My textbook says that we can infer from Kepler's second law that angular momentum is conserved for a planet, and therefore gravity is a central force.
Now I understand how constant angular momentum implies that gravity is a central force. However, I don't see how we know that angular momentum is conserved, based on Kepler's second law.
My textbook describes Kepler's second law as follows: $$ \int_{t_1}^{t^2}rv_\phi\,\mathrm dt=C\int_{t_1}^{t_2}\mathrm dt=C(t_2-t_1), $$ where $C$ is a constant.
We see that $rv_\phi=r^2\dot\phi=C$. We also know that $|\vec{L}|=|\vec{r}\times\vec{p}|=rmv\sin\theta=mr^2\omega\sin\theta.$
Right, so we can assume $m$ is constant, and $r^2\omega$ as well, by Kepler's second law. What about $\theta$ though? How do we know $\theta$ is constant?
For circular orbits, I can see that $\theta=\frac{1}{2}\pi$, but how about elliptical orbits?
EDIT
Okay, I think I got it. We're considering a solid object (planet) rotating about a fixed axis of rotation, so technically, we should be using $\vec L=I\vec\omega$. But I guess we can approximate the moment of inertia for a planet as $mr^2$, considering the spatial dimensions we're working with. And therefore we get $|\vec L|=I|\vec \omega|=mr^2\omega=$ constant. Given that a planet doesn't 'turn' suddenly, we can also assume the direction of $\vec \omega$ being constant.