22: Kepler's Laws, Elliptical Orbits, and Change of Orbits

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Topics covered: Kepler's Laws, Elliptical Orbits, Change of Orbits, and the famous passing of a Ham Sandwich. Kepler's three Laws summarize the motion of the planets in our solar system. Following Newton's law of universal gravitation, the conservation of angular momentum and mechanical energy allow us to calculate the semimajor axis of the elliptical orbits, the orbital period and other orbital parameters. All we have to know is one position and the associated velocity of a planet and the entire orbit follows.

Instructor/speaker: Prof. Walter Lewin

Date recorded: November 3, 1999

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  • Kepler's Laws and Elliptical Orbits
    A review of equations for the period, velocity and mechanical energy of a circular orbit is given. Kepler's 3 Laws, and the planetary data that led him to his third law are introduced. The equations for elliptical orbits are discussed and compared with the equations for circular orbits.

  • Elliptical Orbit from Initial Conditions
    The elliptical orbit follows from the initial conditions. Using the conservation of mechanical energy, you can find the semimajor axis, and this, in combination with Kepler's 3rd law, enables you to calculate the orbital period. A numerical example for a high eccentricity orbit of an Earth orbiting satellite, is worked out. Using the conservation of angular momentum, one can determine the distances to apogee and perigee, and the satellite's velocity at apogee and perigee.

  • Changing from Circular to Elliptical Orbits
    By firing a rocket on board a spacecraft that is in a circular orbit, the spacecraft's velocity (vector) will change, and this leads to an elliptical orbit and a change in orbital period. The new elliptical orbits are sketched along with the original circular orbit, setting the stage for how astronauts in different spacecrafts can pass a sandwich.

  • Astronauts Pass a Ham Sandwich
    Peter and Mary are astronauts in different spacecrafts but in the same circular orbit. Peter wants to throw a ham sandwich to Mary. The question is: how to do that? There is a large family of solutions which are discussed.

  • Simulations of the Passing of the Sandwich
    A computer model for finding solutions to how astronauts Peter and Mary can pass a sandwich is introduced and exercised by its author, Dave Pooley. GREAT FUN!

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  • English - US (SRT)