Ranking the periods of four satellites

Problem

Ranking. Four satellites orbit planets with mass $M$ and orbital radius $r$: (A) $M$, $r$; (B) $4M$, $r$; (C) $M$, $4r$; (D) $2M$, $2r$. Rank the periods $T \propto \sqrt{r^3/M}$ from smallest to largest.

Solution

Setting up. From Kepler’s third law $T = 2\pi\sqrt{r^3/(GM)}$, the period scales as $T \propto \sqrt{r^3/M}$. We fix the unit with case (A) and calculate the factor $\sqrt{r^3/M}$ for each satellite: $\begin{aligned} \text{(A)}\ M,\,r:\quad & \sqrt{r^3/M} = 1 \\ \text{(B)}\ 4M,\,r:\quad & \sqrt{r^3/(4M)} = \tfrac{1}{2} = 0{,}5 \\ \text{(C)}\ M,\,4r:\quad & \sqrt{(4r)^3/M} = \sqrt{64} = 8 \\ \text{(D)}\ 2M,\,2r:\quad & \sqrt{(2r)^3/(2M)} = \sqrt{8/2} = 2 \end{aligned}$

Ranking. Comparing the values $0{,}5 < 1 < 2 < 8$: $\ev{T_B < T_A < T_D < T_C}$

The radius matters more than the mass: increasing $r$ (exponent $3$ under the root) strongly lengthens the period, while increasing $M$ only shortens it as $1/\sqrt{M}$.

Links

Topics: Gravitazione Concepts: Leggi di Keplero · Orbite Skills: Ragionamento di ranking · Impostazione simbolica Objects: Satellite