.1. Star formation1) Very faint galaxies Using (A3), (B7) and (D5) (since these galaxies have only consumed a very small fraction of their gas content) we get: Moreover, since these galaxies have a small stellar content we have , hence: 2) Faint galaxies We obtain: 3) Bright galaxies Since these galaxies have lost most of their gas, which has already formed stars, we now have , and using we get: ## G.2. Mass/Light ratio and Tully-Fisher relationFor a constant star-mass/luminosity ratio we obtain for the 3 regimes we described above the scaling relations: 1) Very faint galaxies: 2) Faint galaxies: 3) Bright galaxies: ## G.3. Metallicity1) Very faint galaxies: Since we noticed earlier that , we obtain: 2) Faint galaxies: We still have , so we get: The increase of the slope of , as compared to the previous regime, is due to the fact that the evolution time-scale gets smaller for more luminous galaxies (their density increases, which implies that their dynamical time decreases), and for a constant galactic age (of the order of the age of the universe) it means that the halo is more evolved, hence more enriched. 3) Bright galaxies: These galaxies have already converted most of their initial gas content into stars, which implies that . Most of the gas is in the dense component , since supernovae are not very efficient and , hence and we recover the usual one component closed-box model, with . Thus, in our case we obtain . Since we have: ## G.4. Slope of the luminosity functionUsing the PS approximation, we get for an initial spectrum (that gives results quite close to those obtained with CDM initial conditions) the following behaviour. 1) Very faint galaxies: 2) Faint galaxies: 3) Bright galaxies: Using the non-linear scaling approximation, we write for , where , and for with . 1) Very faint galaxies: 2) Faint galaxies: 3) Bright galaxies: © European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |