PX155 - E9 - orbital angular momentum

a satellite of mass $m_{s}$ orbiting the earth
gravity provides centripetal force, acts towards the centre of rotation of circular orbit
angular momentum of satellite $L = I ω = m_{s} R^{2} ω$
$F = \frac{G M_{E} m_{s}}{R^{2}} = m_{s} \frac{v^{2}}{R} ⟹ v = (\frac{G M_{E}}{R})^{\frac{1}{2}}$
what torque does this generate?
- $\vec{τ} = \vec{R} \times \vec{F} = R F \sin θ = 0 (∵ θ = π)$
centripetal force generates no torque about $O$
$\frac{d \vec{L}}{d t} = 0$ about $O$ , $∴ \vec{L}$ is conserved
- holds true for elliptical orbital as well