Conservation of angular momentum and Kepler's second law

The angular momentum $\vec{L}$ of a planet with respect to the star is conserved, and the reason is geometric: the gravitational force is always directed along the line joining the planet and the star. Taken about the star as the pole, this force has zero lever arm and therefore produces no torque. With no external torque, the angular momentum stays constant throughout the orbit.

Kepler's second law as a consequence

Conservation of $\vec{L}$ is equivalent to Kepler’s second law: the planet sweeps out equal areas in equal times. At the points of closest approach (perihelion) and greatest distance (aphelion) the velocity is perpendicular to the line joining planet and star, so the lever arm is $d$ and $L = m\,v\,d$.

This relation has a powerful, intuitive consequence: where the planet is closer to the star it must move faster, and where it is further away it moves slower, because the product $v\,d$ must remain constant.

At the extreme points of an orbit

$L_\text{perielio} = m v_p d_p \qquad L_\text{afelio} = m v_a d_a$ Equating the two (same $L$): $\;v_p d_p = v_a d_a$. Closer means faster.

Angular momentum is thus the second conserved quantity of the orbit, alongside energy. Together, energy and angular momentum are enough to solve almost all orbital problems without having to integrate the equations of motion.

Links

Topics: Gravitazione Concepts: Momento angolare · Conservazione del momento angolare · Leggi di Keplero Skills: Scelta del polo

Related exercises: Comet at perihelion and aphelion · True or false on energy and orbits · Elliptical orbit — perihelion, aphelion and the variation of speed