Kepler's Laws of Planetary Motion

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

Kepler's first law

The orbit of every planet is an ellipse with the sun at one of the two foci.

Kepler's Second law

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Kepler's third law

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Kepler's third law formula

\(\large{ P^2 = a^2   }\)

Where:

\(\large{ P }\) = planet's distance from the sum

\(\large{ a }\) = semi-major axis of the planet's orbit

\(\large{ G }\) = universal gravitational constant

\(\large{ t }\) = satellite orbit period (time)

\(\large{ r }\) = satellite mean orbital radius

\(\large{ m }\) = planet's mass

Solve for:

\(\large{ t = \sqrt {   \frac {4\; \pi^2 \;r^3} {G\;m} }  }\)

\(\large{ r = \sqrt {   \frac {t^2 \;G\; m} {4\; \pi^2} }  }\)

\(\large{ m = \frac {4\; \pi^2\; r^3} {G\;t^2}  }\)

 

Tags: Equations for Gravity Equations for Motion