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A star of mass $M$ and radius $R$ is made up of gases. The average gravitational pressure compressing the star due to gravitational pull of the gases making up the star depends on $R$ as
$1 / R^{4}$
$1 / R$
$1 / R^{2}$
$1 / R^{6}$
Solution
$(a)$ For a spherical shell of radius $r$ and thickness $d r$, weight of layer is balanced by pressure on layer.
$\therefore \quad$ Weight $=\{p-(p+\Delta p)\} 4 \pi r^{2}$
$\Rightarrow \quad m g=-\Delta p \cdot 4 \pi r^{2}$
$\Rightarrow 4 \pi r^{2} \cdot d r \cdot \rho \cdot g=-\Delta p \cdot 4 \pi r^{2} \Rightarrow-\Delta p=\rho g d r$
$\Rightarrow \quad-\Delta p=\rho^{2} \cdot \frac{4}{3} \pi R d r$
So, $\quad p_{ av }=\frac{1}{R} \int \limits_{0}^{R} \Delta p=\frac{4}{3} \pi \rho^{2} R \int \limits_{0}^{R} d r$
$=\frac{4}{3} \frac{\pi M R \cdot R}{(V)^{2}}$
$=\frac{4}{3} \pi \cdot \frac{M R^{2}}{\left(\frac{4}{3} \pi R^{3}\right)^{2}}$
$\Rightarrow \quad p_{ av }=\frac{3}{4 \pi} \cdot \frac{M}{R^{4}}$
$\Rightarrow \quad p_{ av } \propto \frac{1}{R^{4}}$
