Angular Momentum in Orbits: Why Planets Speed Up Near the Sun

Gravity pulls the planet straight at the Sun, so the position vector r and the force F_g lie along one line pointing opposite ways — the angle between them is 180° at every instant. Torque about the Sun is τ = r F sin 180° = 0. The Sun pulls the planet but never twists it about itself, so angular momentum about the Sun cannot change.

Watch the graph: r and v are mirror images, and L = m v r sin θ stays pinned even on a scale that would show a 2% wobble. At the two turning points the velocity is perpendicular to r, so sin θ = 1 and the equation reduces to L = m v r. Equal values at both ends mean v_p / v_a = r_a / r_p. At e = 0.60 the planet is four times closer at perihelion (0.40 AU against 1.60 AU), so it travels four times faster there.

Now drive PLANET MASS m from 1.0 to 10.0. The path, the speed and the timing do not move at all — mass cancels out of the acceleration — while L scales with m exactly. Set e = 0 and the trade disappears: a circular orbit holds r and v constant, so every line on the graph goes flat.