Monday, January 16, 2012

Lesson 21 Kepler's Three Laws

Studying the orbit of Mars around the Sun, Johannes Kepler discovered its path could be explained only if the planet traveled on an ellipse. This was the first of Kepler's three laws.

Eccentricity literally means somewhat off center while focus was used to describe a fireplace. Kepler utilized these words to describe characteristics of ellipses.

Kepler set out for the most accurate astronomical models and therefore sought Tycho Brahe. Brahe's family attempted to withold  Tycho's studies after his death, but stole them in order to "wage his war on mars," and develop his three laws to interpret the universe. Kepler embraced the Copernicun system and had to study the orbit of Mars on a planet that was not central and stationary, but instead in motion with all the other planets; a planet that spins on its axis in a non-circular fashion with constantly varying velocity, a planet with an unknown center.

Kepler studied the orbit of the Earth by keeping track of Mars's orbit year after year an using triangulation.

Kepler discoered that Mars moved faster when closer to the Sun and more slowly when farther away. This later coincided with Kepler's second law.

Every two years the Sun, Earth, and Mars are in opposition. In other words, whether seen from the Earth or the Sun, the position of Mars is the same. With observations year after year, Kepler triangulated the orbit of Mars. A circular orbit could not fit all the points of the orbit of Mars even though it could be off center of the Sun. In fact, the orbit of Mars was an ellipse with a very small eccentricity of .09. Any circle viewed obliquely is an ellipse.

Circles, ellipses, hyperbolas, and parabolas are all conic sections. A moving point traces out a conic section if its distance from some fixed point (focus) and its directrix are contant. The ratio of distances is eccentricity, where an ellipse has an eccentricity that is less than one, a circle has an eccentricity equal to zero, a parabola has an eccentricity equal to one, and a hperbola has an eccentricty that is greater than 1.

All conic sections can be described using an algebraic equation that is also useful for studying planetary motion:

r = ed/ ecos(theta)+1

Kepler utilized this equation and his study of the conic sections in order to develop his three laws.

Kepler's Three Laws:

1st - r = ed/ ecos(theta)+1

Each planet moves in an ellipse with the Sun at one focus.

2nd - dA/dt = constant

A line drawn from the Sun to a planet sweeps out equal areas in equal times.

3rd - T^2 = (4pi^2/GM) a^3

The square of the period of a planet's orbit is proportional to the cube of the length of the semi - major axis.

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