Kepler's Third Law - with Radial Acceleration

Description

In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1.The orbit of a planet is an ellipse with the Sun at one of the two foci.
2.A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3.The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are almost circles, so it is not apparent that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, also have elliptical orbits.

Kepler’s work improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets’ speeds varied, and using elliptical orbits rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler’s would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation.

Kepler’s laws are part of the foundation of modern astronomy and physics.

Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the “music of the spheres” according to precise laws, and express it in terms of musical notation. So it was known as the harmonic law.

The modern formulation, with the constant evaluated and in relation to the radial acceleration, is shown here.

The value M+m changes for each planet, so the proportionality constant is not truly the same. Nevertheless, given that m is so small relative to M for planets in our solar system, the approximation M+m~M is good.

Related formulas

Variables

ωradial velocity (1/s)
GNewtonian constant of gravitation
Mmass of larger body (kg)
rdistance between centers of mass (m)