Analytic Solutions for Equilibrium of Rotating Isothermal Clouds: One-Parameter Family of Axisymmetric and Conformal Configurations

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Abstract
Equilibrium states of rotating axisymmetric isothermal gas clouds are studied and it is found that, if the linear rotation velocity is constant throughout a cloud, there exists a one-parameter family of conformal solutions which describe the equilibrium exactly. The parameter depends only on the ratio of the rotation velocity to the sound velocity and represents the flatness of equi-density contours in the direction of the rotation axis. Two extreme values of the parameter correspond to a non-rotating spherical configuration and a rapidly-rotating thin disk. Specific angular momentum on a cylindrical surface around the rotation axis is found to be proportional to the amount of mass contained inside the cylinder considered. The stability of the equilibrium configurations is examined analytically as well as numerically and it is found that nearly spherical configurations are unstable to contraction and expansion as a whole, very flat configurations are unstable to ring formation and intermediate configurations are likely to be stable. Finally, these equilibrium configurations are compared with the results of numerical computations which have so far been performed by many people on the collapse of rotating isothermal clouds.

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