2. Modelling the shock
In the galaxy formation canonical scenario, a galaxy forms in the gravitational potential well of a DM halo of mass . Following Silk & Rees (1998), we consider that each forming galaxy, which is fully described by the mass , hosts a super-massive BH. The fraction of seeded proto-galaxies is thus 100%. For the sake of simplicity, we assume that all proto-galaxies are spheroids. A more refined and accurate description would take into account morphological segregation (Balland et al. 1998). However, these effects do not introduce significant differences. We assume that the BH radiates during its lifetime a fraction (Natarajan & Sigurdsson 1999) of its Eddington luminosity (). The latter is fixed by the BH mass, :
where is the proton mass and G the gravitational constant. The mass of the BH should be directly related to the mass of the proto-galaxy it seeds, and thus to the fraction of baryonic matter locked up in the spheroid in which the central BH collapses. We assume a simple relation of proportionality () between the mass of the BH and the mass of the spheroid . Haehnelt & Rees (1993) give and they assume that declines with mass. However, observations of galactic nuclei (Kormendy et al. 1997; Magorrian et al. 1998) indicate that is relatively constant. Here, we take the value as advocated by Magorrian et al. (1998) and Silk & Rees (1998). We assume that the medium is instantaneously ionised, which is very likely due to the intense radiation emitted by the central BH (Voit 1996). We therefore study the propagation of a strong shock driven by a mechanical energy which represents some fraction of the BH luminosity. The first analytic solutions of spherically symmetric explosions were given by Taylor (1950) and Sedov (1959) and applied to several astrophysical problems such as stellar winds and supernovae explosions (Ikeuchi et al. 1983; Bertschinger 1986; Koo & McKee 1992a,b; Voit 1996). Following Voit (1996), we study the expansion of the shock and its effects on the gas within the seeded proto-galaxy. We characterise the shock in an expanding universe by a set of physical properties (its radius, velocity and temperature). Adapting Voit's solutions to our study, the shock at a redshift , where is the redshift at which the BH switches on, has a radius which reads as:
It propagates at a velocity
In the two previous equations, represents the mechanical energy which drives the shock. Very little is known about the fraction which can be on the order of 0.5 or even higher (Natarajan & Sigurdsson 1999). For the lifetime of the BH in the bright phase, we choose the recent value derived by Haiman & Loeb (1997): years (our results, however, are not very sensitive to the exact value of ). is the mean over-density of the proto-galaxy (assumed to be spherical), is the Hubble constant, is the density parameter and is the baryon density in the universe. Zero subscripts denote present day values. In the following and throughout the paper, we use and (Walker et al. 1991) and give the results as a function of , for and .
Following Natarajan & Sigurdsson (1999), we assume that the BH radiates 10% of its Eddington luminosity, i.e. and that half of the corresponding energy is mechanical. Furthermore, we consider that galaxies form at sufficiently high redshifts () so that only hydrogen and helium are present in substantial amounts. We compute both the size and velocity of the shock and find that the host proto-galaxy is always embedded in the shocked region.
2.1. Cooling mechanisms
The temperature of the shocked medium can be directly derived from the equipartition of energy:
Here, is the mean molecular weight of a plasma with primordial abundances. The galactic matter can be shock-heated up to very high temperatures. For the most massive galaxies, we find that the matter is heated up to a few K, a value comparable to the temperature of the intracluster medium in galaxy clusters. For these temperatures and redshifts (), the main cooling process at play is bremsstrahlung. Therefore, the shocked gas looses heat with a cooling rate given by a temperature-dependent cooling function . We have compared three different cooling functions (Bertschinger 1986; Koo & McKee 1992a; Voit 1996) given in the literature for temperatures between a few and K. We find that the condition required for efficient cooling is always satisfied in our redshift range. This means that the cooling time, given by with the hydrogen number density, is always smaller than the age of universe. Cooling is thus very efficient in our picture. The comparison between the three cooling functions shows that they all give essentially the same final temperature after cooling. We follow Bertschinger (1986) and we find that the shocked matter cools down to K at .
© European Southern Observatory (ESO) 2000
Online publication: May 3, 2000