Kepler's laws of planetary motion

Johannes Kepler's primary contribution to astronomy/astrophysics were the three laws of planetary motion. Kepler derived these laws, in part, by studying the observations of Brahe. Isaac Newton would later verify these laws with his laws of motion and universal gravity. The generic term for an orbiting object is "satellite".

Kepler's Laws of Planetary Motion

• Kepler's First Law (1609): The orbit, of a planet about a star, is an ellipse with the star at one focus.

• Kepler's Second Law (1609): A line joining a planet and its star, sweeps out equal areas during equal intervals of time.

• Kepler's Third Law (1618): The square of the sidereal period, of an orbiting planet, is directly proportional to the cube of the orbit's semimajor axis.

Kepler's First Law

• Kepler's First Law (1609): The orbit, of a planet about a star, is an ellipse with the star at one focus.

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.

Kepler's Second Law

• Kepler's Second Law (1609): A line joining a planet and its star, sweeps out equal areas during equal intervals of time.

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)

• Kepler's Third Law (1618): The square of the sidereal period, of an orbiting planet, is directly proportional to the cube of the orbit's semimajor axis.

• • P² = a³
  • P = object's sidereal period in years
  • a = object's semimajor axis, in AU

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:

$P^{2}={\frac {4\pi ^{2}}{G(m_{1}+m_{2})}}a^{3}$

where:

• P = object's sidereal period in years
• a = object's semimajor axis, in AU
• G = Gravitational Constant
• m₁ = mass of object 1
• m₂ = mass of object 2