Kepler S Third Law Of Motion Formula

Kepler discovered that the size of a planet s orbit the semi major axis of the ellipse is simply related to sidereal period of the orbit.

Kepler s third law of motion formula.

Murray and dermott solar system dynamics cambridge university press 1999 isbn 0 521 57597 4. In orbital mechanics kepler s equation relates various geometric properties of the orbit of a body subject to a central force. The position vector from the sun to a planet sweeps out area at a constant rate. T 2 a 3.

Kepler s third law kepler s third law of planetary motion. If the size of the orbit a is expressed in astronomical units 1 au equals the average distance between the earth and the sun and the period p is measured in years then kepler s third law says. The square of the period of a planet around the sun is proportional to the cube of the average distance between the planet and the sun. Kepler s third law sometimes referred to as the law of harmonies compares the orbital period and radius of orbit of a planet to those of other planets.

In satellite orbits and energy we derived kepler s third law for the special case of a circular orbit. See for example pages 161 164 of meriam j l. Encyclopædia britannica inc patrick o neill rileythe usefulness of kepler s laws extends to the motions of natural and artificial satellites as well as to stellar systems and extrasolar planets. Kepler s third law of planetary motion the square of the period of any planet about the sun is proportional to the cube of the planet s mean distance from the sun kepler s 3rd law equation.

Let us prove this result for circular orbits. Consider a cartesian coordinate system with the sun at the. Orbital velocity formula is used to calculate the orbital velocity of planet with mass m and radius r. Equation 13 8 gives us the period of a circular orbit of radius r about earth.

It was first derived by johannes kepler in 1609 in chapter 60 of his astronomia nova and in book v of his epitome of copernican astronomy 1621 kepler proposed an iterative solution to the equation. Kepler s third law states that the square of the period is proportional to the cube of the semi major axis of the orbit. Using the equations of newton s law of gravitation and laws of motion kepler s third law takes a more general form. P 2 frac 4 pi 2 g m1 m2 a 3 where m1 and m2 are the masses of the orbiting objects.

A derivation of kepler s third law of planetary motion is a standard topic in engineering mechanics classes. Kepler s third law is generalised after applying newton s law of gravity and laws of motion. The equation has played an important role in the history of both.