Consider a spherical gaseous cloud of mass density ρr in a free space where r is the radial distance from its centre. The gaseous cloud is made of particles of equal mass m moving in circular orbits about their common centre with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρr is constant with time. The particle number density nr $$=$$ ρ$$r/m$$ is:[ G is universal gravitational constant]

Question

Consider a spherical gaseous cloud of mass density ρr in a free space where r is the radial distance from its centre. The gaseous cloud is made of particles of equal mass m moving in circular orbits about their common centre with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρr is constant with time. The particle number density nr $$=$$ ρ$$r/m$$  is:[ G is universal gravitational constant]

Infinity Learn · Accepted Answer

$$M(r)$$ $$=$$ mass of gas in a sphere of radius r.   For any particle of mass m located at radius r, $$Fg=Fcp$$ ⇒$$GMrmr2=mv2r$$⇒GM (r) m $$=$$ $$2(K)$$ r           here $$K=12mv2$$ ⇒$$Mr=2KGmrOn$$ differentiating w.r.t r,  $$dMdr=2KGm$$ ................ (1)Here$$ dM $$=$$ mass of gas in a spherical shell of radius r and thickness dr.∴$$dM=$$ρ$$rdV=$$ρ$$r4$$$$π$$$$r2dr$$  ............... $$(2) Substitute (2) in (1)ρ$$r4$$$$π$$$$r2drdr=2KGm$$ ∴ρ$$r=K2$$$$π$$$$r2Gm$$ ⇒$$nr=$$ρ$$rm=K2$$$$π$$$$r2Gm2$$

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