Physics

Problem

Astronomers discover four moons orbiting the exoplanet Zeus. Their measured orbital data are listed below.

Moon	Orbital radius $R$ ( $\times10^{6}\,\text{m}$ )	Orbital period $T$ (days)
A	1.50	0.80
B	2.30	1.46
C	3.60	3.10
D	5.10	5.75

Assume the moons travel in nearly circular orbits and that the mass of each moon is negligible compared with the mass

M_Z

of Zeus. (a) Transform the data so that a single straight-line graph can be produced that tests the relationship between

T

and

R

for circular orbital motion. State clearly what quantities you would plot on each axis and explain why this transformation yields a linear graph. (b) Using the transformed data, calculate the numerical value of the slope of the best-fit line. Show all working, including any unit conversions required. (c) Derive an expression that relates the slope obtained in part (b) to the mass

M_Z

of Zeus, and then compute

M_Z

. Express your answer in kilograms. (d) Identify two possible systematic errors in the astronomical measurements that could cause the computed mass to differ from the true mass. For each error, indicate whether it would tend to make the calculated mass too large or too small, and justify your reasoning.

Intuitive explanation

This problem tests Kepler's third law, which relates orbital period to orbital radius for objects orbiting the same central body. The key insight is that $T^2 \propto R^3$ — so if you plot the data the right way (or check the ratio $T^2/R^3$ ), all four moons should give you the same constant, confirming they orbit Zeus. This lets you extract the planet's mass and verify the moons follow the same gravitational rules.

Intuitive explanation

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