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In order to model the formation and evolution of galaxies on top of the dark matter structure evolution, the SAM code includes physical prescriptions for gas cooling, re- ionization, star formation, supernova feedback, metal evolution, black hole growth and AGN feedback. In the following, the implementations of the different mechanisms are briefly summarized. In table 4.1, a summary of the different galaxy formation parameters is provided.

1. Radiative coolingThe rate of gas condensation via atomic cooling is computed

based on the model proposed by White & Frenk (1991). At each radius the cooling time can be computed according to

tcool =

3/2µmpkT

ρg(r)Λ(T, Zh)

. (4.1)

Here, T is the virial temperature, µmp is the mean molecular mass, ρg(r) is the radial density profile of the gas andΛ(T, Zh)is the cooling function (temperature and metallicity dependent). The cooling time is the time required for the gas to radiate away all its energy starting at the virial temperature. The gas density

profile ρg(r) is assumed to follow an isothermal sphere: ρg(r) =mhot/(4πrvirr2). Putting this expression in Eq. 4.1one can solve for a cooling radiusrcool. Within

the cooling radius all gas can cool within the cooling time tcool. The cooling rate

for the mass within rcool is

dmcool dt = 1 2mhot rcool rvir 1 tcool . (4.2)

FollowingSpringel et al.(2001a) andCroton(2006), it is assumed that the cooling time is equal to the halo dynamical time tcool = tdyn = rvir/Vvir. In order to account for cold gas flows and hot gas accretion as found by simulations, two different modes of accretion are distinguished: the rapid and the slow cooling regime (= cold and hot mode cooling). In the rapid cooling regime, where the cooling radius is larger than the virial radius rcool > rvir, the cooling rate is equal to the gas accretion rate, governed by the mass accretion history. Slow mode cooling occurs, whenever the cooling radius is smaller than the virial radius

rcool < rvir. Here, the cooling rate is calculated according to equation 4.2.

2. Photo-ionizationPhoto-ionization heating causes halos below a certain, critical

filtering massMF to have a lower baryon fraction than the universal average. The collapsed baryon fraction as a function of redshift and halo mass is parameterized by the expression:

fb,coll(z, Mvir) =

fb

[1 + 0.26MF(z)/Mvir]3

, (4.3)

where fb is the universal baryon fraction. The filtering mass is a function of redshift and depends on the re-ionization history of the universe.

3. Star formation The SAM distinguishes between quiescent star formation and

merger-driven starbursts. The quiescent star formation is based on the the em- pirical Schmidt-Kennicutt relation (Kennicutt, 1989, 1998). The star formation rate surface density ΣSFR is calculated according to

ΣSFR=AKSΣNgasK (4.4)

with AKS = 1.67×10−4,NK = 1.4, and Σgas is the surface density of cold gas in the disk. The normalisation is tuned for a Chabrier IMF (Chabrier, 2003). The gas follows an exponential disk (proportional to the scale-length of the stellar disk) and only gas above a critical surface density threshold Σcrit (= 6M"/pc2)

is available for star formation. Computing the radius rcrit within which the gas density exceeds the critical value the total star formation rate is

˙

m∗ = ' rcrit

0

Star formation during starbursts is driven by merger events. The star formation rate is assumed to be a function of the mass ratio and the combined cold gas content of the merging galaxies, the bulge to total stellar component and burst timescale (exponential decline). The burst continues until the cold gas reservoir is exhausted and the burst SFR will decline exponentially. In the case that a merger occurs, whenever a starburst from a former merger is still going on, the new cold gas is added to the current fuel reservoir.

4. Supernova feedback Cold gas is reheated through the energy release from

supernovae explosions and the cold gas may be ejected from the galaxy by supernova-driven winds. The heating rate of the cold gas is calculated with

˙ mrh =0SN0 # Vdisk 200km/s $αrh ˙ m∗, (4.6) where 0SN

0 and αrh are free parameters. The circular velocity Vdisk is assumed to be the maximum rotation velocity of the dark matter halo,Vmax. The heated gas

can either stay within the dark matter halo as hot gas, or, if the supernova-driven winds are strong enough, it can be ejected from the halo into the intergalactic medium (IGM). The fraction of reheated gas, which is ejected from the halo, is given by feject(Vvir) = ! 1 + # Vvir Veject $αeject"−1 , (4.7)

with αeject = 6 and Veject is a free parameter (≈ 100−150km/s). This ejected heated gas can re-collapse onto the halo at later times and then is available for cooling. As in Springel et al. (2001a) and De Lucia & Blaizot (2007) the rate of reinfall of rejected gas is given by

˙ mreinfall=χreinfall # meject tdyn $ . (4.8)

Here, χreinfall is a free parameter, meject is the mass of the ejected gas outside of the halo and tdyn=rvir/Vvir is the dynamical time of the halo.

5. Metal enrichment To track the production of metals, it is assumed that, to-

gether with a parcel of new stars dm∗, a certain mass of metals dMZ = ydm∗

is created and instantaneously mixed with the cold gas in the disc. The yield is assumed to be constant y = 1.5 and is treated as a free parameter. Whenever new stars are formed, they are assumed to have the metallicity of the cold gas at this time step. When metals are ejected from the disc due to SN-winds, either the metals are mixed with the hot gas or ejected from the halo into the ’diffuse’ IGM in the same proportion as reheated cold gas. Note that only metal enrichment due to Supernovae TypeII is tracked.

6. Black hole growth with AGN-driven winds and Radio-mode feedback

Black holes form at the centers of the galaxies and are thought to grow by two channels: quasar mode and radio mode. The quasar mode is the bright mode of black hole growth observed as optical or X-ray bright AGN radiating at a sig- nificant fraction of their Eddington limit (L_≈(0.1₋1)LEdd; Vestergaard, 2004;

Kollmeier et al., 2006). Such bright AGN are believed to be fed by optically thick, geometrically thin accretion disks (Shakura & Sunyaev,1973). In the next Section, the physical processes during the quasar mode and the corresponding implementations in the code will be discussed in more detail. In contrast, AGN activity in the radio-mode is much less dramatic. A large fraction of massive galaxies are detected at radio wavelengths (Best et al., 2007) without showing characteristic emission lines of classical optical or X-ray bright quasars (Kauff- mann et al., 2008). Their accretion rates are believed to be a small fraction of the Eddington rate and they are radiatively extremely inefficient. Even if AGN spend most of their time in the radio-mode, they gain most of their mass during the short, Eddington limited episodes of quasar phases which in the model are assumed to be triggered by merger events. The energy being released during the rapid growth of the black holes can drive powerful galactic scale winds. By relat- ing the momentum of the radiative energy from the accreting black hole with the momentum of the outflowing wind, the following expression for the mass outflow rate can be obtained:

dMout dt =0windηrad c vesc ˙ macc, (4.9)

where 0wind is the effective coupling efficiency, ηrad is the conversion efficiency of matter into radiation,Mout is the ejected gas mass andvesc is the escape velocity. In contrast to the quasar mode, the radio mode has low-Eddington ratio accretion rates and thus, is radiatively inefficient and associated with efficient production of radio jets that can heat gas in a quasi-hydrostatic hot halo solving the overcooling problem of massive galaxies at low redshifts. Assuming Bondi-Hoyle accretion combined with an isothermal cooling flow solution (Nulsen & Fabian, 2000) the accretion rate in the radio mode can be calculated by the following formula:

˙ mradio =κradio ! kT Λ(T, Zh) " # M• 108_M " $ . (4.10)

Thereby, κradio is a free parameter, namely the efficiency for gas accretion in the radio mode. Note that the central black hole accretes at this rate whenever hot halo gas is present (hot ’cooling’ mode, rcool > rvir). The energy that effectively couples to and heats the hot gas is given by Lheat =κheatηradm˙radioc2. Assuming that all the hot gas is at the virial temperature of the halo, the rate of the heated gas mass is given by

˙ mheat= Lheat 3/4v2 vir . (4.11)

The net cooling rate is then the usual cooling rate minus the heating rate from the radio-mode.