Question

Consider a planet moving around a star in an elliptical orbit with period $T$. The area of the elliptical orbit is proportional to,

• A. $T^{\frac{4}{3}}$
• B. $T$
• C. $T^{\frac{2}{3}}$
• D. $T^{\frac{1}{2}}$

Answer 1

Answer: (A)

Area of ellipse
$A=\pi r_1{r}_2$
$\because r_1=a-ae=a(1-e)$
and $r_2=a+ae=a(1+e)$
$A=\pi \{a(1-e)\}\{a(1+e)\}$
$=\pi a^2(1-e)(1+e)$
$=\pi a^2\left(1^2-e^2\right)$
$\because e^2<<a^2$ then
$A=\pi a^2$
So, $a\propto A^{\sqrt{2}}$
According to Keplar's III law
$T^2\propto a^3$
$T^2\propto {\left[(A{)}^{1/2}\right]}^3$
$T^2\propto A^{3/2}$
$A\propto {\left(T^2\right)}^{2/3}$
$A\propto T^{4/3}$