Kepler’s Second Law

Kepler's Second Law states that a line segment joining a planet to the Sun sweeps out equal areas in equal intervals of time. This means that a planet does not travel at a constant speed along its orbit — it speeds up when closer to the Sun and slows down when farther away. Published by Johannes Kepler in 1609, this law was derived empirically from the meticulous astronomical observations of Tycho Brahe, long before Newton provided a physical explanation rooted in gravity and angular momentum. The physical reason behind Kepler's Second Law is the conservation of angular momentum. Because gravity always acts along the line joining the planet to the Sun (it is a central force), it exerts no torque on the planet. With no torque acting, the planet's angular momentum cannot change — it is conserved. This conservation directly forces the areal velocity (the rate at which area is swept out) to remain constant throughout the orbit, whether the planet is near or far from the Sun. This law applies to any object orbiting under a central gravitational force, not just planets. Comets with highly elliptical orbits provide spectacular demonstrations: they zip through the inner solar system at enormous speed, sweeping large areas quickly near the Sun, then crawl slowly through the outer solar system, sweeping the same area but over a much longer time.

Key takeaways
  • Defining Areal Velocity
  • Linking Area to Angular Momentum
  • Why Angular Momentum is Conserved
  • Speed Variation Around the Orbit

Defining Areal Velocity

Imagine the planet moves a tiny distance along its orbit in a short time Δt. The planet, the Sun, and the planet's new position form a thin triangle. The area of this triangle is ½ × base × height. As Δt shrinks to zero, this area divided by Δt gives the areal velocity: dA/dt. Kepler's Second Law says dA/dt is constant throughout the orbit.

Linking Area to Angular Momentum

The angular momentum of a planet of mass m moving with velocity v at distance r from the Sun is L = m·r·v⊥, where v⊥ is the component of velocity perpendicular to the radius vector. The tiny triangular area swept in time Δt is ΔA = ½·r·(v⊥·Δt). Dividing by Δt gives dA/dt = ½·r·v⊥ = L/(2m). Since L and m are both constant, dA/dt is constant — the law is a direct consequence of angular momentum conservation.

Why Angular Momentum is Conserved

Gravity is a central force: it always points from the planet directly toward the Sun. Torque is defined as τ = r × F. Because r and F are parallel (both along the same line), their cross product is zero, so τ = 0. Newton's second law for rotation tells us dL/dt = τ = 0, which means L is constant in time. No torque → no change in angular momentum → constant areal velocity.

Speed Variation Around the Orbit

At perihelion (closest approach, distance r_p), the planet's speed v_p is greatest. At aphelion (farthest point, distance r_a), its speed v_a is smallest. Since angular momentum is conserved and at both these points v⊥ = v (the velocity is exactly perpendicular to the radius), we get m·r_p·v_p = m·r_a·v_a, or equivalently r_p·v_p = r_a·v_a. A planet at half the aphelion distance must be moving twice as fast — an elegant and testable prediction.

The Key Equation

dAdt=L2m=constant\frac{dA}{dt} = \frac{L}{2m} = \text{constant}
  • \frac{dA}{dt} — Areal velocity (m² s⁻¹): The rate at which the line from the Sun to the planet sweeps out area.
  • L — Angular momentum (kg m² s⁻¹): The planet's rotational momentum about the Sun; conserved because gravity exerts no torque.
  • m — Mass of the planet (kg): The planet's mass, which is constant.

Worked Example

Problem. Earth orbits the Sun in an ellipse. At perihelion (closest point), Earth is r_p = 1.471 × 10¹¹ m from the Sun and travels at v_p = 3.029 × 10⁴ m s⁻¹. At aphelion (farthest point), Earth is r_a = 1.521 × 10¹¹ m from the Sun. Use conservation of angular momentum to find Earth's speed at aphelion.

va=rp vpra=(1.471×1011)(3.029×104)1.521×1011≈2.93×104 m s−1v_a = \frac{r_p \, v_p}{r_a} = \frac{(1.471 \times 10^{11})(3.029 \times 10^4)}{1.521 \times 10^{11}} \approx 2.93 \times 10^4 \text{ m s}^{-1}

Answer. Earth's speed at aphelion is approximately 2.93 × 10⁴ m s⁻¹ (29.3 km s⁻¹).

Figure
Diagram
SunPlanet ataphelionPlanet atperihelionA₂ (large area,slow speed)A₁ (small area,fast speed)A₁ = A₂ swept in equal time intervalsv (slow)v (fast)
Origins · 1 minJohannes Kepler and how this was discovered
JK
Johannes Kepler
1571–1630 · German

Kepler formulated his three laws of planetary motion between 1609 and 1619, derived from years of painstaking analysis of Tycho Brahe's precise observations. His Second Law — equal areas in equal times — was the first mathematical description of how a planet's speed varies along its elliptical orbit, published in 'Astronomia Nova' (1609). Kepler achieved this without knowledge of calculus or Newtonian mechanics, relying purely on geometry and arithmetic. Kepler initially tried to fit planetary orbits to nested Platonic solids (cubes, tetrahedra, etc.) inscribed within spheres — a beautiful but incorrect idea. It was only after this geometric model failed to match Brahe's data that he abandoned circular orbits and discovered the ellipse.

  • Newton's Law of Gravitation — Newton's gravitational force law, acting as the central force directed toward the Sun, is the physical cause of angular momentum conservation and therefore directly underlies Kepler's Second Law.
  • Kepler's First Law — Kepler's First Law establishes that planetary orbits are ellipses with the Sun at one focus — the geometric framework within which the Second Law's speed variation occurs.
  • Kepler's Third Law — Kepler's Third Law relates orbital period to semi-major axis and, together with the Second Law, gives a complete description of planetary motion and speed at any point.
  • Orbital Energy and Escape Velocity — Conservation of energy in orbits complements angular momentum conservation: together they determine the exact speed at every point of an elliptical orbit.