# Planetary Motion (Kepler's Laws)

Kepler's laws describe the motion of planets around the sun and were published between 1609 and 1619 by Johannes Kepler.

• Kepler’s first law

Every planet moves along an ellipse, with the sun located at a focus of the ellipse. To be more precise: sun and planets orbit their barycenter.

• Kepler’s second law

An imaginary line joining any planet to the sun sweeps out equal areas in equal times. This law is illustrated in Figure 1. The time it takes a planet to move from position 1 to 2, sweeping out area 'A' is exactly the time taken to move from position 3 to 4, sweeping area 'B', these areas are the same, A=B. As can be shown, this law is a consequence of conversation of angular momentum.

• Kepler’s third law

The square of the period of any planet is proportional to the cube of the semi-major axis of the orbit, i.e. T2 ~ a3 meaning that capital letter 'T' squared is proportional to small letter 'a' cubed, with 'T' denoting the period and a the semi-major axis of the orbit (see Figure 1). For the special case of a circular orbit (a=r) where 'a' is equal to 'r' this can be shown by equating the gravitational force with the centripetal force and substituting the orbital velocity, seen on figure 1, under the illustration.

Figure 1: Illustration of the first and second law. Note, that all planets, except Mercury, have nearly circular orbits and that the given illustration is highly exaggerated.