A satellite of mass $m$ is in a circular orbit of radius $2R_E$ about the Earth (where $R_E$ is the

Question

A satellite of mass $m$ is in a circular orbit of radius $2R_E$ about the Earth (where $R_E$ is the Earth's radius). How much energy is required to transfer it to a circular orbit of radius $4R_E$? (Let $g$ be the acceleration due to gravity at the Earth's surface)

Professor PC Thomas Classes · Answer

Total energy in a circular orbit of radius $r$ is $E = -\frac{GMm}{2r}$. Energy required $\Delta E = E_f - E_i = -\frac{GMm}{2(4R_E)} - (-\frac{GMm}{2(2R_E)}) = \frac{GMm}{8R_E}$. Since $g = \frac{GM}{R_E^2}$, this simplifies to $\Delta E = \frac{mg R_E}{8}$.