The Formation of Galactic Disks
The disk components of galaxies are usually assumed to form later than the spheroidal components but how much later is still an open question. The following argument, based on the spin-up of collapsing protogalaxies, suggests that the disk components formed recently (Fall and Efstathiou 1980, Fall 1985). Since the specific angular momentum of a disk today is likely to be nearly the same as that in the protogalaxy, the initial size can be calculated as a function of the initial rotation velocity. The initial size of the protogalaxy then determines the free-fall time and hence the maximum redshift of collapse. The initial rotation is most conveniently expressed in terms of the dimensionless spin parameter, λ ≡ J | E |1/2 G -1 M -5/2, where J, E and M are respectively the total angular momentum, energy and mass of the protogalaxy, including any dark matter it may contain. For an exponential disk with a scale length α-1 embedded in an isothermal halo with a circular velocity vc, one finds that the free-fall time is τ f f = 1.9(αv c λ)-1. The values of α-1 and v c can be derived from observation, and for all theories that invoke hierarchical clustering. the values of λ can be derived from N-body simulations. The distribution of spin parameters turns out to depend only weakly on the spectrum of perturbations at recombination and always has a median value near λ ≃ 0.05 (Barnes and Efstathiou 1987, Quinn, Salmon and Zurek 1987). I have used these results to show that the median redshift of disk formation is z d ≤ 2 for q 0 = 1/2 and z d ≤ 7 for q 0 = 0 (Fall 1987).