View source for Kepler's laws of planetary motion

[[de:Keplersche Gesetze]] [[fr:Lois de Kepler]] [[pl:Prawa Keplera]] [[zh:&#24320;&#26222;&#21202;&#23450;&#24459;]]
[[Johannes Kepler]]'s primary contribution to [[astronomy]]/[[astrophysics]] were the '''three laws of planetary motion'''. Kepler derived these laws, in part, by studying the [[observation]]s of [[Tycho Brahe|Brahe]]. [[Isaac Newton]] would later verify these laws with his [[Newton's laws of motion|laws of motion]] and [[Law of Universal Gravitation|universal gravity]]. The generic term for an orbiting object is "[[satellite]]".

==Kepler's Laws of Planetary Motion==

* <b>Kepler's First Law (1609):</b>  ''The [[orbit]], of a [[planet]] about a [[star]], is an [[ellipse]] with the star at one [[focus]]. ''

* <b>Kepler's Second Law (1609):</b> ''A [[line]] joining a planet and its star, sweeps out equal [[area]]s during equal intervals of [[time]].''

* <b>Kepler's Third Law (1618):</b>  ''The [[square]] of the [[orbital period|sidereal period]], of an orbiting planet, is directly [[proportion]]al to the [[cube]] of the orbit's [[axis|semimajor axis]]. ''

==Kepler's First Law==

* <b>Kepler's First Law (1609):</b>  ''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==

* <b>Kepler's Second Law (1609):</b> ''A [[line]] joining a planet and its star, sweeps out equal [[area]]s during equal intervals of [[time]].''

This is also known as the '''law of equal areas'''. Suppose a planet takes 1 [[day]] to travel from [[point]]s ''A'' to ''B''. During this [[time]], an imaginary line, from the Sun to the planet, will sweep out a, roughly, [[triangle|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 '''[[periapsis|perihelion]]''' and more slowly when at '''[[apoapsis|aphelion]]'''.

This law was developed, in part, from the observations of [[Tycho Brahe|Brahe]]; which, indicated that the velocity, of planets, was not constant.

==Kepler's Third Law (Harmonic Law)==

* <b>Kepler's Third Law (1618):</b>  ''The [[square]] of the [[orbital period|sidereal period]], of an orbiting planet, is directly [[proportion]]al to the [[cube]] of the orbit's [[axis|semimajor axis]]. ''

**P<sup>2</sup> = ''a''<sup>3</sup>
**P = object's sidereal period in years
**''a'' = object's semimajor axis, in [[astronomical unit|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==

[[Isaac Newton|Newton]], understanding that his [[Newton's laws of motion|third law of motion]] was related to Kepler's third law of planetary motion, devised the following:

:<math>P^2 = \frac{4\pi^2}{G(m_1 + m_2)}a^3</math>

where:
*P = object's sidereal period in years
*''a'' = object's semimajor axis, in AU
*G = [[physical constants|Gravitational Constant]]
*''m''<sub>1</sub> = mass of object 1
*''m''<sub>2</sub> = mass of object 2