EQUILIBRIUM-MODELS FOR SELF-GRAVITATING INVISCID DISKS RESULTING FROM THE COLLAPSE OF ROTATING CLOUDS

A method is presented for the construction of velocity and surface density profiles of infinitely thin, self-gravitating disks formed from the collapse of spherical clouds, under the condition that the angular momentum of each material parcel is conserved. The distributions of mass and angular momentum of a progenitor cloud may be arbitrary functions of the spherical radius R. Results are presented for initially uniformly rotating clouds with density distributions varying as R-n and n = 0, 1, 2. The resulting disk surface density distributions are well represented by power laws in to, the cylindrical radius, over most of their extent. Moreover, the form of the zeroth-order approximation, used in the iteration procedure to obtain the results, provides a useful representation of the final surface density function near the center for all three cases, and for the n = 2 case this result holds throughout the disk. The result, however, is not generally true for the velocity functions. We examine exact solutions first discussed by Mestel, confirm some of his inferences regarding the nature of a disk formed from a uniformly rotating, constant-density sphere, and obtain a modest generalization of his results. Uniformly rotating clouds that collapse to infinitely thin configurations (i.e., configurations that retain no internal energy) are incapable of forming rings, in contrast to the results of axisymmetric numerical collapse calculations, in which ring formation is common. This fact draws attention to the importance of the energy budget in all dynamical collapse calculations. We suggest that an accurate representation of accretion shock radiation is crucial in determining the true nature of the collapsed configuration. In situations where thin disks do occur, our results indicate that gravitational instabilities occur very early in the formation stage.