Kepler's Second Law of Planetary Motion

Kepler's second law states that a planet's radius vector sweeps out equal areas in equal time intervals. If a planet travels from point A to A1 in the same time it travels from B to B1, the sector areas covered are equal. This means planets move faster when closer to the Sun and slower when farther away, maintaining constant sector velocity throughout their elliptical orbit.

Sector velocity equals half the product of the planet's radial distance squared and its angular velocity. Mathematically, this equals the planet's angular momentum divided by twice its mass. Since sector velocity remains constant in an elliptical orbit, Kepler's second law directly implies that angular momentum is conserved throughout the planet's orbital motion.

In an elliptical orbit, a planet's distance from the Sun varies. At perigee, the planet is closest to the Sun; at apogee, it is farthest. Since sector velocity must remain constant to maintain equal areas in equal times, the planet must move faster when closer to the Sun and slower when farther away. This speed variation ensures the area swept by the radius vector stays constant.

The sector area dA swept in time dt equals half the product of radial distance r and angular displacement rdθ. The sector velocity dA/dt equals half of r²(dθ/dt), where dθ/dt is angular velocity. This constant sector velocity across different orbital positions demonstrates that the planet's angular momentum remains conserved throughout its elliptical orbit.

In an elliptical orbit, a planet's total energy remains conserved throughout its revolution around the Sun. This energy conservation explains why planets slow down at apogee, where they are farthest from the Sun and have maximum potential energy, and speed up at perigee, where they are closest and have maximum kinetic energy. The trade-off between kinetic and potential energy maintains constant total energy.

Angular momentum conservation means the quantity mr²ω remains constant throughout a planet's elliptical orbit. Since sector velocity is proportional to angular momentum, constant sector velocity directly implies constant angular momentum. This conservation law is a fundamental consequence of Kepler's second law and reflects the absence of external torques acting on the planet-Sun system.

Kepler's second law uses the equal areas-equal times principle to reveal how planets orbit. By comparing sector areas swept during identical time intervals at different orbital positions, the law shows that planets maintain constant sector velocity. This geometric principle elegantly demonstrates that orbital motion is governed by conservation of angular momentum and energy, fundamental to understanding planetary dynamics.