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PhyHardAdv 2019 • Paper 1

Particle Number Density in Spherical Gaseous Cloud Under Mutual Gravitation

Consider a spherical cloud of gaseous particles with mass density ρ(r)\rho(r) in free space, where rr indicates the radial distance from the center. The cloud comprises particles of identical mass mm undergoing circular orbits around the central point with uniform kinetic energy KK. The governing force on the particles is their mutual gravitational attraction. Provided ρ(r)\rho(r) remains constant over time, the particle number density n(r)=ρ(r)/mn(r) = \rho(r)/m is expressed as (GG is the universal gravitational constant):

Options

A

K2πr2m2G\frac{K}{2 \pi r^2 m^2 G}

Correct
B

Kπr2m2G\frac{K}{\pi r^2 m^2 G}

C

3Kπr2m2G\frac{3K}{\pi r^2 m^2 G}

D

K6πr2m2G\frac{K}{6 \pi r^2 m^2 G}

Topics & Concepts

GravitationGravitation

Step-by-Step Solution

For a particle of mass mm undergoing a circular orbit of radius rr with velocity vv and uniform kinetic energy KK, the kinetic energy is given by: K=12mv2  ⟹  mv2=2KK = \frac{1}{2}m v^2 \implies m v^2 = 2K

The centripetal force required for this circular orbit is provided by the gravitational attraction of the mass M(r)M(r) enclosed within a sphere of radius rr: mv2r=GM(r)mr2\frac{m v^2}{r} = \frac{G M(r) m}{r^2}

Substituting mv2=2Km v^2 = 2K into the equation above: 2Kr=GM(r)mr2  ⟹  M(r)=2KrGm\frac{2K}{r} = \frac{G M(r) m}{r^2} \implies M(r) = \frac{2 K r}{G m}

The total enclosed mass M(r)M(r) in terms of the mass density ρ(r)\rho(r) is: M(r)=∫0r4πr′2ρ(r′) dr′M(r) = \int_{0}^{r} 4\pi r'^2 \rho(r') \, dr'

Differentiating M(r)M(r) with respect to rr: dM(r)dr=4πr2ρ(r)\frac{d M(r)}{dr} = 4\pi r^2 \rho(r)

From M(r)=2KrGmM(r) = \frac{2 K r}{G m}, taking the derivative with respect to rr yields: dM(r)dr=2KGm\frac{d M(r)}{dr} = \frac{2K}{G m}

Equating the two expressions for dM(r)dr\frac{d M(r)}{dr}: 4πr2ρ(r)=2KGm  ⟹  ρ(r)=K2πr2Gm4\pi r^2 \rho(r) = \frac{2K}{G m} \implies \rho(r) = \frac{K}{2\pi r^2 G m}

The particle number density n(r)n(r) is defined as n(r)=ρ(r)mn(r) = \frac{\rho(r)}{m}: n(r)=K2πr2m2Gn(r) = \frac{K}{2\pi r^2 m^2 G}

Thus, the correct option is (A).

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