Question

Explain the third law of Kepler.

Answer 1

Kepler discovered that the size of a planet's orbit (the semi-major axis of the ellipse) is simply related to the sidereal period of the orbit. If the size of the orbit (a) is expressed in astronomical units (1 AU equals the average distance between the Earth and the Sun) and the period (P) is measured in years, then Kepler's Third Law says:
$P^2\propto a^3$
After applying Newton's Laws of Motion and Newton's Law of Gravity we find that Kepler's Third Law takes a more general form:
$P^2=\frac{4{\pi }^2}{G(M_1+M_2)}{a}^3$
where $M_1$ and $M_2$ are the masses of the two orbiting objects in solar masses. Note that if the mass of one body, such as $M_1$, is much larger than the other, then $M_1+M_2$ is nearly equal to $M_1$. In our solar system $M_1=1$ solar mass and this equation become identical to the first.
Kepler's Third Law