A line that connects a planet to the sun sweeps out equal areas in equal times.-physicsknow

The Law of Areas-PhysicsKnow

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the sun, then slower when farther from the sun. Kepler's second law states that the blue sector has constant area.

Kepler-second-law

This is one of Kepler's laws.This empirical law discovered by Kepler arises from conservation of angular momentum. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time.

Developing Kepler's Law of Areas-PhysicsKnow Law of Areas: Considering the area of an elliptical orbit, an infinitesemal area element can be expressed as Integrating over r from the focus outward gives So the time rate of change of the area swept out is The velocity is in the plane of the ellipse and can be divided into radial and angular components: Note that the angular component is proportional to the rate of change of area so that Note that the radius r and the angular velocity are perpendicular to each other so that their product is equal to the magnitude of their vector product. But it was shown in the development of the Law of Orbits that this vector product is proportional to the angular momentum L. It was also shown there that the angular momentum L is a constant, so that establishes that the rate of change of area is a constant for all parts of the orbit, the Law of Areas.

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