Friday, December 27, 2013

SIDEBAR: Kepler

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Sidebar: Johannes Kepler

A Short Biography. Johannes Kepler was born at 2:30 PM on December 27, 1571, in Weil der Stadt, Württemburg, (in now Germany). Though sickly and from a poor family, his obvious intelligence earned him a scholarship to the University of Tübingen to study for the Lutheran ministry. There, he learned and delighted in the ideas of Copernicus.

During his life, Kepler worked as a teacher and mathematician, along with duties as court astrologer and astronomer. However, he is best known for his work with the last great astronomical observer of the pre-telescope age, Tycho Brahe. Brahe spent years carefully measuring the position of astronomical objects, including Mars. His goal was to understand the motions of the planets.

Tyco spent over twenty years meticulously collecting observational data on all the visible planets. Tyco's data were the best available before the invention of the telescope. While his celestial measurement skills were superb, Brahe lacked math skills to analyze his collected data. This gave Kepler an opportunity to use his considerable mathematical expertise.

Kepler Works on Motion of Mars. Brahe gave Kepler the data for Mars, including measurements of Mars at successive oppositions over a period of many years. Of all the planets, the predicted position of Mars had the largest errors and therefore posed the greatest problem. Mars's motion never quite fit that of a circular orbit, nor an orbit of epicycles (multiple circles upon circles, a model favored by Brahe and other contemporary astronomers). After ten years of work, Kepler determined the orbit of Mars to be an ellipse.

Kepler's calculations were based on observational data and added mathematical credibility to Copernicus's Solar centric concept. This marks the beginning of modern, scientific astronomy. He based his explanations upon observations, rather than making the observations fit an assumed model of the universe. Today, we call this the scientific method.

In 1601, Kepler inherited Tycho's post as Imperial Mathematician. He spent the rest of his life working with Brahe’s data and publishing many works. Johannes Kepler died in Regensburg in 1630, while on a journey from his home in Sagan. His grave was demolished within two years because of the Thirty Years War. Though Kepler lived and died in turbulent times, his scientific works had a far greater effect on humanity then all the historical events during his time.
Kepler's Three Laws of Planetary Motion. Kepler discovered three relationships, now called "Kepler's laws" that describe the orbital motion of the planets. Prior to Kepler, planets were believed to orbit Earth in circular paths. The many variations of their motions were accommodated by adding more and more complex sets of circles upon circles (epicycles) to rationalize the observations. Although this worked moderately well, the future position of planets could only be roughly predicted.

1. Law of Ellipses (1609). The orbit of each planet is an ellipse, with the Sun at one focus. Earth is closest to the Sun in January and farthest from the Sun in July as it travels along its elliptical orbit.

2. Law of Equal Areas (1609). A line from the planet to the Sun sweeps out equal areas in equal times. This geometric description describes the fact that a planet's orbital velocity varies in a regular way -- the farther the planet is from the Sun, the more slowly it moves along its orbit. For example, Earth moves fastest in January (when it’s further from the Sun) and slowest in July (when it’s closest).

3. Harmonic Law (1618). The square of the sidereal period of a planet is directly proportional to the cube of its orbit’s semi-major axis. This is described by following equation:
T2= a3 * k
T2= a3 * 4 π2 / μ
T = 2 π (a3/μ)
For Solar orbits:
μ = G * MSol = (6.667 x 10-11 N m2/kg2) * (2 x 1030 kgs)
μ = 132,712,440,018 km3/sec2 = 13.2 x 1010 km3/sec2
Remarks: Period (T) doesn't care about shape, e, or circumference, C. However, it does care about a, semimajor axis. For example, any orbit with a= 1.0 AU will take exactly one year to complete. Thus, the first item in Table 2, a (semimajor axis), relates directly to the last item, T, period.

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