Astronomers discover four moons orbiting the exoplanet Zeus
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
Astronomers discover four moons orbiting the exoplanet Zeus. Their measured orbital data are listed below.
Moon
Orbital radius R (×106m)
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 MZ 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 MZ of Zeus, and then compute MZ. 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.