Lecture 22: Kepler's Laws - Elliptical Orbits - Satellites - Change of Orbits - Ham Sandwich
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1. 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.
2. 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.
3. 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.
4. 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.
5. 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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