Use Newton's form of Kepler's third law to calculate a rough estimate for the mass of Charon, Pluto's moon. Take the semimajor axis of Charon's orbit to be 19,640 km and the period to be 6.39 days. The mass of Pluto is about 1.2 x 10²² kg. This gives the mass of Charon to be about: A)10¹⁰ kg. B)10¹⁷ kg. C)10²¹ kg. D)10²³ kg.

Question

Quizplus · Accepted Answer

Newton's form of Kepler's third law states: 
$M_{pluto} + M_{charon} = \frac{4\pi^{2}a^{3}}{Gp^{2}}$ where $M_{charon}$ is the mass of Charon, $a$ is the semimajor axis of Charon's orbit, $p$ is the period of Charon's orbit, $G$ is the gravitational constant, and $M_{pluto}$ is the mass of Pluto.
Plug in the given values:
$1.2 	imes 10^{22} kg + M_{charon} = \frac{4\pi^{2}(19640	imes 10^{3})^{3}}{(6.39 	imes 24 	imes 60 	imes 60 s)^{2} 	imes 6.67 	imes 10^{-11} \frac{Nm^{2}}{kg^{2}}}$
Simplifying the right side of the equation gives approximately $1.97 	imes 10^{20} kg$.
Solving for $M_{charon}$ gives approximately $1.21 	imes 10^{21} kg$, which is equivalent to $10^{21.08}$ kg, or about $10^{21}$ kg to the nearest order of magnitude. Therefore, the answer is choice C, $10^{21} kg.$

Use Newton's Form of Kepler's Third Law to Calculate a Rough