Kepler's Laws of Planetary Motion

The first person to understand exactly How the planets moved was Johannes Kepler (it would remain for Newton and gravity to understand Why). Before we talk about Kepler and his Laws of Planetary Motion, we must first talk about:

Tycho Brahe (1546 - 1601)

Yes, he did have a gold nose! (some historical accounts say it was silver not gold)

Kepler's Laws

Using Tycho's data (especially for Mars), Kepler discovered his three laws of planetary motion essentially by trial and error, i.e. he just kept changing the shape of the orbit and the planet's speed until he got something that matched the observational data. Kepler arrived at his first two laws in 1609; while the third law would wait until 1619.

Kepler's First Law

Geometry of an Ellipse:

How do we characterize the size and shape of an ellipse?

Size is given by the Semimajor Axis =

Shape is given by the Eccentricity =

Note:

So, for Mars, Kepler's First Law says:

Note: in reality the eccentricity of the Martian orbit =

If I drew such an ellipse, you would have trouble distinguishing it from a circle. It was only the extremely precise data of Tycho and the genius of Kepler that permitted the distinction between perfect circles and ellipses of small eccentricity.

Kepler's First Law gives the shapes of the planetary orbits, but how does the planet move in its orbit?

Kepler's Second Law

What the hell does that mean? Believe it or not, this statement precisely defines the way the speed of a planet varies in its elliptical orbit around the Sun. Let's see if we can understand how by considering the planetary orbit below:

IF:

and IF:

THEN:

Therefore, Kepler's Second Law means:

(For those of you who may have seen a little physics, Kepler's Second Law is equivalent to the Conservation of Angular Momentum.)

Kepler's Third Law

Where:

Period =

So, symbolically, Kepler's Third Law says:

NOTE: this is a proportionality; however, IF we use period in years and semimajor axis in AU's, THEN it is an equality.

FOR EXAMPLE: it was well known at the time of Kepler that the orbital period of Mars was:

Calculations like this could be easily done for all of the then known planets; thus, laying out the scale of the solar system.

Some Remarks about Kepler's Laws:

Kepler's Laws

Kepler's Laws

Kepler's Laws

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