Elliptical Orbits: Energy, Escape Speed & Eccentricity
1 · Predict
A satellite is launched sideways at radius r about a central mass, at a speed below the circular-orbit speed. Does it trace a closed ellipse, hold a perfect circle, or fly off and never return?
- It settles into a closed, elongated ellipse and returns each orbit.
- It holds a perfect circle at the launch radius.
- It escapes the central mass and never comes back.
2 · Set Up
- Open the elliptical-orbit simulation and press Reset. The central mass sits at the middle of the stage with gravitational parameter GM = 40 m³/s².
- Enable the energy and eccentricity probes on the satellite so both invariants are shown live.
- For each trial below, place the satellite at radius r on the x-axis, give it a purely sideways (tangential) launch speed v, press Play, and let it complete at least one orbit before recording the settled probe readings.
3 · Collect Data
| Gravitational parameter GM (m³/s²) | Launch radius r (m) | Tangential launch speed v (m/s) | Escape speed v_esc (m/s) | Specific orbital energy ε (J/kg) | Eccentricity e |
|---|---|---|---|---|---|
| 40 | 2 | 4 | |||
| 40 | 3 | 3 | |||
| 40 | 4 | 2.4 |
Plot the measured eccentricity e (y-axis) against the specific orbital energy ε (x-axis). As ε climbs toward zero, what happens to the orbit's shape?
4 · Analyze
- Compute the specific orbital energy ε = v²/2 − GM/r for each row and confirm the sign. What does a negative ε tell you about whether the satellite is bound to the central mass?
- Compare each launch speed v to the escape speed v_esc you computed. The launch is tangential, so h = r·v; show that e = √(1 + 2εh²/GM²) reproduces the eccentricity you read off the sim, and explain why a launch nearer escape speed gives a more elongated ellipse.
5 · Extend
- For each launch radius, what tangential speed would turn the ellipse into a perfect circle? (Hint: v_circ = √(GM/r).) Is your table's launch speed above or below that value, and how does that explain the orbit's shape?
- Long-period comets have eccentricities close to 1. What does that imply about their speed at closest approach relative to the local escape speed — and why don't they quite escape the Sun?
The Physics Behind This Experiment
Escape Speed
The minimum tangential speed at radius r that would let the satellite escape the central mass. A launch below this speed keeps ε negative and the orbit bound — an ellipse rather than an unbound trajectory.
Circular Orbital Speed
The tangential speed that produces a perfect circle at radius r. Launching slower than this (as in every trial here) drops the satellite inward after release, tracing an ellipse with the launch point as its far apoapsis.