Gravitational Fields, Circular Orbits and Kepler

Gravitational Field The Force of gravity per unit of mass at a given point It is the same as “local gravity” for a planet

Finding g (0,100) m = 10,000 kg What is g at the origin? m = 20,000 kg (80,0)

Speed in a Circular Orbit

Tycho Brahe

Johannes Kepler

Kepler’s First Law of Planetary Motion The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus http://home.cvc.org/science/kepler.gif

Speed in an Elliptical Orbit At Apogee: R = maximum and v = minimum At Perigee: R = minimum and v = maximum

Ellipse vs. Circle

Kepler’s Second Law of Planetary Motion A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time This occurs because angular momentum is conserved http://home.cvc.org/science/kepler.gif

Kepler’s Third Law of Planetary Motion The squares of the periods of the planets are proportional to the cubes of their semi-major axes If the distance R is measured in AU and the period T is measured in years then the third law for orbits about the Sun becomes

Comet Axis? What is the length of the semi-major axis of the orbit of a comet that takes 76 years to revolve around the sun?