Kepler's Third Law

We have seen that the radius vector connecting our planet to the origin sweeps out area at the constant rate $dA/dt=h/2$. [See Equation (1.314).] We have also seen that the planetary orbit is an ellipse. Suppose that the major and minor radii of the ellipse are $a$ and $b$, respectively. It follows that the area of the ellipse is $A=\pi\,a\,b$. Now, we expect the radius vector to sweep out the whole area of the ellipse in a single orbital period, $T$. Hence,

$$T = \frac{A}{(dA/dt)} = \frac{2\pi\,a\,b}{h}.$$ (1.322)

It follows from Equations (1.301), (1.302), and (1.321) that

$$T^{2} = \frac{4\pi^{2}\,a^{3}}{G\,M}.$$ (1.323)

In other words, the square of the orbital period of our planet is proportional to the cube of its orbital major radius. This is Kepler's third law of planetary motion. As is clear from Table 1.4, Kepler's third law very accurately describes the orbits of the planets in the solar system.

1pt

Table: 1.4 Orbits of planets in the solar system. $a$ - major radius; $e$ - eccentricity; $T$ - orbital period. Here, an astronomical unit (AU) is $1.496\times 10^{11}\,{\rm m}$.

Planet	$a({\rm AU})$	$e$	T( yr)	$T^2/a^3$
Mercury	0.3871	0.20564	0.241	1.0013
Venus	0.7233	0.00676	0.615	0.9995
Earth	1.0000	0.01673	1.000	1.0000
Mars	1.5237	0.09337	1.881	1.0002
Jupiter	5.2025	0.04854	11.87	1.0006
Saturn	9.5415	0.05551	29.47	0.9998
Uranus	19.188	0.04686	84.05	1.0000
Neptune	30.070	0.00895	164.9	1.0001