The mass density of a spherical galaxy varies | vedclass

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(C) The mass of the galaxy within a radius $R$ is calculated by integrating the density $\rho(r) = \frac{K}{r}$ over the volume.$dm = \rho(r) \cdot 4\

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$T^2 \propto R$

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(C) The mass of the galaxy within a radius $R$ is calculated by integrating the density $\rho(r) = \frac{K}{r}$ over the volume.$dm = \rho(r) \cdot 4\pi r^2 dr = \left(\frac{K}{r}\right) \cdot 4\pi r^2 dr = 4\pi K r dr$$M(R) = \int_{0}^{R} 4\pi K r dr = 4\pi K \left[ \frac{r^2}{2} \right]_{0}^{R} = 2\pi K R^2$For a star of mass $m$ in a circular orbit of radius $R$,the gravitational force provides the necessary centripetal force:$\frac{G M(R) m}{R^2} = \frac{m v^2}{R}$Substituting $M(R) = 2\pi K R^2$:$\frac{G (2\pi K R^2) m}{R^2} = \frac{m v^2}{R} \Rightarrow 2\pi G K m = \frac{m v^2}{R} \Rightarrow v^2 = 2\pi G K R$$v = \sqrt{2\pi G K R}$The time period $T$ is given by $T = \frac{2\pi R}{v}$.$T = \frac{2\pi R}{\sqrt{2\pi G K R}} = \sqrt{\frac{4\pi^2 R^2}{2\pi G K R}} = \sqrt{\frac{2\pi R}{G K}} \propto \sqrt{R}$Squaring both sides,we get $T^2 \propto R$.

The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance $r$ from its centre. In that region,a small star is in a circular orbit of radius $R$. Then the period of revolution,$T$ depends on $R$ as

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