2019MNRAS.485.3224D

In this paper, we continue to investigate the energy conservation equation obtained in our previous paper. We set ourselves three new goals: (i) to rewrite the main equations in terms of density profile in order to give more physical insight; (ii) to investigate the significance of two new terms in the energy conservation equation that originate from the gravity of the outer shells of cloud and the masses outer to the cloud, respectively; (iii) to investigate the main equation when the kinetic turbulent term scales according to Larson's law and it is independent, formally, of the accretion, in contrast to the previous work. The combination of supersonic turbulence and spherical symmetry raises a caveat that we comment on in our conclusions. We have obtained two solutions for the density profile, which scale with slopes of -2 and -3/2, respectively. The energy balance for the second solution is the same as in our previous paper: this is a free-fall. For the first solution, there are two cases: (1) if the turbulent term does not scale, then it could be important for the energy balance of the cloud; (2) if the turbulent term does scale, then it is not important for the energy balance of the cloud. The two new gravitational terms do not affect the existence of the two solutions, but the gravitation of the outer masses calibrates the energy balance for the first solution.