# 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 semi-major axis of its orbit.

#### Key Points

• Kepler's third law can be represented symbolically as <equation contenteditable="false">$P^{2} \propto a^{3}$, where P is the orbital period of the planet and a is the semi-major axis of the orbit (see Figure 1).

• 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

• 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.

• The mean distance from the Earth to the Sun (the semi-major 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.-->

#### Figures

1. ##### 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 semi-major axis of the orbit.

2. ##### Understanding Kepler's 3 Laws and Orbits

In this video you will be introduced to Kepler's 3 laws and see how they are relevant to orbiting objects.

## 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 semi-major axis of its orbit Figure 2.

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

$P^{2} \propto a^{3}$,

where P is the orbital period of the planet and a is the semi-major axis of the orbit (see Figure 1).

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 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,

$F_{net}=ma_{c}=m\frac{v^{2}}{r}$.

The net external force on mass m is gravity, and so we substitute the force of gravity for Fnet:

$G \frac{mM}{r^{2}}=m \frac{v^{2}}{r}$.

The mass m cancels, as well as an r, yielding

$G\frac{M}{r}=v^{2}$.

The fact that m cancels out is another aspect of the oft-noted 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,

$v=\frac{2 \pi r}{P}$.

Substituting this into the previous equation gives

$G \frac{M}{r} = \frac{4 \pi r^{2}}{P^{2}}$.

Solving for P2 yields

$P^{2}=\frac{4 \pi ^{2} a^{3}}{GM}$.

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

$\frac{P_{1}^{2}}{P_{2}^{2}}=\frac{r_{1}^{3}}{r_{2}^{3}}$.

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 P2=4π2GM/r3 for the ratio r3/P2. We obtain a relationship that can be used to determine the mass M of a parent body from the orbits of its satellites:

$M=\frac{4 \pi^{2} r^{3}}{G P^{2}}$.

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 r3/T2 should be a constant for all satellites of the same parent body (because r3/T2=GM/4π2).

#### Key Term Glossary

acceleration
The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).
##### Appears in these related concepts:
Acceleration
the rate at which the velocity of a body changes with time
##### Appears in these related concepts:
astronomical unit
The mean distance from the Earth to the Sun (the semi-major 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.-->
##### Appears in these related concepts:
average
The arithmetic mean.
##### Appears in these related concepts:
axis
An imaginary line around which an object spins (an axis of rotation) or is symmetrically arranged (an axis of symmetry).
##### Appears in these related concepts:
centripetal
Directed or moving towards a center.
##### Appears in these related concepts:
circular motion
Motion in such a way that the path taken is that of a circle.
##### Appears in these related concepts:
circumference
The line that bounds a circle or other two-dimensional figure
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
##### Appears in these related concepts:
gravity
Resultant force on Earth's surface, of the attraction by the Earth's masses, and the centrifugal pseudo-force caused by the Earth's rotation.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
mass
The quantity of matter which a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
motion
A change of position with respect to time.
##### Appears in these related concepts:
period
The duration of one cycle in a repeating event.
##### Appears in these related concepts:
Period
The period is the duration of one cycle in a repeating event.
##### Appears in these related concepts:
planet
A large body which directly orbits any star (or star cluster) but which has not attained nuclear fusion.
##### Appears in these related concepts:
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.