Keplers Laws of Planetary Motion

Kepler developed, using Tycho Brahe’s observations, the first kinematic description of orbits. Newton will develop a dynamic description that involves the underlying influence (gravity) 1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus. Ellipses that are highly flattened have high eccentricity.

Ellipses that are close to a circle have low eccentricity. 2nd law (law of equal areas): a line connecting the Sun and a planet (called the radius vector) sweeps out equal areas in equal times. Objects travel fastest at the low point of their orbit, and travel slowest at the high point of their orbit.

3rd law (law of harmonics): The square of a planet’s orbital period is proportional to its mean distance from the Sun cubed. The 3rd law is used to develop a “yardstick” for the Solar System, expressing the distance to all the planets relative to Earth’s orbit by just knowing their period (timing how long it takes for them to go around the Sun).

newton2 copyThe mathematical way to describe Kepler’s 3rd law is: P2 α R3 where the α symbol means ‘proportional to.’ Proportions are expressions that imply there exists some constant, k, that relates the period, P, and the radius, R, such that P2 = kR3 We can determine k by expressing the formula in units of the Earth and its orbit around the Sun, such that: (1 yr)2 = k (1 A.U.)3 so k is equal to one, as long as we use units of years and A.U.’s (the Astronomical Unit, i.e . the distance from the Earth from the Sun). With k=1, then kepler’s 3rd law becomes P2 = R3

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