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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

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.

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.<! do not include encyclopaedic information; this is not necessary to the definition: By definition, when used to describe the motion of bodies within solar systems, the heliocentric gravitational constant is equal to (0.017 202 098 95)² AU³/d² (IERS Conventions (1996), D. D. McCarthy ed., IERS Technical Note 21, Observatoire de Paris, July 1996); hence 1 AU = 149,597,870,691±30 km.>
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 ).
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 this related concepts: Physics and Other Fields, Damped Harmonic Motion, and Models, Theories, and Laws
 acceleration
 Appears in this related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and Applications and ProblemSolving
 axis
 Appears in this related concepts: Area Between Curves, Components of a Vector, and Conservation of Angular Momentum
 centripetal
 Appears in this related concepts: Centripetal Force, Banked and Unbacked Highway Curves, and Overview of NonUniform Circular Motion
 circular motion
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 circumference
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 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
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 gravity
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 mass
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 motion
 Appears in this related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 period
 Appears in this related concepts: Number of Periods, Atomic Size, and Frequency of Sound Waves
 planet
 Appears in this related concepts: Dynamics of UCM, Mars and a Biosphere, and Ocean Floor
 velocity
 Appears in this related concepts: RootMeanSquare Speed, Tangent and Velocity Problems, and Distribution of Molecular Speeds and Collision Frequency
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Source: Boundless. “Kepler's Third Law.” Boundless Physics. Boundless, 27 Jun. 2014. Retrieved 30 Apr. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/uniformcircularmotionandgravitation5/keplerslaws56/keplersthirdlaw26711197/