Kepler's Laws of Planetary Motion
Prove Kepler's Laws visually. Alter orbital eccentricity and watch as planets sweep out perfectly equal geometric areas in equal times, regardless of whether they are slingshotting past perihelion or crawling through aphelion.
WHAT ARE KEPLER'S LAWS?
Johannes Kepler described how planets orbit the sun using three laws. 1. **Ellipses**: Orbits are elliptical with the sun at one focus. 2. **Equal Areas**: A line connecting a planet to the sun sweeps out equal areas in equal times (conservation of angular momentum). 3. **Harmonic Law**: The square of the orbital period () is proportional to the cube of the orbital radius ().
THE CORE FORMULA
Kepler's Third Law for circular orbits derived from Newton's Universal Law of Gravitation: T^2 = left( rac{4pi^2}{GM} ight) r^3 Where is the mass of the central body.
HOW TO USE THIS VISUALIZATION
1. **Set Orbital Distance**: Move the planet closer or farther from the star. 2. **Adjust Eccentricity**: Change the orbit from circular to highly elliptical. 3. **Observe Velocity**: Watch the planet speed up at perihelion and slow down at aphelion. 4. **Check Areas**: Toggle the area sectors to see Kepler's Second Law in action.
CORE FORMULAS
AP EXAM CONNECTION
Unit: Unit 2: Force and Translational Dynamics (Topic 2.4)
Learning Objective: 2.4.2
COMMON MISCONCEPTIONS
- Thinking planets move at constant speed in elliptical orbits.
- Confusing the mass of the planet with the mass of the star in orbital equations.
- Forgetting that gravity provides the centripetal force for orbits.
KEY TAKEAWAYS
- Orbits follow elliptical paths.
- Angular momentum is conserved during an orbit.
- Period and distance have a power-law relationship.
- Gravity is the universal centripetal force.
PRACTICE QUESTIONS
Q1 (QUANTITATIVE): A planet moves from a distance to from its sun. By what factor does its orbital period change?
Show Answer & Explanation
Answer:
Explanation: , so if doubles, increases by . increases by .
