Isothermal Distribution of Gravitating Gas with Uniform Expansion

Toshihiko Tsuneto
1968 Progress of theoretical physics  
In this note we point out a solution of Vlasov equation for mass points with identical mass m moving in the self-consistent averaged gravitational field, which may be of interest in discussing evolutions of irregularities due to gravitation.1) It represents a spherical isothermal distribution in the presence .of the universal expansion or contraction in the Newtonian approximation. With obvious notations we have a set of equations: where Since the general discussion of these equations are well
more » ... nown,2) we shall only sketch our result. Assuming spherical symmetry for simplicity, we suppose f=/(Q) , Q=a[v-u(r, t)J2+ d (r, t). (4) Substituting in (1) and equating terms with each power' of v to zero, one obtains da au (ft-2a-ar =0, (5) ad ad at ar ar 2a ar -. The coefficient a must be independent of r and may be expressed as a= [R(t) / RoJ 2. Integrating (7) and absorbing the integration constant in rp, we get Hence the second term must be independent of r. We put R02 2n G () 2 1> -2R2d=3 Pro t r (10) and o,btain the well-known equation which determines the uniform expansion of the universe corresponding to uniform density Pro (t) :1)
doi:10.1143/ptp.40.672 fatcat:6y7fixeb55gutnw56lb2lkp2bi