Mass of an unknown planet

Problem

A planet has a satellite at $r = 10^8$ m with period $T = 86\,400$ s. Using Kepler’s third law, calculate the mass of the planet.

Solution

Kepler’s third law, obtained by equating the gravitational force to the centripetal force, relates the orbital period and radius to the mass of the central body: $T^2 = \frac{4\pi^2}{G M}\, r^3$ Isolating the mass $M$: $M = \frac{4\pi^2\, r^3}{G\, T^2}$ Calculating the factors: $r^3 = (10^8)^3 = 10^{24}$ m$^3$; $T^2 = (86\,400)^2 = 7{,}46\cdot 10^9$ s$^2$; $4\pi^2 = 39{,}5$. So: $M = \frac{39{,}5\cdot 10^{24}}{6{,}67\cdot 10^{-11}\cdot 7{,}46\cdot 10^9} = \frac{3{,}95\cdot 10^{25}}{4{,}98\cdot 10^{-1}} \approx 7{,}94\cdot 10^{25}~\text{kg}$ Comparing with the Earth’s mass $M_T \approx 5{,}97\cdot 10^{24}$ kg, the planet has about $13$ Earth masses. $\ev{M \approx 7{,}94\cdot 10^{25}~\text{kg} \ (\sim 13\ M_\text{Earth})}$

Links

Topics: Gravitation Concepts: Kepler’s laws · Orbits Skills: Symbolic setup Objects: Satellite