Kepler's First Law: Exploring Its Derivation and Problems

[UJS]

The textbook I'm using states Kepler's first law in the following form: all planets move in elliptical paths with the sun at one of the foci. If I'm understanding this claim correctly, I've got some problems with it..

This conclusion was reached using a potential that depends only on the distance between the two objects. In that case (no external forces), the center of mass of the two-body system shouldn't accelerate. But with one stationary object and another circling around it, this can never be the case. It seems like an approximation in which one of the two objects (the sun) is much more massive than the other, but I don't see that assumption appearing anywhere in the derivation.

The kinetic energy is first expressed in terms of the velocity of the center of mass and the relative velocity of the objects. In polar coordinates the Lagrangian leads to three equations of motion, and filling in the 1/r potential immediately gives elliptical paths.

What's going on?

 

[HallsofIvy]

The textbook I'm using states Kepler's first law in the following form: all planets move in elliptical paths with the sun at one of the foci. If I'm understanding this claim correctly, I've got some problems with it..

This conclusion was reached using a potential that depends only on the distance between the two objects. In that case (no external forces), the center of mass of the two-body system shouldn't accelerate. But with one stationary object and another circling around it, this can never be the case. It seems like an approximation in which one of the two objects (the sun) is much more massive than the other, but I don't see that assumption appearing anywhere in the derivation.

Yes, it is true that the "elliptic path" with the sun at one focus is based upon the approximation that the sun is much more massive than the orbiting body. It "appears" in the derivation with the assumption that the sun is a fixed point.

The kinetic energy is first expressed in terms of the velocity of the center of mass and the relative velocity of the objects. In polar coordinates the Lagrangian leads to three equations of motion, and filling in the 1/r potential immediately gives elliptical paths.

What's going on?

The "kinetic energy" of what? I suspect you mean the kinetic energy of the orbiting object. With the sun assumed stationary (typically at the origin of a coordinate system) then its kinetic energy is 0.

 

[pkleinod]

For two bodies with mass m1 and m2 respectively interacting via a central force, the problem separates into two single-particle problems, one involving a mass M=m1+m2 and one involving a "reduced mass" mu=m1m2/(m1+m2). The motion of particle 1 as viewed from particle 2 is the same as if particle 2 were fixed and particle 1 had mass mu. Kepler's law as stated in the textbook is a statement of what he observed. It is approximately true because the mass of the sun is so much larger than that of any single satellite.

 

[Shooting Star]

This conclusion was reached using a potential that depends only on the distance between the two objects.

The potential has to be specifically proportional to -1/r, not just depend only on the distance.