GATE 2021 PH – Question 49
A spherical galaxy of total mass $M$ and radius $R$ has a uniform mass distribution. In this idealized situation, which graph best describes the orbital speed $v$ of a star of mass $m\ll M$ as a function of its distance $r$ from the galactic centre? $G$ is the gravitational constant.
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Correct answer: (A) Figure (A)
Explanation
For $r<R$, the enclosed mass is $M(r)=M(r/R)^3$. Circular balance gives $v^2=GM(r)/r=GMr^2/R^3$, so $v=\sqrt{GM/R^3}\,r$, rising linearly. For $r>R$, all mass is enclosed and $v=\sqrt{GM/r}$, decreasing as $r^{-1/2}$. Both branches meet at $v(R)=\sqrt{GM/R}$. This is A.