Summary of Kepler's Laws

Universal Laws

  • The laws presented can be applied to other stellar systems but are simplified for our solar system.

Kepler's First Law: Law of Ellipses

  • All planets orbit the sun in elliptical orbits.
  • Illustration of an Ellipse:
    • Draw an ellipse using thumbtacks (formal points) and string.
    • Foci: The two thumbtacks represent the foci of the ellipse, where one foci is empty and the other holds the sun.
  • Center of the Ellipse:
    • Located at the middle of the ellipse.
  • Types of Axes:
    • Semimajor Axis (a): Half of the longer axis.
    • Semiminor Axis: Half of the shorter axis.
    • Major and minor axes can be defined as the full lengths across the ellipse.
  • Characteristics:
    • If the two foci overlap, the shape becomes a circle (the case for circular orbits).
  • An elliptical orbit is defined by the distance between its foci.
Important Notes on Ellipses
  • All circles are ellipses but not all ellipses are circles.
  • Eccentricity (e): Defines how elliptical an orbit is.
    • Range from 0 (circle) to 1 (line).
    • Example Eccentricities:
    • e = 0.02 → Almost a perfect circle
    • e = 0.5 → Moderately elliptical
    • e = 0.9 → Very elongated ellipse
  • For the planets in our solar system:
    • None exceed an eccentricity of 0.5, implying they are very close to circular orbits.
    • Example: Pluto has the most extreme orbit.

Kepler's Second Law: Law of Equal Areas

  • A planet sweeps out equal areas in equal times.
  • **Understanding Movement: **
    • Over one month, a planet sweeps out an equal imaginary area regardless of deviation.
  • Speed Variation:
    • The speed of the planet changes based on its distance from the sun (center of the orbit).
    • Perihelion: Closest approach to the sun, maximum speed.
    • Aphelion: Farthest point from the sun, minimum speed.
    • The time taken to travel these areas is constant, but sectors may vary in shape.
  • Mathematical Relationships:
    • Speed is related to the distance traveled in a fixed time (Δx/Δt).
  • Visualizing the Concept:
    • Distances and areas (triangles of motion) change appearance but remain equal in area.
    • Change in speed happens as a planet travels faster when closer to the sun and slower when farther away.

Application of Newton's Law of Gravity (upcoming topic)

  • The relationship between distance to the Sun and planetary speed will be explored next week with Newton's law of gravity in conjunction with Kepler’s observed laws.

Homework & Group Activity

  • Students are encouraged to pull out their books for further exercises related to these concepts and to seek help from the instructor as needed.