homework and exercises - Based on measurements of a moon's orbit with respect to the planet, what can one calculate?

Question:

"We observe a very small moon orbiting a planet and measure the moon's min. and max. distances from the planet's center, and the moon's maximum orbital speed. Which of the following CANNOT be measured?"

A) Mass of moon

B) Mass of planet

C) Minimum Speed of the moon

D) Period of the orbit

E) Semimajor axis of the orbit

ANSWER:

Based on Kepler's third law, we can calculated D) and E). From this, it's not impossible to measure C).

But how would I go about calculating the planet itself from these measurements?

The correct answer is apparently A), because "an object in free fall in a vacuum will fall at the same acceleration regardless of its mass." Here, space is the vacuum.

Answer

You can use Kepler's Third Law which says $T^2/a^3$ is a constant. Making further calculations using Newtonian gravity for the two-body problem, we can actually compute the theoretical result of this constant: $$ \frac{T^2}{a^3} = \frac{4\pi}{G}\frac{1}{M + m} \approx \frac{4\pi}{GM} $$

Where $M$ is the mass of planet and $m$ is the mass of moon. Assuming the planet mass will be far more than moon mass (after all, its a planet, and its a moon), the approximation is valid. Now, you have the mass of the planet.