Orbital Motion: Kepler’s Laws and the Dance of Planets

Estimated Time: 45–60 minutes Materials: Internet-connected device, Orbital Motion & Kepler’s Laws Simulation, scratch paper or digital notebook.

Teacher Notes

Phenomenon: In 2006, Pluto was reclassified as a dwarf planet. Its highly elliptical orbit was one reason — it even crosses inside Neptune’s orbit! What makes some orbits nearly circular and others wildly elliptical?

NGSS Alignment (HS-ESS1-4):

Evidence Statements Addressed:

  1. Representation: Students construct and interpret representations of trajectories of orbiting bodies, including eccentricity $e = f/d$ (Kepler’s first law).
  2. Mathematical modeling: Students use the relationship $T^2 \propto a^3$ (Kepler’s third law) to model how orbital period relates to semi-major axis.
  3. Analysis: Students analyze motion using Kepler’s second law (equal areas in equal time), predict how orbital distance or period changes, and use Newton’s law of gravitation $F = G\frac{m_1 m_2}{r^2}$ to predict acceleration variation.

Engage

In 2006, the International Astronomical Union reclassified Pluto from a planet to a dwarf planet. One of the key reasons was Pluto’s unusual orbit: it is highly elliptical and actually crosses inside Neptune’s orbit for about 20 years out of its 248-year orbit. Meanwhile, Earth’s orbit is nearly circular.

  1. What do you think determines whether an orbit is a perfect circle or a stretched-out ellipse? Write your initial idea below. _________________

  2. Based on your intuition, do you think planets closer to the Sun move faster or slower than planets farther away? Why? _________________

Explore

Investigation 1: Orbital Shape and Eccentricity (Kepler’s First Law)

Open the Orbital Motion & Kepler’s Laws Simulation. You will see three sliders: Star Mass, Initial Distance, and Velocity Multiplier.

  1. Set Star Mass to 1.0, Initial Distance to 1.0 AU, and Velocity Multiplier to 1.00. Click Reset Orbit.
    • What shape does the orbit appear to be? Record the Eccentricity (e) reading.
    • Eccentricity = _____
  2. Keep Star Mass = 1.0 and Initial Distance = 1.0 AU. Increase the Velocity Multiplier to 1.20 and click Reset Orbit.
    • What happens to the shape of the orbit?
    • New eccentricity = _____
  3. Now try Velocity Multiplier = 0.70. Click Reset Orbit.
    • What happens to the shape? Is the orbit still stable?
    • New eccentricity = _____
  4. Fill in the table below by testing different Velocity Multiplier values:
Velocity Multiplier Eccentricity (e) Orbit Shape Description
0.70    
0.85    
1.00    
1.20    
1.40    
  1. Kepler’s first law states that planets orbit in ellipses with the Sun at one focus. The eccentricity $e = f/d$, where $f$ is the distance from center to focus and $d$ is the semi-major axis.
    • What value of Velocity Multiplier produces a circular orbit ($e \approx 0$)? _____
    • What happens when $e \ge 1$? (Try Velocity Multiplier = 1.50.) _____

Investigation 2: Equal Areas in Equal Time (Kepler’s Second Law)

  1. Set Star Mass = 1.0, Initial Distance = 1.0 AU, and Velocity Multiplier = 1.25. Click Reset Orbit.
  2. Watch the planet’s motion carefully. Observe its speed when it is close to the star (perihelion) versus far from the star (aphelion).
    • Where is the planet moving fastest? _____
    • Where is it moving slowest? _____
  3. Kepler’s second law says: a line from the planet to the star sweeps out equal areas in equal time intervals.
    • Explain in your own words why the planet must speed up when it is closer to the Sun. _________________
    • Use Newton’s law of universal gravitation: $F = G\frac{m_1 m_2}{r^2}$. How does the gravitational force change as the planet gets closer to the star? How does this explain the change in speed? _________________

Investigation 3: The Harmonic Law — $T^2 \propto a^3$ (Kepler’s Third Law)

  1. Reset the simulation to Star Mass = 1.0, Initial Distance = 1.0 AU, Velocity Multiplier = 1.00. Click Reset Orbit.
  2. Click Log Data to record this data point. Observe the values for semi-major axis ($a$), period ($T$), $a^3$, and $T^2$.
  3. Now change Initial Distance to 2.0 AU (Velocity Multiplier = 1.00). Click Reset Orbit, then Log Data.
  4. Repeat for the following distances, always keeping Velocity Multiplier = 1.00:
Semi-major Axis $a$ (AU) Period $T$ (Years) $a^3$ (AU³) $T^2$ (Years²) $a^3 / T^2$
1.0        
1.5        
2.0        
2.5        
3.0        

  1. Look at your data. What do you notice about the ratio $a^3 / T^2$ for each row? _________________

  2. Kepler’s third law states: $T^2 \propto a^3$. Did your data support this relationship? Explain. _________________

  3. Look at the T² vs a³ scatter chart in the simulation. What shape does the trend form? What does this tell you about the relationship? _________________

  4. Challenge: If you discovered a new planet in our solar system with a semi-major axis of 4.0 AU, what would you predict its orbital period to be? Use Kepler’s third law.

    Show your work: $T^2 = a^3$, so $T = \sqrt{a^3}$

    Predicted period for $a = 4.0$ AU: _____ Years

    Verify by testing in the simulation.

  5. Now set Star Mass to 2.0 (with Initial Distance = 1.0 AU, Velocity Multiplier = 1.00). Click Reset Orbit and Log Data.

    • What is the period now? _____
    • Compare this to the period when Star Mass = 1.0. How does the mass of the star affect the orbital period? _________________

Explain

Use your data and observations from the Explore section to answer the following:

  1. Kepler’s First Law: Describe the relationship between velocity multiplier and eccentricity. What does eccentricity tell us about the shape of an orbit? _________________

  2. Kepler’s Second Law: A comet travels in a highly elliptical orbit around the Sun. Using $F = G\frac{m_1 m_2}{r^2}$, explain why the comet moves fastest at perihelion (closest approach) and slowest at aphelion (farthest point). _________________

  3. Kepler’s Third Law: State the mathematical relationship between orbital period ($T$) and semi-major axis ($a$). If Planet X is 4 times farther from its star than Planet Y, how many times longer is its orbital year? _________________

Elaborate

Pluto has a semi-major axis of 39.5 AU and an orbital period of 248 years. Its eccentricity is 0.248 — noticeably higher than Earth’s 0.017.

  1. Use the simulation with Star Mass = 1.0. Can you find a velocity multiplier that produces an eccentricity close to Pluto’s ($e \approx 0.25$) at $a \approx 40$ AU?
    • Velocity Multiplier needed: _____
    • Resulting eccentricity: _____

  2. Why do you think Pluto’s eccentric orbit was one factor in its reclassification as a dwarf planet? Consider what might happen if a planet’s orbit crosses another planet’s path. _________________

  3. The semi-major axis ($a$) and eccentricity ($e$) together define the size and shape of an orbit. The closest approach (perihelion) distance is $a(1-e)$, and the farthest (aphelion) distance is $a(1+e)$.
    • For Pluto ($a = 39.5$ AU, $e = 0.248$):
      • Perihelion = _____ AU
      • Aphelion = _____ AU
    • Neptune orbits at about 30 AU. Does Pluto ever get closer to the Sun than Neptune? Does this confirm the phenomenon we started with? _________________

Evaluate (Deliverable)

Construct an Evidence-Based Scientific Explanation answering the driving question:

Why do some orbits (like Earth’s) remain nearly circular while others (like Pluto’s) become highly elliptical, and how do orbital distance and period relate?

Your explanation must include: