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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 (249)]. 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,
\begin{displaymath}
T = \frac{A}{(dA/dt)} = \frac{2\pi\,a\,b}{h}.
\end{displaymath} (257)

It follows from Equations (236), (237), and (256) that
\begin{displaymath}
T^2 = \frac{4\pi^2\,a^3}{G\,M}.
\end{displaymath} (258)

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.

Note that for an elliptical orbit the closest distance to the Sun--the so-called perihelion distance--is [see Equations (236) and (255)]

\begin{displaymath}
r_p = \frac{r_c}{1+e} = a\,(1-e).
\end{displaymath} (259)

Likewise, the furthest distance from the Sun--the so-called aphelion distance--is
\begin{displaymath}
r_a = \frac{r_c}{1-e} = a\,(1+e).
\end{displaymath} (260)

It follows that the major radius, $a$, is simply the mean of the perihelion and aphelion distances,
\begin{displaymath}
a = \frac{r_p+r_a}{2}.
\end{displaymath} (261)

The parameter
\begin{displaymath}
e = \frac{r_a-r_p}{r_a+r_p}
\end{displaymath} (262)

is called the eccentricity, and measures the deviation of the orbit from circularity. Thus, $e=0$ corresponds to a circular orbit, whereas $e\rightarrow 1$ corresponds to an infinitely elongated elliptical orbit.

As is easily demonstrated from the above analysis, Kepler laws of planetary motion can be written in the convenient form

$\displaystyle r$ $\textstyle =$ $\displaystyle \frac{a\,(1-e^2)}{1-e\,\cos\theta},$ (263)
$\displaystyle r^2\,\dot{\theta}$ $\textstyle =$ $\displaystyle (1-e^2)^{1/2}\,n\,a^2,$ (264)
$\displaystyle G\,M$ $\textstyle =$ $\displaystyle n^2\,a^3,$ (265)

where $a$ is the mean orbital radius (i.e., the major radius), $e$ the orbital eccentricity, and $n=2\pi/T$ the mean orbital angular velocity.


next up previous
Next: Orbital Energies Up: Planetary Motion Previous: Kepler's First Law
Richard Fitzpatrick 2011-03-31