Kepler’s Second Law (Law of Areas)

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Kepler’s Second Law states:

“A line joining a planet and the Sun sweeps out equal areas in equal times.”

This law describes how a planet’s orbital speed varies as it moves along its elliptical path.

Areal Velocity

The areal velocity is defined as the rate at which area is swept out by the radius vector:

\frac{dA}{dt} = \frac{1}{2} r^2 \frac{d\theta}{dt}

Kepler’s Second Law asserts:

\frac{dA}{dt} = \text{constant}

This means the planet moves faster when it is closer to the Sun (periapsis) and slower when farther away (apoapsis).

Physical Origin: Conservation of Angular Momentum

Kepler’s Second Law is a direct consequence of angular momentum conservation in a central force field:

\mathbf{L} = m , \mathbf{r} \times \mathbf{v}

For central forces:

\frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{F} = 0

Thus, angular momentum \mathbf{L} is conserved. For planar motion:

L = m r^2 \frac{d\theta}{dt} = \text{constant}

Dividing both sides by 2m:

\frac{1}{2} r^2 \frac{d\theta}{dt} = \frac{L}{2m} = \text{constant}

This confirms equal areas are swept in equal times.

Geometric Interpretation

  • Near periapsis (closer to the Sun): r is small → \frac{d\theta}{dt} must be large → planet moves faster.
  • Near apoapsis (farther from the Sun): r is large → \frac{d\theta}{dt} is smaller → planet moves slower.

Example Calculation

If the planet sweeps an area A over time \Delta t:

A = \frac{L}{2m} \Delta t

This formula allows calculation of orbital time intervals given areas swept.

Significance

Kepler’s Second Law captures the non-uniform motion of planets along elliptical orbits. It reflects the conservation of angular momentum and provides a precise rule for predicting orbital speeds at different points. This law was crucial in moving away from the idea of uniform circular motion toward an accurate description of planetary dynamics.

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