Heating of galactic discs by infalling satellites

We develop an analytic model to calculate the rate at which galaxy discs are heated by dark matter substructures orbiting in their haloes. The model takes into account the internal structure, mass function and accretion rate of satellites expected in the LambdaCDM cosmology, as well as the growth of the disc by accretion and mergers, but it ignores resonant heating of the disc and the dynamical effects of spiral arms and bars. We calibrate this model against N-body simulations and demonstrate that it is able to reproduce the N-body heating rates to within a factor of 3 in the majority of cases. Our model gives the distribution of disc scaleheights for galaxies of different luminosities. For L-* spiral galaxies, it predicts a median disc thickness of only 5 per cent of the radial scalelength if substructure is the only source of heating. The median disc thickness increases to nearly 20 per cent of the radial scalelength when heating due to gravitational scattering of stars by molecular clouds is also included. The latter value is close to the thickness estimated observationally for the disc of the Milky Way galaxy. The distribution of disc thickness predicted by the model is also consistent with a recent observational determination for sub-L galaxies by Bizyaev & Mitronova. Thus, the observed thickness of the stellar discs of spiral galaxies seems to be entirely compatible with the abundance of substructure in dark matter haloes predicted by the standard Lambda-dominated cold dark matter model of structure formation. In an Omega(0) = 1 universe, our best model of galaxy formation produces similar scaleheights, a consequence of the fact that similar amounts of substructure are accreted by haloes during the lifetime of the disc in Omega(0) = 1 and 0.3, Lambda(0) = 0.7 cold dark matter cosmologies.