Answer key: Kepler’s Laws Orbit Worksheet
For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Kepler’s Laws Orbit Worksheet.
Goal: Use Kepler’s laws to connect an orbit’s size and shape to its period and to the planet’s changing speed.
Completed results table
| Setting | Orbital period | Current speed | Star distance |
|---|---|---|---|
| Earth-like | 1 years | 30.3 km/s | 0.98 AU |
| Mars-like | 1.88 years | 26.51 km/s | 1.38 AU |
| a = 4 AU, e = 0 | 8 years | 14.89 km/s | 4 AU |
| Stretched orbit, 0% progress | 5.2 years | 51.59 km/s | 0.6 AU |
| Stretched orbit, 50% progress | 5.2 years | 5.73 km/s | 5.4 AU |
| a = 1 AU, e = 0, star mass 2 | 0.71 years | 42.12 km/s | 1 AU |
Answers
Kepler’s third law says P² = a³ for a Sun-like star (P in years, a in AU). Predict the period for a = 4 AU before checking your table. Show your work.
P² = 4³ = 64, so P = √64 = 8 years. The lab agrees.
How long is a year on Mars, in Earth years? Do planets farther from the Sun take more or less time per orbit?
About 1.88 Earth years (√(1.524³) ≈ 1.88). Farther planets take longer: their path is longer and they move more slowly.
In the stretched orbit, how fast is the planet moving at its closest point and at its farthest point? About how many times faster is it when close?
About 51.6 km/s at the closest point (0.6 AU) and 5.73 km/s at the farthest (5.4 AU), so it moves about 9 times faster near the star.
Kepler’s second law says the line from star to planet sweeps out equal areas in equal times. Use it to explain your answer to question 3.
Near the star the line is short, so the planet must travel farther each second to sweep the same area. Far away the line is long, so a shorter, slower step sweeps the same area. Gravity is also stronger close to the star and speeds the planet up as it falls inward.
The stretched orbit (a = 3 AU) takes about 5.2 years. Predict: if you set Eccentricity to 0 and keep a = 3 AU, will the period change? Test it.
No. The period stays at about 5.2 years (√(3³) ≈ 5.20) because it depends only on the semi-major axis and the star’s mass, not on the orbit’s shape.
What happened to the period when the star’s mass doubled? Why?
It dropped to about 0.71 years. A more massive star pulls harder, so the planet must move faster to stay in the same orbit and completes each lap sooner: P = √(a³ ÷ M) = √(1 ÷ 2) ≈ 0.71.