# [Physics] Kepler’s second law implies angular momentum is constant

angular momentumconservation-lawsnewtonian-gravitynewtonian-mechanicsorbital-motion

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.

The Kepler's second law states that the radius vector from the Sun to the planet sweeps equal areas in equal times. In another words, the rate of change $\frac{dA}{dt}$ is constant. Consider the figure below,
The are element is $dA=\frac{1}{2}r^2d\theta$ so in the time interval $dt$ we have $$\frac{d\theta}{dt}=\frac{2}{r^2}\frac{dA}{dt},$$ On the other hand the angular momentum magnitude (with respect to $O$) is $L=mr^2\dot\theta$. Thus, $$L=2m\frac{dA}{dt},$$ which is constant.
However this does not prove that the vector $\vec L$ is constant. To prove that the vector does not change its direction one has to assumeeither the first Keppler's law (which implies the orbit lies in a plane) or that the force is central (which automatically implies in the angular momentum conservation).