Kepler's Laws

Kepler's Laws

First Law - Planets orbit the the sun in elliptical orbits, using the sun as one of the ellipses two foci.

Second Law - The areas swept over by a vector (line) drawn from the sun to a planet are proportioned to the times of describing them.

Third Law - The squares of the times of revolution of two planets are in the ratio of the cubes of their mean distances. From data obtained, as orbital radius (R) increases, the period (on cycle, or T), gets bigger. The relationship between R and T were found to be:

                       K = R3
                           T2

Where:

R = orbital radius (m) - center of body 1 to center of body 2
T = period (s) - time per orbit
K = Kepler's Constant (m³/s²) - Constant only in the system it was measured in.

Just a small addition and a small correction to the Third Law..

Since planets travel in an elliptical orbit there is no constant radius, and for the purposes of Kepler's Third Law, R, the orbital radius takes the value of the semi-major axis of the elliptical path. K is a constant used in planetary mechanics usually denote GMm, in which G is Newton's Gravitational Constant, M is the mass of the larger body, and m is the mass of the smaller body. In this instance, the constant of proportionality referred to by WarMachine as K is defined as (4*pi^2)/GM, where G and M are defined earlier in my write-up.

First Law: The orbit of a planet about the sun is an ellipse with the Sun at one focus.

Second Law: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Third Law: The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit.

In the case of our solar system, the Third Law can be stated mathematically as:

P² = a³

where
P = sidereal period of planet (in years)
a = planet's semimajor axis (in AU)

Newton's formulation of the Third Law is more complicated but allows calculations for any pair of bodies, not just the Sun and the planets. It is stated as:

P² = (4pi²a³)/(G(m₁ + m₂))

where
P = sidereal period of orbit (in seconds)
a = semimajor axis of orbit (in meters)
m₁, m₂ = masses of two bodies (in kg)
G = gravitational constant (approximately 6.67 x 10^-11 m³kg^-1s^-2

This can be used to calculate the period of any pair of orbiting bodies. However, this is usually easily determined by empirical observation. A more useful formulation is:

m₁ + m₂ = (4pi²a³)/(GP²)

This allows the combined mass of any pair of bodies to be calculated from their orbital distance and sidereal period. As the ratio of their masses can be calculated from the position of their centre of gravity, the individual masses can hence be determined.