Asri-unix.735
net.space
utzoo!decvax!ucbvax!ARPAVAX:C70:sri-unix!Lynn.ES@PARC-MAXC
Tue Feb  9 11:16:48 1982
Re: Re: sri-unix.707: Horseshoe Orbits
Nice try on the orbital explanation, but it's unfortunately not right (even
ignoring the ground speed business).  Both linear velocity and angular velocity
increase for satellites closer to the body they are orbiting.  Take the moon (radius
238,000 miles, period 27+ days => 2200 mph, 0.04 rev/day) versus a low earth
orbit satellite (radius 4000 miles, period 1.5 hrs => 17,000 mph, 16 rev/day).  The
angular velocity decreases with the 1.5 power of distance, the linear velocity
decreases with the 0.5 power (square root) of distance.  

The actual explanation of why adding speed moves a satellite away from the
body is this: With additional speed, the satellite tends to go in more of a straight
line (has less time to fall toward the parent body), and increases its orbital
distance.  While increasing its distance, it is slowed by gravitation.  It reaches
equilibrium at a greater distance and a slower orbital speed than it originally
had.  

This can be viewed another (equivalent) way: an orbiting body, given extra
speed, has too much kinetic energy for that orbit, so it exchanges some of its
kinetic energy for potential energy.  The equilibrium is reached when the extra
speed we gave it and some of its original speed are exchanged, ending up quite a
bit higher and moving a little slower than originally.  

Point of view is important in understanding the name "horseshoe".  Imagine two
horseshoes, of slightly different size, mouth to mouth, on a plate.  The plate
spins quickly.  As seen by a viewer on the plate, the satellite on the inner orbit
is moving slowly counterclockwise along the smaller horseshoe, while the outer
one is moving slowly clockwise on the larger horseshoe.  The spinning of the
plate represents the average rate of revolution of the two satellites, and so the
viewer on the plate sees only the DIFFERENCE between a satellite's orbital speed
and the average (plate's) speed.  At encounter, we switch the sizes of the
horseshoes, and the satellites (still seen from our spinning point of reference)
seem to each reverse direction and traverse their (now slightly larger or smaller)
respective horseshoes in the opposite directions.   

/Don Lynn