Angular Momentum in Orbits

Gravity from the Sun always points straight at the Sun, along the line connecting the two bodies. A force that points along the radius vector has zero torque about that center, because torque is r x F, and the cross product of a vector with something parallel to it is zero. No torque means no change in angular momentum. So for any orbit, no matter how stretched out the ellipse is, L = mvr sin(theta) stays fixed, where theta is the angle between the velocity and the radius vector.

At the two special points of the orbit, perihelion (closest) and aphelion (farthest), the velocity is perpendicular to the radius, so sin(theta) = 1 and the formula simplifies to L = mvr. Setting the perihelion value equal to the aphelion value gives v_peri r_peri = v_aph r_aph. Closer to the Sun, r is small, so v must be large to keep the product fixed. That is why a comet whips through perihelion and crawls through aphelion.

This same conservation law, restated in terms of geometry instead of speed, is Kepler's second law: the line from the Sun to the planet sweeps out equal areas in equal times. A thin wedge swept in time dt has area proportional to r times the component of velocity perpendicular to r, which is exactly the quantity angular momentum is built from. And the physics is identical to a skater pulling her arms in: reducing r forces v to rise so that the product mvr does not change, whether the central force is gravity or the skater's own muscles.