Elliptical Orbits and Angular Momentum

Picture a comet on a long elliptical path. At its closest approach, perihelion, it rips past the Sun; out at aphelion, the far point, it barely crawls. Nothing mysterious is adding or removing energy at those two points, speed alone is changing with distance. The reason is angular momentum, L = m v r for a point moving perpendicular to its position vector, which happens to be true exactly at perihelion and aphelion since velocity is perpendicular to the radius there. Since L stays constant, v r stays constant: get closer, and speed must rise; move farther out, and speed must fall.

Why does L stay constant at all, given that gravity is clearly speeding the comet up and slowing it down? Torque, not force, changes angular momentum, and torque is force times the lever arm distance from the pivot. Gravity always points from the comet straight at the Sun, directly along the radius line. A force pointing straight at the pivot has zero lever arm, so it produces zero torque about the Sun no matter how strong it is. No torque means no change in angular momentum, full stop, even while the force itself is busy changing speed and direction. This is also the reasoning behind Kepler's equal-areas law: in a short time dt, the swept-out triangle has area proportional to r times the perpendicular velocity component, which is proportional to L. Constant L forces constant sweep rate, so the orbit sweeps equal areas in equal times, fast thin triangles near the Sun, slow fat ones far away.

This sits beside the rest of the unit as angular momentum conservation applied to a continuously changing radius, rather than a sudden pull-in like a spinning skater.