To derive the expression for Kepler's second law, we can start with the conservation of angular momentum, which states that the product of the moment of inertia and the angular velocity of a planet around the sun remains constant.
Mathematically, we can express this as:
r^2 * θ' = h
where r is the distance from the sun to the planet, θ is the angle between the planet's position and a reference direction, θ' is the rate of change of θ with respect to time, and h is a constant known as the specific angular momentum.
If we divide both sides of this equation by 2, we get:
r^2 * θ' / 2 = h / 2
which tells us that the area swept out by the planet in time t is:
A = 1/2 * r^2 * θ
If we differentiate this expression with respect to time, we get:
dA/dt = 1/2 * r^2 * θ'
But since r^2 * θ' is equal to h, we can rewrite this as:
dA/dt = h/2
which tells us that the rate at which the planet sweeps out area is constant. In other words, the planet moves faster when it is closer to the sun and slower when it is farther away, but it always sweeps out equal areas in equal times.
This is the expression for Kepler's second law.