Answer
(a) $\frac{1}{2}c$
(b) $\frac{1}{\sqrt 2}c$
Work Step by Step
(a) We can find the orbital speed of the star as follows:
$\frac{mv^2}{r}=\frac{GMm}{r^2}$
This can be rearranged as:
$v=\sqrt{\frac{GM}{r}}$
We are given: $r=2R$
But $R=\frac{2GM}{c^2}$
$\implies v=\sqrt{\frac{GM}{2(2GM/c^2)}}$
$v=\frac{1}{2}c$
(b) We can find the orbital speed of the star as follows:
$\frac{mv^2}{r}=\frac{GMm}{r^2}$
This can be rearranged as:
$v=\sqrt{\frac{GM}{r}}$
But $R=\frac{2GM}{c^2}$
$\implies v=\sqrt{\frac{GM}{2GM/c^2}}$
$v=\frac{1}{\sqrt2}c$
