The Collapse of Rotating Gas Clouds

A two-dimensional hydrodynamical calculation method of self-gravitating rotating system is presented. The equation of fluid motion is treated in a Lagrangian form, while the Poisson equation of the gravitational potential is solved by the Finite-Element-Method. The results show that, in disagreement with the results of previous authors, there is no evidence of the ring formation along the equator, in case the fluid is initially rotating as a rigid body. The discrepancy is, presumably, due to the exact conservation of the angular momentum in the Lagrangian method and the inexact conservation in the Eulerian method. In this connection, it is demonstrated that a ring-like structure is actually formed under the circumstances where the angular momentum is transferred inwardly or a differential rotation of the fluid is assumed in the initial state. In the collapse of a gas cloud towards a protostar, it is well known that the angular momentum plays an important role. However, most calculations of the protosteller evolution, so far carried out by many authors, have neglected the angular momentum of gas clouds, under the restriction of the spherical symmetry. In recent years, fully two-dimensional hydrodynamical calculations of the axisym-metric collapse of rotating gas clouds with assumed equatorial symmetry have been made by Larson,v Black and Bodenheimer,"> Nakazawa et al. 3 > and Tscharnuter. 4 > The numerical methods of their calculations are essentially Eulerian, though different kinds of coordinates, formalisms and numerical techniques have been adopted. Tscharnuter indicated that the final configuration of rotating gas clouds is an approximate rotating spheroid, though his calculation may be erroneous as pointed out by Nakazawa et al. 3 > The other authors showed that, at about one free fall time after the collapse of gas clouds started, rings are formed on the equatorial plane. Their results are qualitatively similar in spite of the three different boundary conditions: fixed, pressure constant and free. Nakazawa et al. s> showed that in a model of somewhat high initial temperature, a ring is not formed but a density inversion appears. In this model, however, a large fraction of the total mass of the cloud is expelled out as a consequence of the adopted free boundary. Under the fixed boundary condition, a ring is actually formed. 5 > The main purposes of this paper are to present a new two-dimensional numerical calculation method of self-gravitating hydrodynamical system, and, with this method, to make the fundamental studies …