Time evolution of density profile

[throneoo]

Is it possible to work out analytically how the mass density profile ρ(r) of a ball of gas (spherically symmetric) evolve with time given the initial profile ρ₀(r)? The assumption here is that the particles collapse only under the influence of gravity. I thought of this question in the process of deriving the free-fall collapse time. I know the trajectory of each spherical shell yet I don't know how to determine the mass distribution at any given moment during the collapse (quantitatively)

 

[Ken G]

It sounds like you are talking free fall, so imagine two thin shells of test particles, and solve their trajectory. Treat the distance between the shells as small compared to the radius. Find the distance between the shells as a function of time. The mass between the shells stays constant, and the volume between the shells scales in proportion to r² dr, where dr(t) is what you figure out when you figure out r(t) for the two shells. The density is inverse to the volume.