Self-similar solutions and the stability of collapsing isothermal filaments

Self-similar solutions which describe collapsing isothermal cylinders with self-gravity are derived. The solutions are parameterized by their line masses. Their stability is also investigated by two different methods in the linear regime. One is the approximate separation of variables as an eigenvalue problem and the other is direct numerical integration of the evolution of perturbations. It is found that a self-gravitating cylinder is unstable to axisymmetric perturbations with wavelengths greater than about two times the diameter, when its line mass is nearly the same as that for equilibrium. In this case fragmentation is expected with separations of about four times the diameter. When the line mass of the cylinder greatly exceeds the value for equilibrium, perturbations do not grow much and the entire cylinder only collapses toward the axis. Therefore fragmentation is not expected as long as its collapse is isothermal. Subsequent evolution in this case is also discussed, and fragmentation is expected after or during a change in the equation of state.