Kepler's Second Law
A line from a planet to the Sun sweeps equal areas in equal times.
Formula
Variables
Example
A planet speeds up near the Sun and slows down when far away.
How it works
Kepler's Second Law says that the rate at which the radius vector from a planet to the Sun sweeps out area stays constant during the orbit. This happens because gravity is a central force, so it exerts no torque about the Sun; as a result, angular momentum is conserved and the swept area per unit time remains fixed. The planet therefore moves faster when it is closer to the Sun and slower when it is farther away, but the area rate stays the same throughout the orbit.
When to use it
Use this law when analyzing motion in elliptical orbits, especially to compare a body's speed at different points along its path. It is also useful in celestial mechanics and any problem where conservation of angular momentum is being connected to orbital area swept per unit time.
Worked example, step by step
A spacecraft in an elliptical solar orbit sweeps out 12 km^2 of area in 3 s. What is its areal velocity? Use dA/dt = const, so areal velocity = area ÷ time A = 12 km^2 t = 3 s (dA/dt) = 12 km^2 / 3 s (dA/dt) = 4 km^2/s Final answer: the spacecraft's areal velocity is 4 km^2/s. If it later sweeps 20 km^2, the time required is t = A / (dA/dt) = 20 / 4 = 5 s.
Common mistakes to avoid
- Confusing swept area with orbital distance traveled along the path
- Thinking the planet's speed is constant instead of the area rate being constant
- Forgetting that the law applies to motion under a central force about the focus of the ellipse, not about any arbitrary point
Did You Know?
A planet sweeps equal areas in equal times, so it races fastest when closest to the Sun.
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