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5.5

Semi-analytic modelling of galaxy formation

In the semi-analytic models in Kauffmann et al. (1999) and Springel et al. (2001), outputs of N-body simulations are used to follow the evolution and formation of dark matter halos. The merging process of the halos is realized in the simulation, and some physically motivated simple rules are applied to model the dynamics and evolution of the gas in the halos. For each dark halo in the simulation outputs, we assign among others the following halo properties: the virial radiusRvir, the virial massMvir, and the circular velocity

V2

vir =GMvir/Rvir.

5.5.1 Gas cooling

Gas cooling is modelled as in White & Frenk (1991). We assume that the hot gas within a dark halo is distributed like an isothermal sphere with density profile ρg. The local cooling time of the gas, tcool, can be defined as the ratio

of the specific thermal energy content of the gas to the cooling rate per unit volume: tcool(r) = 3 2 kT ρg ¯ µmpn2e(r)Λ(T, Z) , (5.1)

where ¯µis the mean molecular weight,mpis the proton mass,neis the electron

number density,T is the gas temperature which we approximate with the virial temperature of the halo T = 35.9(Vvir/kms−1)2 K, and Λ(T, Z) is the cooling

rate. We employ the cooling functions in Sutherland & Dopita (1993), as in our SPH simulations.

We define the cooling radius rcool such that the cooling time at this radius

is equal to the time for which the halo has been able to cool quasi-statistically. We approximate this time with the halo’s dynamical time Rvir/Vvir.

5.5.2 Stripped down semi-analytic model

Our SPH simulations do not include star formation and subsequent supernova feedback. In order to make a direct comparison with the simulations, we construct a ‘stripped down’ version of the semi-analytic modelling. In the stripped-down model, the star formation and feedback are switched off, and the cooling cut-off for halos with circular velocity larger than 350 km/sec employed in the semi-analytic models in Springel et al. (2001) is not implemented. In

principle, the cooling rate strongly depends on the metallicity of the gas Z. However, as star formation is suppressed in our stripped-down semi-analytic model, the gas does not enrich with metals but stays at primordial metallicity. Therefore the primordial cooling functions are used which is again consistent with what is done in the SPH simulations.

To further approximate the two methods we artificially reduce the reso- lution of the semi-analytic model by allowing gas to cool only in halos with more than a certain threshold number of particles, Nres. This threshold num-

ber is determined by carrying out the following experiment. We found that, for small Nres, there are many SAM-galaxies whose host halos do not con-

tain any SPH-galaxy. Starting from Nres = 32, we progressively increase the

threshold number. We then look for the best value for Nres such that there

are almost equal number of SAM-galaxies that have no SPH counterpart and SPH-galaxies that have no SAM counterpart. We found that fixing Nres= 56

satisfies the above condition. We therefore take this value as theeffective reso- lutionof our SPH cooling simulation and implement the resolution cut-off with this value in the stripped-down model. We note that for this test we turned off the merging of satellite galaxies in our semi-analytic model, so that we can trace the formation history of all the SAM galaxies and their correspondences to SPH-galaxies. For the actual anaylsis in the following, we let SAM satellite galaxies to merge with central galaxies as described in the next section.

5.5.3 Galaxies in the semi-analytic model

Galaxies in our stripped-down semi-analytic model do not possess any prop- erties such as luminosity or morphological type. There are basically only two populations of galaxies. In our model, each dark halo carries exactly one cen- tral galaxy and its position is given by the most-bound particle of the halo. Many halos have also one or moresatellitegalaxies. Satellites are galaxies that had been central galaxies in earlier outputs, but their host halos merged at some time to form a larger halo in which they now reside. Satellite galaxies are assumed to merge with the central galaxy in a time characterised by the dynamical friction time. In our implementation, only the central galaxy is supplied with additional gas that cools within the halo, hence the mass of a satellite galaxy is kept constant.

5.5. Semi-analytic modelling of galaxy formation 135

Figure 5.6 Distribution of SAM-galaxies (top) and SPH-galaxies (bottom) in the S0-C simulation at z=0. Only galaxies with mass larger than 6_×1010_h−1_M

¯ are plotted in the figures. For this mass range there are 162 SAM galaxies and 178 SPH galaxies.