Self-similar solutions provide good descriptions for the gravitational collapse of spherical clouds or stars when the gas obeys a polytropic equation of state , p = K \rho ^ { \gamma } ( with \gamma \leq 4 / 3 , and \gamma = 1 corresponds to isothermal gas ) .
We study the behaviors of nonradial ( nonspherical ) perturbations in the similarity solutions of Larson , Penston and Yahil , which describe the evolution of the collapsing cloud prior to core formation .
Our global stability analysis reveals the existence of unstable bar-modes ( l = 2 ) when \gamma \leq 1.09 .
In particular , for the collapse of isothermal spheres , which applies to the early stages of star formation , the l = 2 density perturbation relative to the background , \delta \rho ( { \bf r } ,t ) / \rho ( r,t ) , increases as ( t _ { 0 } - t ) ^ { -0.352 } \propto \rho _ { c } ( t ) ^ { 0.176 } , where t _ { 0 } denotes the epoch of core formation , and \rho _ { c } ( t ) is the cloud central density .
Thus , the isothermal cloud tends to evolve into an ellipsoidal shape ( prolate bar or oblate disk , depending on initial conditions ) as the collapse proceeds .
This shape deformation may facilitate fragmentation of the cloud .
In the context of Type II supernovae , core collapse is described by the \gamma \simeq 1.3 equation of state , and our analysis indicates that there is no growing mode ( with density perturbation ) in the collapsing core before the proto-neutron star forms , although nonradial perturbations can grow during the subsequent accretion of the outer core and envelope onto the neutron star .

We also carry out a global stability analysis for the self-similar expansion-wave solution found by Shu , which describes the post-collapse accretion ( “ inside-out ” collapse ) of isothermal gas onto a protostar .
We show that this solution is unstable to perturbations of all l ’ s , although the growth rates are unknown .