Why Do Planets Speed Up When They Get Closer to the Sun?

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

THE SAME AXIS, ALWAYS
r FALLS, v RISES, L HOLDS
Net torque about the Sun τ = 0 ⇒ angular momentum about the Sun L is conserved
ANGULAR MOMENTUM ABOUT THE SUN
L = m · v · r · sin θ
6.0 × 2.094 × 1.6000 × 1.0000 = 20.1
 
DISTANCE r
1.6000 AU
Sun to planet
SPEED v
2.094 AU/yr
along the orbit
SIN θ
1.0000
θ between r and v
TORQUE τ ABOUT SUN
0.000
r F sin 180° = 0
ECCENTRICITY e0–0.8
PLANET MASS mrel.
PHYSICS INSIGHTS

Gravity pulls the planet straight at the Sun, so the position vector r and the force Fg 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 vp / va = ra / rp. 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.

L = m v r sin θ = constant  ·  a = 1.00 AU, GM = 4π² AU³/yr² Explore more physics at thesciencecube.com

Why do planets speed up when they get closer to the Sun?

In short: Gravity pulls the planet straight toward the Sun, along the line joining them, so it exerts no torque about the Sun: τ = rF sin 180° = 0. With zero net torque, the planet’s angular momentum about the Sun, L = mvr sin θ, stays constant. At the closest and farthest points the velocity is perpendicular to r, so L = mvr, and when r gets smaller v must get larger. The speeds at those two points satisfy vp ÷ va = ra ÷ rp.

Common doubts

  • “Gravity pulls on the planet, so why doesn’t angular momentum change?” Because the pull is directed straight at the Sun, there is no twisting effect about it. Gravity changes the planet’s speed and direction, but not its angular momentum about the Sun.
  • “What if the orbit is a perfect circle?” Then r never changes, so v never changes either. The planet speeds up and slows down only on an elliptical orbit.
  • “How much faster is it at the closest point?” By the ratio of the distances. At eccentricity e = 0.60 the planet is four times closer at perihelion than at aphelion, so it moves four times faster there.

Press PLAY in the simulation above and watch r and v change in opposite ways while L holds steady.

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