Kepler's Third Law
Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit.
Learning Objective

Apply Kepler's third law to describe planetary motion
Key Points
 Kepler's third law can be represented symbolically as
$P^{2} \propto a^{3}$ , where P is the orbital period of the planet and a is the semimajor axis of the orbit (see.  The constant of proportionality is
$\frac{P_{planet}^{2}}{a_{planet}^{3}}=\frac{P_{earth}^{2}}{a_{earth}^{3}}=1\frac{yr^{2}}{AU^{3}}$ for a sidereal year (yr), and astronomical unit (AU).  Kepler's third law can be derived from Newton's laws of motion and the universal law of gravitation. Set the force of gravity equal to the centripetal force. After substituting an expression for the velocity of the planet, one can obtain:
$G \frac{M}{r} = \frac{4 \pi r^{2}}{P^{2}}$ which can also be written$P^{2}=\frac{4 \pi ^{2} a^{3}}{GM}$ .  Using the expression above we can obtain the mass of the parent body from the orbits of its satellites:
$M=\frac{4 \pi^{2} r^{3}}{G P^{2}}$ .
Terms

astronomical unit
The mean distance from the Earth to the Sun (the semimajor axis of Earth's orbit), approximately 149,600,000 kilometres (symbol AU), used to measure distances in the solar system.

sidereal year
The orbital period of the Earth; a measure of the time it takes for the Sun to return to the same position with respect to the stars of the celestial sphere. A sidereal year is about 20.4 minutes longer than the tropical year due to precession of the equinoxes.
Full Text
Kepler's Third Law
Kepler's third law states:
The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit .
The third law, published by Kepler in 1619, captures the relationship between the distance of planets from the Sun, and their orbital periods. Symbolically, the law can be expressed as
where P is the orbital period of the planet and a is the semimajor axis of the orbit (see ).
Kepler's Third Law
Kepler's third law states that the square of the period of the orbit of a planet about the Sun is proportional to the cube of the semimajor axis of the orbit.
The constant of proportionality is
for a sidereal year (yr), and astronomical unit (AU).
Kepler enunciated 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. Therefore, it used to be known as the harmonic law.
Derivation of Kepler's Third Law
We can derive Kepler's third law by starting with Newton's laws of motion and the universal law of gravitation. We can therefore demonstrate that the force of gravity is the cause of Kepler's laws.
Consider a circular orbit of a small mass m around a large mass M. Gravity supplies the centripetal force to mass m. Starting with Newton's second law applied to circular motion,
The net external force on mass m is gravity, and so we substitute the force of gravity for F_{net}:
The mass m cancels, as well as an r, yielding
The fact that m cancels out is another aspect of the oftnoted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius r, all masses orbit at the same speed. This was implied by the result of the preceding worked example. Now, to get at Kepler's third law, we must get the period P into the equation. By definition, period P is the time for one complete orbit. Now the average speed v is the circumference divided by the period—that is,
Substituting this into the previous equation gives
Solving for P^{2} yields
Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields
This is Kepler's third law. Note that Kepler's third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body M cancel.
Now consider what one would get when solving P^{2}=4π^{2}GM/r^{3} for the ratio r^{3}/P^{2}. We obtain a relationship that can be used to determine the mass M of a parent body from the orbits of its satellites:
If r and P are known for a satellite, then the mass M of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio r^{3}/T^{2} should be a constant for all satellites of the same parent body (because r^{3}/T^{2}=GM/4π^{2}).
Key Term Reference
 Law
 Appears in these related concepts: Physics and Other Fields, Damped Harmonic Motion, and Models, Theories, and Laws
 Newton's Second Law
 Appears in these related concepts: Momentum, Force, and Newton's Second Law, Matter Exists in Space and Time, and Relationship Between Torque and Angular Acceleration
 acceleration
 Appears in these related concepts: Position, Displacement, Velocity, and Acceleration as Vectors, Scientific Applications of Quadratic Functions, and Centripetial Acceleration
 axis
 Appears in these related concepts: Area Between Curves, Regional Terms and Axes, and Components of a Vector
 centripetal
 Appears in these related concepts: Energy of a Bohr Orbit, Centripetal Force, and Overview of NonUniform Circular Motion
 circular motion
 Appears in these related concepts: Water Waves, Sinusoidal Nature of Simple Harmonic Motion, and B.3 Chapter 3
 circumference
 Appears in these related concepts: Simple Harmonic Motion and Uniform Circular Motion, Radians, and Eratosthenes' Experiment
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 gravity
 Appears in these related concepts: Defining Graviational Potential Energy, Key Points: Range, Symmetry, Maximum Height, and Properties of Electric Charges
 mass
 Appears in these related concepts: Mass Spectrometer, Pop Art, and Mass
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 period
 Appears in these related concepts: Frequency of Sound Waves, Sine and Cosine as Functions, and Tangent as a Function
 planet
 Appears in these related concepts: Dynamics of UCM, Mars and a Biosphere, and Ocean Floor
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
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