Laws of Kepler

Question

 

We have discussed in class that Kepler posed his groundbreaking theory that planets travel in elliptical orbits about the sun, the importance of the conics was affirmed. Also The 3rd Law of Kepler says there is a constant c for all planets such that T2/a3=c , where T is the period of the planet and a is the semi-major axis of its orbit. Also, the line segment that connects a planet to the sun sweeps out equal areas in equal times.

 

 

What do the Laws of Kepler say about their speed as the function of the radius r of their orbit?

 

Also I need some help to prove Kepler's Second Law for particles moving in a central vector field. We are supposed to 

Differentiate the vector product r×v ( r is the position of the particle and v is its velocity, both depend on time t ) with respect to time t and compute the result. Recall the size of r×v is twice the area of the triangle with sides r and v . Notice that, in a short time Δt , the area swept by the radius pointing from the center to the particle is 0.5⋅|r×v|⋅Δt

 

 

If someone could assist me with these problems that would be great! I feel like I have them started but I can't seem to make the connections. 

 

 

Answer 1

The key physical insight for both of your questions is conservation of (angular) momentum. Since you stated Kepler's Third Law in its original proportional form, I'll assume that you're meant to do this without Newton's gravity formula unless otherwise specified.

Starting with the K2 proof: angular momentum is defined as L = r×p, where p = m·v. Taking just the r×v term, differentiate with respect to time using the product rule, making sure to preserve the correct vector order. From there, you can recognize that dr/dt is just v and dv/dt is just the acceleration, which for a central vector field points radially inward and is antiparallel to r. The cross products thus end up being r×a and v×v, both of which are identically zero, and therefore r×v is constant. You can then relate the swept area to that cross product as noted in the problem statement, recognizing that Δx = v·Δt is the short side of a very skinny triangle or parallelogram, with r being the long side.

This also largely addresses your first question: for a body in a given orbit, K2 directly implies that orbital speed is greater at smaller distances. The exact relation for elliptical orbits is described by the vis-viva equation: v² = µ·(2/r - 1/a), where µ = G·M is the gravitational parameter. You can derive this relation directly from conservation of energy and angular momentum, and I would refer you to the Wikipedia article for a full derivation and use cases if you're interested. Hope that helps!