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Astron. Astrophys. 345, 329-362 (1999)

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Appendix A: characteristics of galactic halos

Any dark matter halo of radius R can be characterized by its overall density contrast [FORMULA] or equivalently by the parameter x introduced in (8). We assume that these potential wells have a mean density profile of the form [FORMULA] where [FORMULA] is the slope of the two-points correlation function [FORMULA] (we shall come back to this point below). Since [FORMULA] is close to 2, the halo is close to isothermal and the velocity dispersion [FORMULA] is nearly constant, which agrees with observations. With such a density profile, the velocity dispersion of the dark matter verifies:

[EQUATION]

which leads to

[EQUATION]

The energy equipartition [FORMULA], where [FORMULA] is the mean molecular weight of the gas and [FORMULA] the proton mass, gives for the gas temperature:

[EQUATION]

which also leads to the hydrostatic equilibrium for the gas, using [FORMULA], where [FORMULA] and [FORMULA] are the baryon number density and mass density, if [FORMULA]. 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:

[EQUATION]

where [FORMULA] is the present ratio of the baryon density to the critical density. We shall take [FORMULA] in the case [FORMULA], and [FORMULA] in the case [FORMULA]. Both values are consistent with the bounds given by primordial nucleosynthesis (Walker et al. 1991). Finally, we note:

[EQUATION]

which correspond to halos of primordial composition (with an helium mass fraction [FORMULA]), where [FORMULA] is the electron number density. We note V the circular velocity of the galaxy at its external radius R:

[EQUATION]

We also define a luminous radius [FORMULA], and the circular velocity [FORMULA] at [FORMULA], by:

[EQUATION]

We shall use [FORMULA] to compare our model to the observed Tully-Fisher relation, which relates the luminosity to the circular velocity at a radius [FORMULA]. Since [FORMULA] we have [FORMULA] (the rotation velocity is roughly constant throughout the halo).

The time [FORMULA] 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):

[EQUATION]

The time of turn-around is simply [FORMULA], and the radius of maximum expansion is [FORMULA]. When the overdensity virializes, at [FORMULA], its radius is [FORMULA], and the averaged density within [FORMULA] is [FORMULA]. Thus we obtain:

[EQUATION]

which gives the time of turn-around [FORMULA] of the halo as a function of its density.

The slope [FORMULA] 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 [FORMULA] which correspond to clusters of galaxies (at low z) contain several higher-density sub-units which are identified as distinct galactic halos. In fact, we recognize directly these smaller objects without dealing (in this article) with their larger host halo.

Appendix B: star formation model for an isolated halo

With 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. Model

For each galaxy we divide stars in two populations: a first class of massive stars, with a life-time [FORMULA] shorter than the age t of the galaxy, and a second class of small stars with a life-time [FORMULA] larger than t. Thus, the gas used to form these small stars has not been recycled in the ISM, since these stars are still on the main sequence, while the large stars consist of many successive generations which have continuously recycled part of their mass through supernovae explosions, planetary nebulae and winds. We consider explicitly the mass recycled by SNII, and by stars in the AGB phase, but we neglect the mass ejected by SNI (believed to be associated with white dwarf coalescence and explosion) which is very small as compared to the one involved in the former processes. To include SNI would be straightforward, but it would not change our results as long as we do not consider the history of Si or Fe. We note [FORMULA], [FORMULA] and [FORMULA] the total mass in the form of gas, short-lived stars and long-lived stars. Moreover, we consider two gaseous phases: some low-density diffuse gas [FORMULA] spread over the halo, and some dense gas [FORMULA] within the core of the galaxy in the form of clouds which turns into stars with a time-scale [FORMULA]. Initially, we have [FORMULA] and [FORMULA] with [FORMULA] the mass of baryons in the halo (hence we assume that at the time of virialization the baryon fraction within any halo is representative of the universe: there has been no prior segregation between baryonic and dark matter). The diffuse phase [FORMULA] is continuously replenished by stellar winds and supernovae, which eject and heat part of [FORMULA]. Meanwhile, the diffuse gas settles in the central parts of the galaxy and forms dense clouds over a dynamical time-scale [FORMULA]. Indeed, the gas ejected by supernovae or ionized by stellar winds, or initially hot after virialization, cools over a time-scale [FORMULA] and then falls back into the center of the potential well over a time [FORMULA]. By definition of our halos, the constraints (15) ensure that [FORMULA] hence [FORMULA] is the only one relevant time-scale. In fact we even have [FORMULA] at high redshifts or small temperatures when [FORMULA], in this case most of the gas [FORMULA] is cold. This would be different for galaxies in clusters, where the gas may be spread all over the cluster, cooling is less efficient, and where the potential wells of the galactic halos are expected to be modified by the additional cluster gravitational energy. The properties of galaxy clusters, however, will be treated in a forthcoming paper. Finally, we can write:

[EQUATION]

where [FORMULA] is the mass fraction of the IMF corresponding to short-lived stars, for [FORMULA] of gas which is processed into stars, and [FORMULA] 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 [FORMULA] 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:

[EQUATION]

where [FORMULA] is the fraction of the energy [FORMULA] delivered by supernovae transmitted to the gas, and we defined:

[EQUATION]

using [FORMULA] and [FORMULA] erg. This value of [FORMULA] 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 [FORMULA].

Using the fact that by definition of short-lived stars [FORMULA] we can make the approximation that [FORMULA] evolves in a quasi-static way: [FORMULA]. Hence we obtain:

[EQUATION]

Then we solve numerically (B1) to get the evolution of [FORMULA], [FORMULA] and all other quantities.

B.2. Analytic approximations

For faint galaxies characterized by a small virial temperature [FORMULA] 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 [FORMULA]. 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 [FORMULA], corresponding to matter ejected by supernovae (which involves the star formation rate and the supernovae efficiency parameterized through [FORMULA]), balances the source term due to the infall of gas from the extended halo [FORMULA] (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, [FORMULA] evolves in a quasi-static way and follows closely the mass of the reservoir [FORMULA]:

[EQUATION]

Note that the condition of validity of this approximation implies that [FORMULA]. Then, since [FORMULA] we obtain

[EQUATION]

where [FORMULA] is the age of the galaxy and we defined:

[EQUATION]

Thus, as we explained above, the galactic evolution is governed by [FORMULA] and [FORMULA], the time-scale [FORMULA] does not appear in the global gas mass, or stellar content. The instantaneous star formation rate is

[EQUATION]

since [FORMULA], hence [FORMULA] (we neglect the mass in the form of short-lived stars). The mass within massive short-lived stars is:

[EQUATION]

while the mass in the form of small long-lived stars is:

[EQUATION]

The mass locked in star remnants, white dwarfs or neutron stars, is

[EQUATION]

Appendix C: redshift evolution and merging of galaxies

C.1. Model

In the previous section we considered non-evolving halos, in the sense that the total mass of the galaxy, and its time-scales [FORMULA] and [FORMULA], 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 [FORMULA] (see Fig. 2). Thus, we now have to model the evolution of the matter located on [FORMULA] through its merging history. Similarly to White & Frenk (1991), we can write for the comoving stellar content of the universe at time t:

[EQUATION]

[EQUATION]

where [FORMULA] is the probability (which we do not know) that matter which was in a halo of parameter [FORMULA] at time [FORMULA] will be part of a halo of parameter x - [FORMULA] at t, and [FORMULA] is the star formation rate of the corresponding halo. We can define halos by their parameter x for both PS and non-linear scaling approaches because we noticed in Sect. 2 that the usual linear parameter [FORMULA] is a function of x only, see (13). If individual halos are stable once in the non-linear regime (when [FORMULA] and [FORMULA]), their parameter x remains constant with time, but they accrete some mass as their virialization radius gets larger with time and they may join together to form a larger halo. Hence, we shall make the approximation that the mass which is within halos [FORMULA] at t was within halos of the same range in x at all earlier times. This also satisfies the constraint that mass be conserved since the mass fraction within halos [FORMULA] does not depend on time, and is a function of the sole variable x, see (10). In fact, the conservation of mass implies to use a distribution in x, or to take x constant which should be a good approximation if this distribution is peaked around a particular value. This also ensures that we do not mix the mass of non-evolving galaxies located on [FORMULA] with the halos on [FORMULA], and that the mass at a given time t comes from older less massive halos characterized by a slightly weaker potential well since for a fixed x the virial temperature T is smaller at higher redshifts. This is quite natural since as time goes on potential wells merge to form deeper ones. In other words, we neglect the scatter in all possible "merger trees" and simply use a "mean history" defined by compatibility requirements. This analysis also applies to the PS approach, which would only give a different function [FORMULA], hence a different mass fraction.

In fact, when [FORMULA] one expects that the correlation [FORMULA] disappears, or at least gets weaker, but this is not a problem if the stellar content of a given object at t is dominated by its recent star formation history, which is the case. Moreover, within the framework of our model for star formation small galaxies are regulated by the interplay between supernovae (which eject gas) and the infall of gas from the outer parts of the halo and their luminosity is dominated by recently formed stars which are more numerous. As a consequence the details of their previous stellar history are not very important. On the other hand, very massive and bright galaxies are not affected by the supernovae feedback mechanism (since their potential well is sufficiently deep to retain the gas very efficiently) so that their past history matters. Besides, at low redshifts their star formation rate begins to decline significantly as they exhaust their gas content. This means that their luminosity and stellar properties are governed by old stars which formed during ealier and more active periods. However, in our astrophysical model we assume that these galaxies (located on the cooling curve [FORMULA]) do not evolve significantly any more so that our approximation [FORMULA] becomes correct for these objects. Thus, our approximation is consistent with our model and it should provide a reasonably good description. Moreover, it allows one to get simple analytic insights into the global galaxy formation process, which clearly show the general trends implied by any such model based on hierarchical structure formation supplemented by cooling constraints. Note that using a different galaxy formation model Kauffmann et al. (1998) found that their results were not very sensitive on the details of the merging trees of their halos (they obtained similar results with N-body simulations and an extended Press-Schechter theory for the properties of individual galaxies although the merger trees differ in detail). Using (9) we obtain

[EQUATION]

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 [FORMULA] and [FORMULA], and the virial temperature T, are now functions of time.

C.2. Analytical solutions

Although 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 T decreases at higher redshifts, for a fixed x, all galaxies follow this regime when they are young.

Since [FORMULA], and [FORMULA] in this case, we obtain:

[EQUATION]

Thus we have an equation similar to the one describing halos located on [FORMULA], but now the global star formation time-scale [FORMULA] depends on [FORMULA]. The density [FORMULA] of the halos located on [FORMULA], which virialize at the time [FORMULA], is according to the spherical model:

[EQUATION]

Hence [FORMULA], whatever the value of [FORMULA]. If clustering is stable, for [FORMULA], the quantity [FORMULA] is constant, hence [FORMULA]. Thus, for a constant parameter [FORMULA] we obtain for halos on [FORMULA]:

[EQUATION]

As a consequence, since [FORMULA] for halos such that [FORMULA], see (B7), we get:

[EQUATION]

Hence, if we note [FORMULA] the age of the universe at the redshift we consider, and [FORMULA] the star formation time-scale at this date, we can write:

[EQUATION]

using [FORMULA]. We can note that this integral converges for [FORMULA], and T (which measures the depth of the potential well) decreases at higher redshifts, provided [FORMULA], which corresponds to the range of interest where hierarchical clustering is valid. Hence our analysis applies to all relevant power-spectra [FORMULA]. Thus we obtain for the gas mass fraction at any time t:

[EQUATION]

In the case [FORMULA] we have for the gas mass fraction at time [FORMULA], when we calculate [FORMULA],

[EQUATION]

Thus, we obtain a relation similar to (B6), with an "effective age" [FORMULA] for the galaxy given by [FORMULA]. Hence the time-evolution of [FORMULA] reduced this age by a factor [FORMULA], as compared to (B6), since for [FORMULA] the time-scale [FORMULA] increases at high redshifts, which leads to a less efficient star formation, because the temperature T decreases. This would not be the case for large n, where the dominant effect would be the increase of the density. Note however that the variation with n of the factor [FORMULA] can simply be incorporated into the parameters [FORMULA] and [FORMULA]. Galaxies located on [FORMULA] with a high virial temperature [FORMULA] have a slightly more complicated history. There is a first stage, at small times, where [FORMULA] and they follow the behaviour we have just described, and a second stage where [FORMULA] and [FORMULA]. Hence we divide the integral of (C3) in two parts, and we get a similar result, with an effective age: [FORMULA]. In fact, the relation (B6) does not apply to these massive galaxies, but it still gives a good estimate of their evolution, and this simple result shows that the effective age is increased because the temperature remains constant for some time, as could be expected, but only by a logarithmic term. For massive galaxies located on [FORMULA] we have [FORMULA] constant down to the time [FORMULA] when [FORMULA] and we switch onto [FORMULA]. At small redshifts [FORMULA] the fact that [FORMULA] is constant means that the gas mass fraction follows a simple exponential decline [FORMULA], usually different from the evolution [FORMULA] along [FORMULA]. The time of virialization is simply [FORMULA] and the star formation time-scale at this date is [FORMULA]. Hence these galaxies have an effective age [FORMULA] which is the sum of their effective age at [FORMULA], which we obtained above, and of the time which has elapsed since this date: [FORMULA]. Thus, to sum up, for all galaxies the quasi-stationary approximation gives again the relations of the previous section, (B6) to (B11), with an effective age given by:

[EQUATION]

Note that halos on [FORMULA] satisfy [FORMULA] by definition. Thus, the gas mass fraction within halos at time [FORMULA] is given by a relation of the form [FORMULA], whatever the precise dependence on z of the star formation time-scale, provided the integral of the right-hand side of (C3) converges (star formation does not occur as a sudden burst at high redshifts). Moreover, the variation with n of the factor [FORMULA] in the age of the galaxy, when it is on [FORMULA], is simply incorporated into the parameter [FORMULA] which enters the definition of [FORMULA], see (B7). Thus, as far as star formation processes are concerned, all galaxies have roughly the same age (of the order of the age of the universe), even if they did not exist as distinct entities over this whole period of time: they continuously accreted some mass or merged with neighbours.

We can see that the time evolution of these virialized halos does not depend on [FORMULA]. 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 [FORMULA], in parallel to the relation time-redshift. For instance, if [FORMULA] we have [FORMULA] while [FORMULA] if [FORMULA] and [FORMULA]. Thus, in the case [FORMULA] we have for halos on [FORMULA] defined by a fixed parameter x:

[EQUATION]

where [FORMULA] 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 [FORMULA], the time-scale which governs their merging history is [FORMULA]. Hence, if star formation is also enhanced by gravitational interactions with surroundings and merging with other halos, the natural time-scale is again [FORMULA] which gives another justification for our star formation time-scale [FORMULA]. 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 [FORMULA] we have by definition [FORMULA], 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 [FORMULA] on the history of galaxies. Indeed, since [FORMULA], we can see that the integral on the right-hand side diverges for [FORMULA] if [FORMULA] is a strong power-law of the density: [FORMULA] with [FORMULA]. In fact, there is a cutoff at high redshifts because for a fixed parameter x the temperature gets smaller in the past, and when [FORMULA] K cooling is very inefficient so that star formation is suppressed. However, this large decrease of T by a factor 100 (from [FORMULA] K to [FORMULA] K) means that the redshift of cutoff [FORMULA] corresponding to present galaxies is rather high: [FORMULA]. Thus, such a model for [FORMULA] would imply that most stars formed at [FORMULA] and that the global star formation rate has been very low ever since. This is certainly inconsistent with observations, which show that the star formation rate has not experienced such a dramatic decline since [FORMULA] and that star formation is still active in the present universe. If [FORMULA] varies weakly with [FORMULA]: [FORMULA], the integral converges and star formation is dominated by the latest epochs, and increasingly so for smaller [FORMULA], until all the gas is converted into stars (at some time in the future). The case [FORMULA], which corresponds to our prescription, is intermediary as the divergence would only be logarithmic, which is not a problem because of the cutoff. In fact, the additional temperature dependence [FORMULA] introduces another redshift factor which makes the integral converge. This strongly suggests that the star formation time-scale [FORMULA] should vary as [FORMULA], at least at high redshift, (with a possible additional dependence on T), independently of the arguments presented in the previous section, so that star formation is neither dominated by a sudden burst at high redshift when most of the mass reaches [FORMULA] K, nor by the very recent epochs, since these both cases are inconsistent with observations. Note that it is correct to consider the quasi-stationary regime in this analysis since we are interested in the earliest stages of galactic evolution when [FORMULA].

C.3. Total mass in stars and global star formation rate

We can also consider the evolution with redshift of the total mass in stars per comoving [FORMULA]. The mass of baryons within galaxies is [FORMULA]. Using [FORMULA] we obtain:

[EQUATION]

where [FORMULA] is the stellar density parameter and [FORMULA] is the mass fraction in galaxies of mass between M and [FORMULA], given by (3) and (10) for the PS and non-linear scaling prescriptions. Naturally, we always have [FORMULA], as is clearly seen in (C11), since [FORMULA]. In fact, there is also a low temperature cutoff at [FORMULA] K, but up to [FORMULA] this only involves a very small fraction of the total mass.

We obtain the comoving star formation rate in the same way. We have [FORMULA], for long-lived stars, hence:

[EQUATION]

For the quasi-stationary regime we get for the derivative of the stellar comoving mass density:

[EQUATION]

It first increases with redshift because the gas content of bright galaxies gets higher (depletion term [FORMULA]) and the star formation time-scale [FORMULA] decreases. Indeed, since [FORMULA], see (B7), and [FORMULA] for most galaxies which are close to [FORMULA], we obtain [FORMULA] as long as [FORMULA]. 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 [FORMULA] K), and more importantly because as the virial temperature declines the star formation time-scale [FORMULA] starts to increase (in the case [FORMULA]), see (B7). Naturally, if we neglect the very small apparent mass loss at [FORMULA] K, we have:

[EQUATION]

by construction, as implied by (C2) (and checked numerically).

We can also look at the evolution with redshift of the star formation rate [FORMULA] of individual halos defined by a fixed temperature T. Note that two such halos defined by the same temperature at different redshifts are not necessarily formed by the same matter. Along [FORMULA], we have [FORMULA], hence [FORMULA], and [FORMULA]. For [FORMULA] the age of the universe scales as [FORMULA], and [FORMULA], hence we obtain [FORMULA] constant with time. In a similar fashion, for halos located on [FORMULA] we also get [FORMULA] constant as long as [FORMULA]. Thus, the star formation rate is roughly constant with time for these objects, which is consistent with observations.

Appendix D: stellar properties of galactic halos

We 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 [FORMULA]) 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:

[EQUATION]

where we used [FORMULA], since in the case of usual stellar initial mass functions (IMF) most of the mass is within small long-lived stars, and we have [FORMULA]. 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 [FORMULA] in Fig. 2), hence to a small global star formation time-scale [FORMULA], 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 [FORMULA] and [FORMULA]):

[EQUATION]

as is the case in any reasonable stellar evolution model. We can check that mass is conserved with time in the sense that:

[EQUATION]

In fact we have a slight excess of mass [FORMULA], which is negligible since we noticed above that [FORMULA] for instance. This is due to our quasi-static approximation [FORMULA]. Finally, we also consider that for [FORMULA] of gas converted into stars a fraction [FORMULA] 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 [FORMULA] in the numerical calculations (in fact we could as well use [FORMULA] or [FORMULA], since this parameter has almost no influence as long as it remains small). Then, the mass [FORMULA] of luminous stars is in our model [FORMULA], since we have to remove the part of [FORMULA] formed by dark objects, but this fraction is negligible. Besides, [FORMULA], [FORMULA] and we have already noticed that [FORMULA]. Thus, we obtain:

[EQUATION]

At small times when the gas content of the galaxy is still important we have:

[EQUATION]

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 [FORMULA] of the Milky Way and its age give directly its star formation time-scale [FORMULA] (whence the parameter [FORMULA]).

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 [FORMULA] and [FORMULA] the global luminosity of short-lived and long-lived stars per unit mass, so that the luminosity L of the galaxy is:

[EQUATION]

Now we have to precise the values of the quantities [FORMULA] and [FORMULA]. We shall derive them from the initial mass function (IMF) of stars and mass luminosity and mass life-time relations. For [FORMULA] of matter converted into stars, the number of stars [FORMULA] formed in the mass range [FORMULA] is

[EQUATION]

where m is the star mass in units of [FORMULA] and a the normalization constant. We use [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA], which is similar to the IMFs given by Salpeter (1955) and Scalo (1986). This applies to stellar masses between [FORMULA] and [FORMULA]. We could change somewhat this IMF (for instance choose [FORMULA] for [FORMULA]) without significant variation in our results. Moreover, a fraction [FORMULA] may go into "invisible" compact objects, such as brown dwarfs, so we have

[EQUATION]

which defines the normalisation of [FORMULA]. We note [FORMULA] the mass which separates the two classes of stars we introduced above. Then, we obtain:

[EQUATION]

and

[EQUATION]

The mass [FORMULA] is formed of luminous stars, with [FORMULA], and of dark objects. Next we use the mass luminosity and the mass life-time relations:

[EQUATION]

with [FORMULA], which is consistent with the mean observed mass - B band luminosity relation for stars on the main sequence. Hence we have:

[EQUATION]

and

[EQUATION]

Similarly, we get:

[EQUATION]

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 [FORMULA]. Finally we use [FORMULA], and the mass [FORMULA] which separates the two classes of stars we introduced above is chosen to be given by:

[EQUATION]

so that [FORMULA] 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 [FORMULA] a galaxy like the Milky Way, that is with a circular velocity [FORMULA] km/s, corresponds in the present universe to [FORMULA] years, [FORMULA] and [FORMULA].

Thus we can attach a luminosity to each galaxy, which enables us to get the luminosity function of galaxies from the mass function:

[EQUATION]

We can also use this model to obtain the supernovae rate in galaxies. Thus, we assume that stars more massive than [FORMULA] 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:

[EQUATION]

Thus, in the case of the quasi-stationary regime we obtain for the number of such stars at any time:

[EQUATION]

and the supernova rate is:

[EQUATION]

with

[EQUATION]

In the case [FORMULA], 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: metallicity

Finally, 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 [FORMULA] and [FORMULA] as the fractions of metals within the diffuse gas [FORMULA] and within the gas in the core of the galaxy [FORMULA]. We note [FORMULA] the mean stellar metallicity. Thus, we obtain:

[EQUATION]

where y is the yield. The fractions of metals in both gas phases [FORMULA] and [FORMULA] vary because of the exchange of matter between these two components. In addition, the central gas [FORMULA] is enriched by stellar ejecta. The mass of metals in stars increases as metals are incorporated from the star-forming gas [FORMULA]. In the case of the quasi-stationary regime, we obtain for the gas in the diffuse phase:

[EQUATION]

while we get for the dense central phase:

[EQUATION]

where [FORMULA] 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 [FORMULA], 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 [FORMULA] from stellar ejecta balances the loss due to the exchange of matter with the diffuse phase, which replaces some gas with the metallicity [FORMULA] by some gas falling from the halo with the metallicity [FORMULA]. Since this exchange of gas between both components [FORMULA] and [FORMULA] is driven by supernovae, or stellar winds, whose importance is parameterized by [FORMULA], see (B2), the equilibrium metallicity is smaller for small temperatures (weak potential wells) where these flows of matter are more important. Then [FORMULA] increases linearly with time as [FORMULA] receives some metals from [FORMULA] at a constant rate (the term in [FORMULA] corresponds to the second term in [FORMULA], simply multiplied by a temperature-dependent factor and time). Naturally, when [FORMULA] reaches the "stationary" value described above for [FORMULA], 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:

[EQUATION]

The second term in [FORMULA], which is constant in time, corresponds to the "stationary" regime we described for [FORMULA]. Since stars form from the dense phase [FORMULA], their metallicity quickly reaches the surrounding gas metallicity [FORMULA] which is constant with time. Later, when most baryons are incorporated into stars ([FORMULA]), the stellar metallicity increases to reach y if the previous equilibrium value was smaller than y. Indeed, at very long times when all the gas is converted into stars the mean stellar metallicity [FORMULA] must be equal to the yield by definition, since our system is closed (there is no global mass loss nor gain although the mass of individual galaxies evolves with time).

Appendix F: role of parameters and scalings

The model we constructed in the previous sections has only two main specific parameters: [FORMULA] and [FORMULA], which determine respectively the size (and mass) of galaxies and their star formation rate. For massive galaxies, with [FORMULA], there is also a dependence on [FORMULA], while for faint galaxies there is a dependence on [FORMULA], but this is not critical. Naturally, it also depends on the usual cosmological parameters [FORMULA], and on the power-spectrum [FORMULA] for the multiplicity functions. Now we shall see how we can get the value of [FORMULA] and [FORMULA], for a given set of cosmological parameters and a fixed [FORMULA].

Let us consider a galaxy like the Milky Way located on the curve [FORMULA] (see Fig. 2), defined by a fixed temperature T or circular velocity V. From the relation [FORMULA] we obtain [FORMULA], and then all the characteristics of this galaxy. Thus, we get:

[EQUATION]

and

[EQUATION]

Besides, since [FORMULA] we have [FORMULA], 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 [FORMULA] while for bright galaxies which have already consumed most of their gas content [FORMULA]). Hence, a larger [FORMULA] 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 [FORMULA] 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 [FORMULA] K, give the value of [FORMULA] and [FORMULA], as a function of the cosmological parameters. Thus we obtain:

[EQUATION]

Now, we can use these values of [FORMULA] and [FORMULA] 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 [FORMULA] for the Milky Way gives [FORMULA] 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 [FORMULA] of the Galaxy, and its ratio [FORMULA], lead to [FORMULA]. 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 [FORMULA] and the value of the product [FORMULA]. For [FORMULA], we get [FORMULA]. In Table F1 we consider two types of galaxies: i) massive galaxies like the Milky Way located on the cooling curve [FORMULA], which have a high luminosity and have already consumed most of their initial gas, and ii) small galaxies located on [FORMULA] (hence characterized by the density contrast [FORMULA]), which are faint and have a small stellar content.


[TABLE]

Table F1. Scalings


Thus, the values of the cosmological parameters imply well-defined galaxy characteristics. For instance, for a given [FORMULA], a larger [FORMULA] (that is a higher baryon fraction) leads to a smaller radius R (since the total baryon mass, linked to the luminosity, must not vary), a smaller mass M, whence a larger number of objects (measured by [FORMULA]). Since the radius of a galaxy similar to the Milky Way should be larger than 60 kpc we get a higher limit on [FORMULA], while the observed luminosity function, which gives the number density of galaxies, sets a lower limit. These two limits could have been incompatible, since for [FORMULA] there is no freedom in the choice of the function [FORMULA] which determines the multiplicity function in the non-linear scaling approach. Thus it appears to be a remarkable success that for [FORMULA] it is possible to find a baryon fraction (in our case [FORMULA]) which is compatible with the three constraints provided by 1) the nucleosynthesis predictions, 2) the galaxy masses and 3) their luminosity function. Note however that for this latter constraint we still have the choice of the power-spectrum [FORMULA], that is its local index n and its amplitude parameterized by [FORMULA]. Nevertheless, these two parameters are also constrained by observations and we find that the choice [FORMULA] and [FORMULA] or a CDM power-spectrum (which is in good agreement with observations) provides satisfactory results. In fact, we could also choose a lower value for [FORMULA], but this would create some problems for Lyman-[FORMULA] clouds (Valageas et al. 1998). In the case of a low-density universe [FORMULA], the conditions 1) and 2) imply [FORMULA].

Appendix G: approximate power-law regimes

From 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 [FORMULA] and [FORMULA] K, with a constant density contrast .

2) Faint galaxies, located on [FORMULA]: [FORMULA] kpc and [FORMULA] K, with a nearly constant radius .

3) Bright galaxies located on [FORMULA] kpc and [FORMULA] K, which nearly have exhausted all their gas .

G.1. Star formation

1) Very faint galaxies

Using (A3), (B7) and (D5) (since these galaxies have only consumed a very small fraction of their gas content) we get:

[EQUATION]

Moreover, since these galaxies have a small stellar content [FORMULA] we have [FORMULA], hence:

[EQUATION]

2) Faint galaxies

We obtain:

[EQUATION]

3) Bright galaxies

Since these galaxies have lost most of their gas, which has already formed stars, we now have [FORMULA], and using [FORMULA] we get:

[EQUATION]

G.2. Mass/Light ratio and Tully-Fisher relation

For a constant star-mass/luminosity ratio [FORMULA] we obtain for the 3 regimes we described above the scaling relations:

1) Very faint galaxies:

[EQUATION]

2) Faint galaxies:

[EQUATION]

3) Bright galaxies:

[EQUATION]

G.3. Metallicity

1) Very faint galaxies:

Since we noticed earlier that [FORMULA], we obtain:

[EQUATION]

2) Faint galaxies:

We still have [FORMULA], so we get:

[EQUATION]

The increase of the slope of [FORMULA], as compared to the previous regime, is due to the fact that the evolution time-scale [FORMULA] 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 [FORMULA]. Most of the gas is in the dense component [FORMULA], since supernovae are not very efficient and [FORMULA], hence [FORMULA] and we recover the usual one component closed-box model, with [FORMULA]. Thus, in our case we obtain [FORMULA]. Since [FORMULA] we have:

[EQUATION]

G.4. Slope of the luminosity function

Using the PS approximation, we get for an [FORMULA] initial spectrum (that gives results quite close to those obtained with CDM initial conditions) the following behaviour.

1) Very faint galaxies:

[EQUATION]

2) Faint galaxies:

[EQUATION]

3) Bright galaxies:

[EQUATION]

Using the non-linear scaling approximation, we write [FORMULA] for [FORMULA], where [FORMULA], and [FORMULA] for [FORMULA] with [FORMULA].

1) Very faint galaxies:

[EQUATION]

2) Faint galaxies:

[EQUATION]

3) Bright galaxies:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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