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A printable worksheet for the lesson Orbits, Kepler's laws and resonances (A-level). It prints on A4 without the website around it. The answers are on a separate sheet for the teacher.

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A-level · Ages 16 to 18 (Years 12 and 13)Howardites worksheet

Orbits, Kepler's laws and resonances

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Key words

Asteroid
A rocky or metal-rich object, smaller than a planet, that goes around the Sun.
Orbit
An orbit is the curved path one object follows around another under gravity, such as Earth around the Sun.
Jupiter
Jupiter is the largest planet in the Solar System, whose gravity shapes the asteroid belt and steers meteoroids towards Earth.
Astronomical unit
An astronomical unit is the average distance from Earth to the Sun, about 149.6 million kilometres.
Kirkwood gaps
Narrow zones in the main asteroid belt with very few asteroids, cleared by orbital resonances with Jupiter.
Eccentricity
Eccentricity measures how stretched an orbit is, from 0 for a perfect circle towards 1 for a long, thin ellipse.

Choose the right answer

Circle the letter of the right answer.

  1. Exposure ages of a few to tens of millions of years exceed the ~1 million years of resonant delivery. What does this suggest?
    • ACosmic rays only reach the inner Solar System
    • BMeteorites spend millions of years on Earth before discovery
    • CResonances take billions of years to act
    • DMost travel time is spent drifting before reaching a resonance
  2. An asteroid is in the 5:2 resonance with Jupiter. How many orbits does it complete while Jupiter completes two?
    • ATwo and a half
    • BTwo
    • CSeven
    • DFive
  3. Why does the Yarkovsky thrust come mainly from a rotating asteroid's afternoon side?
    • AThe surface takes time to warm, so it radiates most after noon
    • BSunlight is strongest in the afternoon in space
    • CThe solar wind blows hardest on that side
    • DThe afternoon side always faces Jupiter
  4. Away from any resonance, what happens to Jupiter's gravitational tugs on a main-belt asteroid?
    • AThey always push the asteroid towards the Sun
    • BThey grow steadily until the asteroid is ejected
    • CThey act at changing points and largely average out
    • DThey vanish, because Jupiter is too far away
  5. Why can T² = a³, with a in AU and T in years, not be used for a moon orbiting Jupiter?
    • AThe constant depends on the central mass, here Jupiter's
    • BKepler's law only works for asteroids
    • CJupiter's moons are too close for gravity to matter
    • DMoons follow elliptical orbits, not circular ones

True or false?

Is the answer given right? Tick True or False.

  1. Once a body drifts into a strong resonance, roughly how long does it take for its eccentricity to be pumped up?

    Answer given: About a million years TrueFalse

  2. What is the orbital period of an asteroid in the 3:1 resonance with Jupiter, whose period is about 11.9 years?

    Answer given: About 5.9 years TrueFalse

Write your answer

Answer in a word or a short sentence.

  1. If an asteroid's direction of spin were reversed, what would happen to its Yarkovsky drift?
  2. Which property of an asteroid's orbit does the Yarkovsky effect slowly change, letting it drift into a resonance?
  3. Of the 3:1, 5:2, 7:3 and 2:1 resonances with Jupiter, which lies closest to the Sun?

Task

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.

Howardites · free to copy for use in your classroom · answers are on the separate teacher's answer sheet