Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
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.<! 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.>

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: Mechanical Work and Electrical Energy, Gauss's Law, and Models, Theories, and Laws
 Newton's Second Law
 Appears in these related concepts: Driven Oscillations and Resonance, Momentum, Force, and Newton's Second Law, and Matter Exists in Space and Time
 acceleration
 Appears in these related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and Graphical Interpretation
 axis
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Area Between Curves, and Components of a Vector
 centripetal
 Appears in these related concepts: Centripetal Force, Kinematics of UCM, and Banked and Unbacked Highway Curves
 circular motion
 Appears in these related concepts: Water Waves, Sinusoidal Nature of Simple Harmonic Motion, and XRay Diffraction
 circumference
 Appears in these related concepts: Eratosthenes' Experiment, Simple Harmonic Motion and Uniform Circular Motion, and Using Interference to Read CDs and DVDs
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
 Appears in these related concepts: Work, Force, and Force of Muscle Contraction
 gravity
 Appears in these related concepts: Properties of Electric Charges, Defining Graviational Potential Energy, and Key Points: Range, Symmetry, Maximum Height
 mass
 Appears in these related concepts: Mass Spectrometer, Mass, and Pop Art
 motion
 Appears in these related concepts: Motion with Constant Acceleration, Newton and His Laws, and Motion Diagrams
 period
 Appears in these related concepts: The Periodic Table, Number of Periods, and Atomic Size
 planet
 Appears in these related concepts: Force at an Angle to Displacement, Dynamics of UCM, and Mars and a Biosphere
 velocity
 Appears in these related concepts: Rolling Without Slipping, RootMeanSquare Speed, and Applications and ProblemSolving
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Kepler's Third Law.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 26 Nov. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/uniformcircularmotionandgravitation5/keplerslaws56/keplersthirdlaw26711197/