## Kepler’s Laws Revisited

Previous: Ellipse

Kepler thought that, in the Solar System, the planets rotate in elliptical trajectories with the center of the Sun in one focus. Newtonian mechanics unveiled that Kepler’s assumption was not fully correct.

The Sun is not located in one focus, but a bit further from it. In truth, it is the center of mass (CM) of the system Planet-Sun, which is in the focus.

This was very difficult to notice by simple observation, because in one hand the eccentricity of the orbits of the planets is very small. The orbits are almost circular. And in the other hand the mass of the Sun is much larger than that of the planets, which locates the Sun and the CM of the solar system very close together.

But again, this assessment shows the power of Newton’s Laws to describe the universe.

Kepler’s Laws can be revisited and generalized on the light on Newtonian Mechanics, considering the system planet-Sun as an interacting two-body problem.

1st Law: Both objects in a binary orbit move around the CM in ellipses, with the center of mass occupying one focus of each ellipse.

The trick is: A interacting two-body system can be treated as a one-body system with the reduced mass μ moving around a fixed mass M (fixed at the CM) at a distance r. Or what is the same, a mass μ subject to a central potential V:

Useful formulae to convert an n-body problem into a “center of mass” problem:

Useful formulae for central force problems (like the CM-μ system):

The total force F acting on the CM is zero, thus the CM can be taken as an inertial reference frame, which is very convenient. This means: we can arbitrarily set it to the position R = 0.

The angular momentum L of the reduced mass is constant in this type of systems! Don’t even ask for the angular momentum of the CM…

After some playing around, one reaches the 2nd of Kepler’s laws.

Note to the reader: I omit lots of mathematical developments of the formulae for the sake of brevity. I want the content as straightforwardly as possible and so I just go to the end result. For in-depth step by step obtention of the formulae, check a textbook.

2nd Law: The time rate of change of the area A (that is, dA/dt) swept out by a line connecting a planet to the focus of an ellipse is constant.

Additional things that we find out are:

This v is the relative velocity between masses 1 and 2.

The point being: these magnitudes depend on the parameters of the ellipse (a and e), on the parameters of the system (that is, masses), and on the relative position (which is a function of time).

3rd Law:

We obtain the precise dependence between P and a, it depends only the masses. This is powerful, because we can use it to obtain the masses of objects orbiting around something else (planets, moons, stars, galaxies…).

Virial theorem: For gravitational systems in equilibrium, the total energy E is always negative and one-half of the time-averaged potential energy U.

• If the system is in equilibrium, the total energy must be constant.
• If the total energy is constant, there must be a constant relationship between the kinetic energy K and the potential energy U.

This works for gravitational systems in equilibrium, and is widely applicable, from ideal gases to clusters of galaxies.