Question

How are Newton's laws related to Kepler's laws of planetary motion?

Answer 1

Kepler's 3rd law of planetary motion: $T^2\propto r^3$

Explanation:

We can relate Newton's universal law of gravitation to Kepler's third law of planetary motion.

In fact, Newton's universal law of gravitation states that the gravitational force between a planet and the Sun is:

$F=G\frac{Mm}{r^2}$

where

G is the gravitational constant

M is the mass of the Sun

m is the mass of the planet

r is the distance between the Sun and the planet

SInce the motion of the planet around the Sun is approximately circular, we can equate this force to the centripetal force:

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

where v is the orbital speed of the planet. Simplifying,

$\frac{GM}{r}=v^2$

The orbital speed of the planet is equal to the ratio between the length of the orbit ($2\pi r$) and the orbital period, T:

$v=\frac{2\pi r}{T}$

So the previous equation becomes

$\frac{GM}{r}=\frac{4\pi^2 r^2}{T^2}$

Which can be re-arranged as

$T^2 = \frac{4\pi^2}{GM}r^3$

The term $\frac{4\pi^2}{GM}$ is  constant for all planets, so the equation can be rewritten as

$T^2\propto r^3$

Which is exactly Kepler's third law of planetary motion, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit (here we approximated the orbit as a circle, so r representes the average radius of the orbit).

Learn more about Kepler's laws:

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