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Feedback is an essential component of galaxy formation within CDM models. It is invoked primarily to flatten the luminosity function given the steep mass function of CDM halos. For example, the faint end of the luminosity function in the Local Group is flat over a range that is about 10 magnitudes fainter than M*. By varying the efficiency of feedback with halo mass, it is possible to get a reasonably flat luminosity function as Figure 3 demonstrates. The parameter alpha controls how much gas is ejected from dark matter halos of a given circular velocity and allows one to form systematically less stars in smaller mass halos.

Figure 3

Figure 3. The histogram shows the cumulative distribution of absolute magnitudes of the 11 Galactic satellites. The curves show predictions from the Durham semi-analytic models of galaxy formation (Cole etal 2000) varying the parameter alpha that controls the efficiency of feedback. This can be tuned to give the correct distribution of satellite luminosities which in this case would lie somewhere in between the blue and pink curves.

The problem with a uniform feedback scheme is that the mass to light ratios of galaxies will increase rapidly for fainter galaxies. Thus we find that a satellite halo with absolute magnitude MB = - 10 is predicted to have a circular velocity of 40 km/s, roughly three times that observed for the dSph's. Figure 4 shows the "Tully-Fisher" relation for the 11 Galactic satellites compared with the curves predicted from the semi-analytic models employed in Figure 3. The circular velocities are overestimated by a factor of 3-4.

Figure 4

Figure 4. The Tully-Fisher relation for the Galactic satellites (green points) where the velocity dispersions of the spheroidals have been converted to circular velocity assuming isotropic orbits and isothermal potentials. The curves show the Tully-Fisher relation predicted by the semi-analytic models from Figure 3 where we varied the efficiency of feedback to match the numbers of dwarf galaxies.

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