The Equilibria of Rotating Isothermal Clouds - Part Two - Structure and Dynamical Stability

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Abstract
Numerical results are presented for the structure of differentially rotating, isothermal clouds. The equilibria can be characterized by two parameters: a dimensionless mass M and the rotational parameter β(0 ≤ p ≤ ⅓). Within the class of dynamically stable equilibria, clouds with low M and β have small density contrasts (ratio of density at the center to density at the cloud boundary) and resemble Maclaurin spheroids; clouds with high M and β have larger density contrasts and eccentricities. At any fixed β, there exists a critical mass M*(β) which is marginally dynamically unstable to axisymmetric perturbations. Stability is diagnosed by investigating the change in M as a function of density contrast at fixed β. This "static method" is proved rigorously for the present case by showing that there exists a conserved free energy G for isothermal perturbations. Clouds with M < M*(β) can be either stable or unstable, depending on their density contrasts; clouds with M > M*(β), should they exist, must be unstable. The structure of unstable equilibria with M < M*(β) is displayed. The marginally unstable normal modes of oscillation at the critical masses are obtained numerically over the full range of β. These normal modes prove to be the fundamental ones. Similar cases of instability in the fundamental mode are shown to exist in nonrotating polytropes bounded by a finite external pressure. Finally, a comparison of the stability results for isothermal clouds with the dynamical collapse calculations suggests a simple interpretation for the latter. Ringlike collapses or other runaway situations are obtained with initial conditions such that M > M*(β). Stable, disklike configurations are obtained for subcritical masses. The disks actually reached with the collapse codes agree well with the results of this study.

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