
Kepler's Laws of Planetary Motion
There is no object at the other focus of a planet's orbit. The semimajor axis, a, is the average distance between the planet and its star.
This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a, roughly, triangular area. This same amount of area will be swept every day.
As a planet travels in its elliptical orbit; its distance, from the Sun, will vary. As an equal area is swept, during any period of time; and since, the distance from a planet to it's orbiting star varies; one can conclude that in order for the area being swept to remain constant; that, a planet must vary in velocity. Planets move most rapidly when at perihelion and more slowly when at aphelion.
This law was developed, in part, from the observations of Brahe; which, indicated that the velocity, of planets, was not constant.
Kepler's Third Law (Harmonic Law)
The larger the distance (between a planet and its sun), a, the longer the sidereal period. By understanding this, and the second law, one can determine; that, the larger an orbit is  the slower the average velocity, of an orbiting object, will be (as the satellite will be consistently farther from the object being orbited).
Newton would modify this third law, noting that the period is also affected by the satellite's mass.
Not Just Applicable to Planets
The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a satellite revolving around Earth.
Kepler's Understanding of Said Laws
Kepler did not understand why his laws were correct, it was Isaac Newton who discovered the answer to this.
Newton's Form of Kepler's Third Law
Newton, understanding that his third law of motion was related to Kepler's third law of planetary motion, devised the following:
where:
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