Problems

01

Kepler’s Second Law

This law states that the area swept out by a planet (in sectors drawn from the sun) is proportional to the time of travel (equal time, equal area), regardless of the point in the orbit. Verify this law in two steps:

• (a) Show that $\mathbf{n}=r^2{\theta }^{\prime }(t){\mathbf{e}}_{\mathbf{k}}$ and thus $n=r^2{\theta }^{\prime }(t)$.
• (b) Let $A(t)$ be the area swept out in the time interval $[t_0,t]$ for a fixed $t_0$. By calculating this area in polar coordinates, show that $A^{\prime }(t)=\frac{1}{2}{r}^2{\theta }^{\prime }(t)$, and thus $A^{\prime }(t)=\frac{1}{2}n$, a constant.

02

Kepler’s Third Law

This law states that the total orbit time squared is proportional to the major axis cubed. Verify this law in three steps:

Let $T$ be the total orbit time, and $2a$ the major axis, and $2b$ the minor axis.

• (a) Use the fact that $A^{\prime }(t)=\frac{1}{2}n$ to show $T=\frac{1}{n}2\pi ab$.
• (b) Show that $\frac{n^2}{GM}=\frac{b^2}{a}$ by converting the polar form of an ellipse to the rectangular form.
• (c) Show that $T^2=\left(\frac{4{\pi }^2}{GM}\right)a^3$, and note that $\left(\frac{4{\pi }^2}{GM}\right)$ does not depend on the planet.