Question

A satellite of mass $M_S$ is orbiting the earth in a circular orbit of radius $R_S$. It starts losing energy slowly at a constant rate $C$ due to collisions with dust particles. If $M_e$ and $R_e$ denote the mass and radius of the earth respectively, show that satellite falls on the earth in a time $t$ given by $t=\frac{G{M}_s{M}_e}{2C}(\frac{1}{R_e}-\frac{1}{R_s})$.

Answer 1

$\frac{dE}{dt}=cJ/s$

$E_i=-\frac{GMm}{2r}$

$E_f=-\frac{GMm}{2Re}$

$\Delta E=\frac{GMm}{2}[\frac{1}{Re}-\frac{1}{r}]$

$t=\frac{\Delta E}{c}$

$\Rightarrow t=\frac{GMm}{2C}(\frac{1}{Re}-\frac{1}{r})$

in another representation,

$t=\frac{GMsMe}{2C}(\frac{1}{Re}-\frac{1}{Rs})$