Download Kepler`s third law states that the square of the orbital period of a

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.
LEARNING OBJECTIVE [ edit ]
Apply Kepler's third law to describe planetary motion
KEY POINTS [ edit ]
Kepler's third law can be represented symbolically as P ∝
2
a
3
, where P is the orbital period of
the planet and a is the semi-major axis of the orbit (see.
The constant of proportionality is P
a
2
planet
3
planet
=
P
a
2
earth
3
= 1
yr
2
AU
earth
for asidereal year (yr),
3
and astronomical unit (AU).
Kepler's third law can be derived from Newton's laws of motionand the universal law of
gravitation. Set the force of gravityequal to the centripetal force. After substituting an expression
for the velocity of the planet, one can obtain: G
M
r
=
πr
4
P
2
2
which can also be written P
2
=
4
π
2
a
3
GM
Using the expression above we can obtain the mass of the parent body from the orbits of its
satellites: M
=
4
π
2
r
GP
2
3
.
TERMS [ edit ]
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.-->
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.
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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 .
The third law, published by Kepler in 1619,
captures the relationship between the
distance of planets from the Sun, and their
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orbital periods. Symbolically, the law can be expressed as
P
2
∝a ,
3
where P is the orbital period of the planet and a is the semi-major axis of the orbit (see ).
a
rmin
rmax
Planet
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
2
P planet
3
a planet
P
=
2
earth
3
a earth
= 1
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
v
2
r
.
The net external force on mass m is gravity, and so we substitute the force of gravity for Fnet:
G
mM
r
v
= m
2
2
r
.
The mass m cancels, as well as an r, yielding
G
M
= v
r
.
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 sameacceleration. 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 =
πr
.
2
P
Substituting this into the previous equation gives
G
M
r
πr
4
=
P
2
.
2
Solving for P2 yields
P
2
=
4
π
2
a
GM
3
.
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
2
3
2
P1
P
2
3
=
2
r1
r
.
3
2
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/r3for 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 =
4
π
2
r
GP
3
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).
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.