Kepler's second law implies angular momentum is constant? - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-07-19T11:03:04Z https://physics.stackexchange.com/feeds/question/311386 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/311386 5 Kepler's second law implies angular momentum is constant? Sha Vuklia https://physics.stackexchange.com/users/140553 2017-02-11T15:03:52Z 2017-02-11T16:57:24Z <p>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.</p> <p>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.</p> <p>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.</p> <p>We see that $rv_\phi=r^2\dot\phi=C$. We also know that $|\vec{L}|=|\vec{r}\times\vec{p}|=rmv\sin\theta=mr^2\omega\sin\theta.$</p> <p>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?</p> <p>For circular orbits, I can see that $\theta=\frac{1}{2}\pi$, but how about elliptical orbits?</p> <p><strong>EDIT</strong></p> <p>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.</p> https://physics.stackexchange.com/questions/311386/-/311401#311401 1 Answer by Diracology for Kepler's second law implies angular momentum is constant? Diracology https://physics.stackexchange.com/users/63097 2017-02-11T16:03:20Z 2017-02-11T16:57:24Z <p>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, <a href="https://i.stack.imgur.com/Ze5iF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ze5iF.png" alt="enter image description here"></a></p> <p>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.</p> <p>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).</p> https://physics.stackexchange.com/questions/311386/-/311408#311408 1 Answer by David Hammen for Kepler's second law implies angular momentum is constant? David Hammen https://physics.stackexchange.com/users/52112 2017-02-11T16:26:07Z 2017-02-11T16:26:07Z <blockquote> <p>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.</p> </blockquote> <p>That alone says that the magnitude of angular momentum is constant.</p> <p>Your textbook's $v_\phi$ is the component of the velocity vector that is normal to the radial vector: $\vec v = v_r \hat r + v_\phi \hat \phi$. Thus $\vec L = m \vec r \times \vec v = m r v_\phi\,\hat r \times \hat \phi$. Since since $||\hat r \times \hat \phi|| \equiv 1$, the magnitude of a planet's angular momentum vector is $||\vec L|| = m r v_\phi$. Since mass is constant and since $\int_{t_1}^{t_2} r v_\phi dt = C(t_2-t_1)$, the magnitude of the angular momentum vector is constant.</p> <p>To arrive at the angular momentum vector being constant, we need to know that it's direction is constant as well. This is a consequence of orbits being planar, which is part of Kepler's first law.</p>