# The Physics, Astronomy and Mathematics of the Solar System

## Discovering Conic Sections in the Motion of Heavenly Bodies

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## The Ellipse

The general equation for the ellipse is given as: x^2/a^2 + y^2/b^2 = 1,
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where
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b^2 = a^2 – c^2

Tycho Brahe (around late 1500 AD) built what could be characterized as an early astronomical observatory. He is credited with the most accurate astronomical observations of his time. He created a huge database, which catalogued the stars and planets positions with great accuracy. Redundant observations allowed an accurate track of the heavenly bodies' motions even though the math had not yet been developed or applied in order to make accurate predictions.

Using Brahe's observations Johannes Kepler (around early 1600's AD) would develop his now famous laws of planetary motion. The first law states that the orbits of the planets are elliptical, with the sun at one of the foci. The second law states that a line joining a planet and the sun sweeps out equal areas during equal intervals of time. The third law states, the squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orb.

An ellipse is the set of all points in a plane such that the sum of the distances (focal radii) from two given points (foci) is constant. The parts of the ellipse are shown in Figure 3. F1 and F2 are the foci. Line segment AB is the major axis. Line segment CD is the minor axis. The semi-major axis is denoted by the letter "a "in Figure 3 while the semi-minor axis is denoted by the letter "b". The eccentricity of an ellipse as in Figure 3 may be determined with the equation,

E = (1- b^2/a^2)^1/2

where
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a
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and
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b
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are the semi-major and semi-minor axes respectively.

(image available in print form)

Figure 3.
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source:
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Wikipedia

Students should be able to write and graph equations of ellipses, given their identifying characteristics.