Confusion in explaining Kepler's second law in terms of energy

[thebosonbreaker]

Hello.
As I understand it, Kepler's 2nd law of planetary motion can be explained through conservation of energy or conservation of angular momentum.
I am having trouble with the conservation of energy explanation.

We know that the sum of potential and kinetic energy of a planet in orbit around the parent star (let's assume it's the sun) must be constant (for all points in the orbit) so that energy is conserved. This is fine.

By Kepler's 1st law, the orbit is elliptical. When the planet is closer to the sun, it is moving faster, so it must have more KE. But if it has more KE, it must have less PE, so that PE + KE stays constant.

What I don't understand is why the potential energy is less when the planet is closer to the sun. The potential energy of a body in the gravitational field of another body is given by the equation GMm / r, in which r is the distance between the bodies.

When the planet is closer to the sun, r is smaller. Does this not imply a larger potential energy (not a smaller one) and hence a smaller kinetic energy (not a larger one)? I am confused here!

Any help/clarification would be much appreciated.
Many thanks.

 

[PeroK]

The potential energy of a body in the gravitational field of another body is given by the equation GMm / r, in which r is the distance between the bodies.

It's $-\frac{GMm}{r}$

The minus is important.

 

[nasu]

Kepler's second law is a consequence of conservation of angular momentum. Conservation of energy cannot "explain" it.

 

[Filip Larsen]

When the planet is closer to the sun, r is smaller. Does this not imply a larger potential energy (not a smaller one) and hence a smaller kinetic energy (not a larger one)?

Potential energy means there is a potential or reservoir for kinetic energy "later" and by applying work you can increasing that potential energy, i.e "store" more energy into that reservoir. If you are out walking on a hill, you have to work for it to go upwards (you are increasing your gravitational potential energy) and visa versa going down-hill. This surely means potential energy must increase with height, right?

Also when dealing with potential energy it is also worth to remember that its all about work going in and out so potential energy in a particular system is determined except for a constant (of integration). If you describe the potential energy of a particular system as $V(x)$ then an equally valid potential is $V_C(x) = V(x) + C$, where $C$ is a constant. As PeroK mentioned, for gravity one usually write the potential with a minus and no constant, which effectively implies the convention that gravitational potential energy of a mass is zero when you are infinitely far away from it. If you are still confused try plot the gravitational potential function as a function of $r$ and notice if potential energy is increasing or decreasing when you increase $r$ (i.e. when you go up-hill).