Three satellites are revolving around the earth: one along the circular orbit and the other two along elliptical orbits whose major axes are equal to the diameter of the circular orbit. The masses of the satellites are same. Choose the CORRECT options:

For a satellite revolving around sun in an elliptical orbit, of eccentricity 'e' and semi major axis 'a'

Let r₁ and r₂ be the distances from the Sun, of the perihelion and aphelion points respectively.

${\mathrm{r}}_1=\mathrm{a}(1-\mathrm{e})\text{ and }{\mathrm{r}}_2=\mathrm{a}(1+\mathrm{e})$

$\text{ Total energy }=\mathrm{KE}\text{ at perihelion }+\mathrm{PE}\text{ at perihelion }$

$=\frac{1}{2}m{v}_1^2-\frac{GMm}{r_1}=\frac{1}{2}m{v}_1^2-\frac{GMm}{a(1-e)}$

By angular momentum conservation,

${\mathrm{mv}}_1{\mathrm{r}}_1={\mathrm{mv}}_2{\mathrm{r}}_2\Rightarrow {\mathrm{v}}_1(1-\mathrm{e})={\mathrm{v}}_2(1+\mathrm{e})$...(1)

By energy conservation, 

$\frac{1}{2}{\mathrm{mv}}_1^2-\frac{\mathrm{GMm}}{\mathrm{a}(1-\mathrm{e})}=\frac{1}{2}{\mathrm{mv}}_2^2-\frac{\mathrm{GMm}}{\mathrm{a}(1+\mathrm{e})}$..(2)

using (1) and (2).

${\mathrm{v}}_1=\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(\frac{1+\mathrm{e}}{1-\mathrm{e}}\right)}\text{ and }{\mathrm{v}}_2=\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(\frac{1-\mathrm{e}}{1+\mathrm{e}}\right)}$

Hence total energy $=\frac{1}{2}m{v}_1^2-\frac{GMm}{a(1-e)}=\frac{1}{2}m\frac{GM}{a}\left(\frac{1+e}{1-e}\right)-\frac{GMm}{a(1-e)}=-\frac{GMm}{2a}$.

So the total energy is same for all three orbits.

using the value of ${\mathrm{v}}_1=\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(\frac{1+\mathrm{e}}{1-\mathrm{e}}\right)} \mathrm{and} {\mathrm{r}}_1=\mathrm{a}(1-\mathrm{e})$.

Angular momentum is given by $\mathrm{L}=\mathrm{m}\times \sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(\frac{1+\mathrm{e}}{1-\mathrm{e}}\right)}\times \mathrm{a}(1-\mathrm{e})=\mathrm{m}\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(1-{\mathrm{e}}^2\right)}$.

So the orbit having higher eccentricity will have lower angular momentum.

hence, Angular momentum of satellite-1 is maximum and Angular momentum of satellite-2 is more than satellite-3.