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The answer sheet for the worksheet Orbits, Kepler's laws and resonances. Each answer comes with the reason, so it can be used to mark or to go over the questions with the class.

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A-level · Ages 16 to 18 (Years 12 and 13)Answer sheet for teachers

Answers: Orbits, Kepler's laws and resonances

  1. D: Most travel time is spent drifting before reaching a resonanceSlow Yarkovsky drift to the resonance takes much longer than the final resonant push, matching the long exposure ages.
  2. D: FiveA 5:2 resonance means five asteroid orbits for every two of Jupiter's, so its period is 2/5 of Jupiter's.
  3. A: The surface takes time to warm, so it radiates most after noonAbsorbed sunlight is re-emitted as infrared slightly later, so the warmest surface, and the strongest emission, has rotated to the afternoon side.
  4. C: They act at changing points and largely average outWithout a simple period ratio, the planets line up at different points each time, so the tugs do not build up.
  5. A: The constant depends on the central mass, here Jupiter'sThe constant 4π²/GM equals 1 in these units only when M is the Sun's mass; a different central body needs a different constant.
  6. True (About a million years)Resonances act quickly by astronomical standards, raising eccentricity within about a million years.
  7. False: the right answer is About 4.0 yearsIt completes three orbits per Jupiter orbit, so T = 11.9 ÷ 3 ≈ 4.0 years. 2.5 is its distance in AU, not its period.
  8. The drift would change direction (accept any answer with the same meaning)Reversing the spin puts the afternoon side on the other side of the body, so the thrust, and the change in semi-major axis, reverse.
  9. Its semi-major axis (accept any answer with the same meaning)The tiny thrust gradually changes the semi-major axis, carrying the body inwards or outwards until it reaches a resonance.
  10. 3:1 (accept any answer with the same meaning)The 3:1 resonance has the shortest period (1/3 of Jupiter's), and by T² ∝ a³ the shortest period means the smallest orbit.

Task

Answers will vary. The task comes from the lesson: Calculate the semi-major axes for the 5:2 and 2:1 resonances with Jupiter. Then take a few asteroids from the asteroids section of this site, note their semi-major axes and orbital periods, and check how well T² = a³ holds for each.

Look for work that uses the key words correctly and gives a reason for each point.

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