Class 34: The Orbits Class 34: The Orbits Keplers Laws 1. The Sun is - PowerPoint PPT Presentation

Class 34: The Orbits Class 34: The Orbits

Kepler’s Laws 1. The Sun is at the focus of the orbits (conic sections). 2. The line joining the Sun and a planet sweeps out equal areas in equal times. This is conservation of angular momentum, true for any central force field. g , y dA 1 L  Move     2 r constant Move faster  slower dt 2 2 3.    For bound orbit (E<0), the Area of ellipse ab 2 ab    T dA L L period of revolution is  dt 2 proportional to (semi ‐ major proportional to (semi major   2 2 2 2 G GM L GM G L      2 E - r r - 0 axis) 3/2 . This is true only for   2 r 2 r E 2 E inverse square central force.    r r GM GM   a p    a 2 2E 2 | E | 2 2 2 2 L L L L     b r r a p   2 E 2 | E |   2 GM GM      | E | and L b 2 | E | b 2a a      2 2 ab b 2 2 ab b 2 2     3/2 T a  L 2 GM GM b a

Hohmann Transfer (Tangential Thrust at extreme radii)  GM  a Thrust 2 | E | Thrust Thrust L  b  2 | E | | | d |E| decrease and L increase d | | |E| increase and L decrease d d Hohmann transfer is the most fuel Hohmann transfer is the most fuel Thrust Thrust efficient path, but not in time.

Radial Thrust at extreme radii |E| decrease and L constant Thrust Thrust |E| decrease and L constant Energy U eff E r  E  E

“Slingslot” V f ’ f V f V f ’ V f V planet V planet V i i V i ’ V planet planet V i ’ To the moving planet (prime To an inertial frame where the frame), |v i ’|=|v f ’| planet is moving , |v f ’|> v i ’|