Keplers Laws Keplers Work Tycho Brahe led a
Kepler’s Laws
Kepler’s Work ] ] Tycho Brahe led a team which collected data on the position of the planets (15801600 with no telescopes). Mathematician Johannes Kepler was hired by Brahe to analyze the data. ] ] He took 20 years of data on position and relative distance. No calculus, no graph paper, no log tables. Both Ptolemy and Copernicus were wrong. He determined 3 laws of planetary motion (16001630).
Kepler’s First Law ] The orbit of a planet is an ellipse with the sun at one focus. A path connecting the two foci to the ellipse always has the same length.
Orbital Description ] An ellipse is described by two axes. • Long – semimajor (a) • Short – semiminor (b) ] The area is pab (becomes pr 2 for a circle). b a
Orbital Speed ] The centripetal force is due to gravity. • GMm/r 2 = mv 2/r • v 2 = GM/r ] Larger radius orbit means slower speed. ] Within an ellipse larger distance also gives slower speed. » 2 nd law
Kepler’s Second Law ] The line joining a planet and the sun sweeps equal areas in equal time. Dt The planet moves slowly here. Dt The planet moves quickly here.
Universal Gravitation F = Gm 1 m 2 r 2 G= 6. 67 x 10 -11 Nm 2/kg 2
Orbital Speed and Period ] The speed is related to the period in a circular orbit. • • v 2 = GMsun/r V= C/T (2 pr/T)2 = GMsun/r T 2 = 4 p 2 r 3/GMsun • T = 2 p r 3/GMsun ] ] Larger radius orbit means longer period. Within an ellipse, a larger semimajor axis also gives a longer period (T).
Orbital Speed and Period ] The speed of an object in orbit around Earth v = GME/r T = 2 p r 3/GME ME= 5. 97 x 1024 kg Radius from moon to center of Earth= 385, 000 m Velocity of moon = 1017 m/s or 1 km/s T= 27. 5 days (time it takes moon to orbit Earth) 3 rd Law
Kepler’s Third Law ] The square of a planet’s period is proportional to the cube of the length of the orbit’s semimajor axis. • T 2/a 3 = constant • The constant is the same for all objects orbiting the Sun direction of orbit semimajor axis: a The time for one orbit is one period: T
Hyperbolic Orbits ] Some satellites have so much speed that gravity can’t hold them in an orbit. ] These objects take a hyperbolic orbit that never returns.
Gravitational Field g=GM r 2 G = 6. 67 X 10 -11 Nm 2/kg 2 ME= 5. 97 x 1024 kg Radius of earth = 6. 38 x 106 m
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