Question

A satellite of mass $m$ orbits the earth in an elliptical orbit having aphelion distance $r_{a}$ and perihelion distance $r_{p}$. The period of the orbit is $T$. The semi-major and semi-minor axes of the ellipse are $\frac{r_{a}+r_{p}}{2}$ and $\sqrt{r_{p} r_{a}}$ respectively. The angular momentum of the satellite is:

• A. $\frac{m \pi\left(r_{a}+r_{p}\right) \sqrt{r_{a} r_{p}}}{T}$
• B. $\frac{2 m \pi\left(r_{a}+r_{p}\right) \sqrt{r_{a} r_{p}}}{T}$
• C. $\frac{m \pi\left(r_{a}+r_{p}\right) \sqrt{r_{a} r_{p}}}{2 T}$
• D. $\frac{m \pi\left(r_{a}+r_{p}\right) \sqrt{r_{a} r_{p}}}{4 T}$

Answer 1

Answer: (A)

By Kepler's second law we use
$\frac{L}{2 M}=\frac{\pi a b}{T}$
where $a$ is semi major and $b$ is semi minor axis of elliptical orbit.
$\Rightarrow L=\frac{m \pi\left(r_{a}+r_{p}\right) \sqrt{r_{a} r_{p}}}{T}$