A Quantum Mechanical Approach To A System of Self-Gravitating Particles   And The Problem of Gravitational Collapse
  By making an intuitive choice for the single-particle density of a system of N self-gravitating particles, without any source for the radiation of energy, we have been able to calculate the binding energy of the system by treating these particles as fermions. Our expression for the ground state energy of the system shows a dependence of $N^{7/3}$ on the particle number, which is in agreement with the results obtained by other workers. We also arrive at a compact expression for the radius of a star following which we correctly reproduce the nucleon number to be found in a typical star. Using this value, we obtain the well-known result for the limiting value of the mass, M, of a neutron star $(M \simeq 3.12 M_{\odot}, M_{\odot}$ being the solar mass) beyond which the black hole formation should take place. Generalizing the present calculation to the case of white dwarfs,we have been able to obtain the so called Chandrasekhar limit for the mass, $M_{Ch}$, $(M_{Ch}\simeq 1.44 M_{\odot})$ below which the stars are expected to go over to the white dwarf state. We reproduce this by introducing a radius, equivalent to Schwarzschild radius, at the interface of the neutron stars and white dwarfs. This is justified by considering the fact that it gives rise to the correct value for the degree of ionization $\mu_e (\mu_e\approx 2)$ for heavy nuclei. 
D.N. Tripathy and Subodha Mishra