The Rhythm of the Spheres: Orbital Mechanics

Part 1: Engage

Every 76 years, Halley’s Comet visits the inner solar system. Unlike the planets, which move in nearly perfect circles, Halley’s Comet travels in an extremely elongated (eccentric) path. It spends most of its time in the cold, dark outer reaches of the solar system, but when it nears the Sun, it accelerates to incredible speeds.

The Puzzling Question: Why does the comet change speed? Why doesn’t it just fly away from the Sun when it’s moving so fast, or fall into the Sun when it’s moving so slow?

Write your initial hypothesis:

Part 2: Explore

Open the Orbital Motion & Kepler’s Laws Simulation.

Investigation A: The Shape of an Orbit

  1. Set the Eccentricity to 0. Observe the shape of the orbit.
  2. Slowly increase the Eccentricity (e) from 0 to 0.9.
    • What happens to the location of the Sun (the focus) as the eccentricity increases?
  3. Set the simulation to “Comet” mode (high eccentricity).
  4. Watch the velocity vector as the object approaches the Sun (perihelion) and moves away (aphelion).
    • Where is the object moving the FASTEST? ______
    • Where is it moving the SLOWEST? ______

Investigation B: Orbital Period (Kepler’s 3rd Law)

  1. Observe the Semi-Major Axis (a) and the Orbital Period (T).
  2. Double the distance (a). Does the period (T) also double, or does it increase by more than double?
  3. Record three sets of data (a, T) and calculate $a^3 / T^2$.
    • What do you notice about the relationship between these numbers?

Part 3: Explain

  1. Gravity and Distance: Explain how the strength of gravitational pull changes as a comet moves from aphelion (far) to perihelion (near).
  2. Conservation of Momentum: Explain why the comet must speed up as it gets closer to the Sun to maintain its orbit.
  3. Predicting Motion: If you know the distance of a new planet from the Sun is 4 Astronomical Units (AU), how could you use Kepler’s 3rd Law to predict how long its “year” will be?

Part 4: Elaborate & Evaluate

The Voyager Challenge: NASA wants to send a probe to Neptune. Using the simulation, determine which is more fuel-efficient: A) Launching a probe in a circular orbit that slowly expands. B) Launching a probe in a highly eccentric “Hohmann Transfer” orbit.

Construct an Evidence-Based Argument: Using the “Energy” graph in the simulation, explain why increasing eccentricity is the fastest way to travel to the outer solar system.

(Space for scientific argument)