An approximate analytical solution for radial evolution of homologous collapse

The homologous collapse of a sphere of uniform density from a resting state under its self-gravity has been used to model the formation of astronomical objects. It is well known that the evolution of the radius with time cannot be obtained explicitly because of the need to solve a transcendental equation of cycloidal parameter with respect to time. By combining the Pade approximation and the Schroder formula, we construct an approximate analytical solution of radius as a function of time. Our method is a direct method rather than an iteration method and it requires only solving a cubic equation and evaluating three trigonometric functions. Compared with the existing methods, the accuracy and effectiveness of this method are clearly illustrated.