Kepler’s Laws - Law of Areas or Kepler's Second Law

The area swept by the planet of mass m in a given interval Δt is:

\[\Delta\vec{A}=\frac{1}{2}(\vec{r}\times\vec{v}\Delta t)\]

• \[\vec r\]: Position vector of the planet (distance from Sun).
• \[\vec v\]: Velocity vector of the planet.
• Δt: Time interval.
• \[\vec p\]: Linear momentum (\[\vec p\] = m\[\vec v\])
• \[\vec L\]: Angular momentum (\[\vec L\] = \[\vec r\] × \[\vec p\])

Kepler's Second Law (Law of Equal Areas)

• A line joining the planet and the Sun sweeps out equal areas in equal time intervals.
• When the planet is nearer the Sun (perihelion), it moves faster.
• When the planet is farther from the Sun (aphelion), it moves more slowly.
• This law reflects conservation of angular momentum.

• Kepler’s Second Law helps us understand how planets move around the Sun.
• The law talks about the area covered by the line joining a planet and the Sun in equal time intervals.
• It explains why planets change speed in their orbits.
• This law is also called the Law of Areas.

• Planets do not move with uniform speed around the Sun.
• Planets move faster when nearer to the Sun.
• Planets move more slowly when farther from the Sun.
• The law is explained by the areas swept in equal intervals of time being equal.
• The law is a result of the conservation of angular momentum.
• It applies to any central force.

• Explains the changing speed of planets in their orbits.
• Shows gravity acts as a central force.
• Supports that angular momentum is conserved.
• Applies to all objects under a central force, not just planets.

• Artificial satellites also follow the law of areas as they orbit planets.
• Comets move faster when nearer the Sun and slower when farther away, following this law.