Kepler's second law

Kepler's second law of planetary motion states that the radius vector connecting a planet to the Sun sweeps out equal areas in equal time intervals. This law is actually a manifestation of the conservation of angular momentum, and can be written in the form

$$\frac{d\theta}{dt} = \frac{h}{r^2},$$ (5)

where $r$ and $\theta$ are plane polar coordinates centered on the Sun (see Fig. 3), and $h$ is the planet's constant angular momentum per unit mass (about an axis perpendicular to the plane of the orbit, passing through the Sun). Equations (2) and (5) can be combined to give

$$\int_0^\theta \frac{d\theta}{(1+e\,\cos\theta)^2} = h'\,(t-t_0),$$ (6)

where $t_0$ is the time at which $\theta=0$ (i.e., the time at which the planet reaches its perihelion), and $h'$ is a constant.