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Astron. Astrophys. 347, 1-20 (1999)

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4. Quasar radiative output

In addition to galaxies we also need to describe the radiation emitted by quasars which provide a non-negligible contribution to the background radiation field, especially at the high frequencies [FORMULA] eV which are relevant for helium ionization. We shall again follow the formalism of VS II to obtain the quasar luminosity function, in a fashion similar to Efstathiou & Rees (1988) and Nusser & Silk (1993). We assume that the quasar mass [FORMULA] is proportional to the mass of gas [FORMULA] available in the inner parts of the galaxy: [FORMULA]. Note that for galaxies which have not yet converted most of their gas into stars (i.e. all galaxies except those with [FORMULA] at [FORMULA]) this also implies [FORMULA] where [FORMULA] is the stellar mass. Indeed, for [FORMULA] (where [FORMULA] is the age of the universe) we have:

[EQUATION]

by definition of [FORMULA], see (6), while the mass [FORMULA] of cold central gas [FORMULA] satisfies:

[EQUATION]

The factor [FORMULA] translates the fact that in our model supernovae eject part of the star-forming gas out of the galactic center into the larger dark matter halo (VS II). Hence we get [FORMULA]. Of course, at late times for bright galaxies when most of the gas has been consumed we have [FORMULA]. Then the mass of gas available to feed the quasar declines with time. This leads to a high luminosity cut-off at low z for the quasar luminosity function since in this regime very massive galaxies have less gas than smaller ones which underwent less efficient star formation (see VS II and Sect. 8.1). We shall use [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. Note that observations (Magorrian et al.1998) find [FORMULA] in large galaxies. Next we write the bolometric luminosity [FORMULA] of the quasar as:

[EQUATION]

where [FORMULA] is the quasar radiative efficiency (fraction of central rest mass energy converted into radiation) and [FORMULA] is the quasar life-time. Since we shall assume that quasars radiate at the Eddington limit we have: [FORMULA] yr. Thus, the quasar luminosity attached to a galaxy of dark matter mass M, virial temperature T, is:

[EQUATION]

As seen above in (15), the temperature term comes from the fact that in our galactic model, small galaxies ([FORMULA]) are strongly influenced by supernova feedback which expells part of their baryonic content from the inner regions. Note however that this term does not enter the relation (quasar mass) - (stellar mass) as it cancels out on both sides. Next we obtain the quasar multiplicity function from the galaxy mass function as:

[EQUATION]

The factor [FORMULA] (we use [FORMULA]) is the fraction of galactic halos which actually harbour a quasar while [FORMULA] is the evolution time-scale of galactic halos of mass M defined by:

[EQUATION]

Since the quasar life-time [FORMULA] yr is quite short, this reduces to [FORMULA]. Together with (17) the relation (18) provides the quasar luminosity function. Thus, we only have two parameters: [FORMULA] (which only depends on F, constrained by the observed (quasar mass)/(stellar mass) ratio, for quasars shining at the Eddington luminosity) which enters the mass-luminosity relation, and [FORMULA] which appears as a simple normalization factor in the luminosity function. Hence a larger fraction of quasars [FORMULA] together with a smaller life-time [FORMULA] would give the same results, so that we could also choose [FORMULA]. In a fashion similar to what we did for galaxies we can now derive the quasar radiative output. We first write the radiation emitted by an individual quasar as:

[EQUATION]

where [FORMULA] is the conversion factor from bolometric luminosity to B-band luminosity ([FORMULA] at [FORMULA] eV), taken from Elvis et al. (1994), while [FORMULA] is the local slope of the quasar spectrum. Then, the source term [FORMULA] for the background radiation due to quasars is:

[EQUATION]

where again [FORMULA] is a mean opacity which we shall describe later.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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