Understanding Kepler's 2nd Law: The Proof and Its Implications

In summary, the conversation discusses a proof of Kepler's 2nd law using the area of a sector of a circle or ellipse and the conservation of angular momentum. It is mentioned that the proof is not complete and needs to take into account the time derivative of the radius and theta, as well as the specific force involved.
  • #1
Master J
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Is my proof of Kepler's 2nd law correct?


area of sector of circle/ellipse (that the planet sweeps out): (1/2)(r^2)O

O is theta!

dA/dO = (1/2) (r^2)

dA= (1/2) (r^2) dO

dA/dt = (1/2) (r^2) dO/dt

It can't be that simple? Can it??
 
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  • #2
What have you proven? :) the second law states that dA/dt = const. why should (1/2) (r^2) dO/dt be a cnostant?
 
  • #3
dO/dt=omega is the angular velocity of the orbital motion, so *once* that you prove Kepler's law (for which you need the conservation of angular momentum), your formula says that when you decrease r you increase omega... as in fact happens :)
 
  • #4
Master J said:
Is my proof of Kepler's 2nd law correct?

dA= (1/2) (r^2) dO

dA/dt = (1/2) (r^2) dO/dt

It can't be that simple? Can it??

Is not. The derivative in respect to time is incomplete. The radius is not a constant unless is circular motion and then the problem is trivial anyway.

And the first three equations are a little bit redundant.
You have dA=1/2r^2*d(Theta) to start with. You cannot write the area itself this way (as a triangular segment) but only an infinitely small segment of area, dA.

Then you can take the the time derivative but both r and theta are time dependent.

Kepler's 2-nd law is a consequence of Newton's laws in the case of a central force. The above formula - for areal speed - is valid for any kind of motion, with any force. So it cannot give Kepler's law unless you introduce the specific force.

It's much easier to start with conservation of angular momentum - a consequence of central force motion.
 

1. What is Kepler's 2nd Law?

Kepler's 2nd Law, also known as the Law of Equal Areas, states that a line connecting a planet to the sun sweeps out equal areas in equal time intervals. In other words, a planet will travel faster when it is closer to the sun and slower when it is further away.

2. How was Kepler's 2nd Law proven?

Kepler's 2nd Law was proven using mathematical calculations and observations made by Kepler himself. He studied the movements of planets and observed that the areas swept out by a planet in a given time period were always equal, regardless of where the planet was in its orbit.

3. What are the implications of Kepler's 2nd Law?

Kepler's 2nd Law has major implications for our understanding of planetary motion and the structure of our solar system. It helps us understand why planets move at different speeds at different points in their orbits and provides a better understanding of the forces that govern the motion of planets.

4. How does Kepler's 2nd Law relate to Newton's Laws of Motion?

Kepler's 2nd Law is related to Newton's Laws of Motion in that it provides a physical explanation for why planets move the way they do. Newton's Laws of Motion, specifically the Law of Universal Gravitation, explain the forces at play in planetary motion and help us understand why planets follow elliptical orbits.

5. Can Kepler's 2nd Law be applied to other objects besides planets?

Yes, Kepler's 2nd Law can be applied to any object that follows an elliptical orbit around a central body. This can include comets, moons, and even artificial satellites. As long as the object is orbiting a central body, Kepler's 2nd Law can be used to understand its motion.

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