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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.

Illustration of Kepler’s laws. The blue ellipse with a small blue sphere on its edge, and a bigger yellow sphere placed to the right from the ellipses’ middle represents the motion of a planet around the sun. The initial position of a blue planet is marked by number 1, its second position, a bit above the first one, is marked by number two, and the distance travelled by the planet from point 1 to point 2 is depicted by small letter t. The sun and two position points create a grey area in between depicted by capital letter A. Similarly, but on the other side of the ellipse, the next two planet positions are depicted by numbers 3 and 4, with a distance travelled by the planet expressed as small letter t. The grey area between the sun and position 3 and 4 is marked by capital letter B. The sun is placed closer to the first two positions than to the positions 3 and 4, thus the areas A and B have different shapes. Under the illustration, a derivation for the equation for the period of the orbiting object states that the period squared is equal to four times pi squared, divided by the gravitational force times mass of the big object, times radius of the orbit.

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.

Referred from:

Article
Derivation of the Law of Universal Gravitation