Consider a spherical shell of radius r and radial thickness dr. P & P + dP are pressure at its inner and outer surface. Let gr = gravitational acceleration at distance r (< R)
For equilibrium of this shell - (P+dP)(4πr2)+(4πr2dr)ρgr=(P)(4πr2) {ρ=
3M
4πR3
=Densityofsphere} ⇒ dP=−ρgrdr ∵ gr=
4
3
πGρr ⇒ dP=−
4
3
πGρ2rdr {(–) ve sign indicates that pressure is decreasing with radius} ⇒