Kepler's Laws: Definition & Equations

[Greg Bernhardt]

Definition/Summary

I. Each planet moves in an ellipse which has the Sun at one of its foci
II. The radius vector of each planet passes over equal areas in equal intervals of time.
III. The cubes of the mean distances of any two planets form the Sun are to each other as the squares of their periodic times.

Equations

$$\frac{a_1^3}{a_2^3} ~::~\frac{P_1^2}{P_2^2}$$

Extended explanation

The Second law is also known as the "Law of areas"

The Third law is also known as the "Harmonic law"

The Third law is only approximate and only closely holds if the Sun is vastly more massive than the planets.

If is not, then the relative masses of the Sun and planets must be taken into account and the relationship becomes:

$$\frac{a_1^3}{a_2^3} ~::~\frac{P_1^2(M+m_1)}{P_2^2(M+m_2)}$$

where M is the mass of the Sun and m₁ & m₂ are the masses of the respective planets.

Newton's law of gravitation:

Kepler's laws combined with centripetal acceleration ($-\omega^2r$) enabled Newton (and others) to obtain the inverse-square law of gravitation:

Kepler: $$\omega^2 :: 1/T^2 :: 1/r^3$$

Newton: $$F :: \omega^2r :: r/r^3 = 1/r^2$$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Aaron Paulo]

Thanks for the overview of Kepler's Laws!