Kepler and the elliptic

orbits

By Marta Ibáñez Díaz-Peco1ºC Bach.

BIOGRAPHY:

Johannes Kepler was one of the greatest astronomers of the 17th century. He was born in Weil der Stadt in southern Germany on 27th December 1571. His family was poor but in 1588 he won a scholarship to study theology at the University of Tubingen, but while he was at university, he became interested in astronomy. He spent several years in Austria, where he became a math teacher and married with his first wife, Barbara. But in 1600 he moved to Prague because of the Catholic Counter-Reformation. In 1611, Kepler’s wife and son died. The following year he moved to Linz, where he married with his second wife, Susanna in 1614.Finally, on 15th November 1630, Kepler died of fever, and he was buried in Regensburg.

It’s interesting to mention that, throughout his life, Kepler published several books, in which he explained his works and observations.He also wrote a story about a trip to the Moon called The Dream (in 1634), and it could be called the first science fiction story.

WHICH ASPECTS OF MATHEMATICS WAS HE INTERESTED IN?

His investigations are very extensive, but we can highlight the following:

• He explained correctly the planetary motion, thereby, becoming founder of celestial mechanics and the first "natural laws" in the modern sense; being universal, verifiable, precise.

• He investigated the formation of pictures with a pin hole camera.• He explain the process of vision by refraction within the eye.• He explained the principles of how a telescope works.• He discovered and described the properties of total internal reflection.• He explained that the tides are caused by the Moon.• He used stellar parallax caused by the Earth's orbit to measure the

distance to the stars.• He derived logarithms purely based on mathematics, independent of

Napier's tables published in 1614.

HOW IS HIS WORK RELATED TO THE MATHS?

In his book “Epitome Astronomiae Copernicanae”, the first astronomy textbook based on the Copernican model, Kepler introduced what is now known as Kepler's equation for the solution of planetary orbits, using the eccentric anomaly E, and the mean anomaly M.

The mean anomaly M is the angular distance from perihelion which a (fictitious) planet would have if it moved on the circle of radius a with a constant angular velocity and with the same orbital period T as the real planet moving on the ellipse. By definition, M increases linearly (uniformly) with time.

The term anomaly (instead of angle), which means irregularity, is used by astronomers describing planetary positions. The term originates from the fact that the observed locations of a planet often showed small deviations from the predicted data.

Operating with radians Kepler's equation is: E(t) - e*sin[E(t)] = M(t)

or, using degrees: E(t) - (180°/π)*e*sin[E(t)] = M(t)

The equation can be derived from Kepler's second law.

A SHORT DESCRIPTION OF PLANETARY MOTION AND THE ELLIPTIC ORBITS

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1.The orbit of a planet is an ellipse with the Sun at one of the two foci.2.A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.3.The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are almost circles, and careful observation and calculation is required in order to establish that they are actually ellipses. Using calculations of the orbit of Mars, whose published values are somewhat suspect, which indicated elliptical orbits, Johannes Kepler inferred that other heavenly bodies, including those farther away from the Sun, also have elliptical orbits.

Kepler's work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles. Isaac Newton showed in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation.

Kepler's laws are part of the foundation of modern astronomy and physics.

BIBLIOGRAPHY AND SOURCES

http://www.localhistories.org/kepler.html

http://simplementeeluniverso.blogspot.com.es/2012/05/introduccion-las-matematicas-de-las.html

http://kepler.nasa.gov/Mission/JohannesKepler/

https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

http://www.jgiesen.de/kepler/kepler.html