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.