Orbital Mechanics I

Discussion

circular orbits

Newton's laws only. Nothing about energy or momentum. Centripetal force and gravitational force.

     
Fc  =  Fg
     
mv2  =  GMm
rp rp2
v = √  Gm
r

Kepler's third law. Derive Kepler's third law of planetary motion (the harmonic law) from first principles.

v = √  Gm  =  r  
r T
  Gm  =  2r2
r T2
r3  =  Gm  = constant
T2 2
r3  ∝  T2  
 

The "constant" depends on the object at the focus. Although formulated from the data for objects orbiting the sun, Newton showed that Kepler's third law can be applied to any family of objects orbiting a common body.

orbit families

[slide]

A "snapshot" of the earth and about 500 of its artificial satellites generated one summer evening in 2002. Nearly all of them are GEOs or LEOs. Satellites on the ring are in geosynchronous earth orbit (GEO). Those clustered near the earth are in low earth orbits (LEO). Scattered in between are satellites in medium earth orbits (MEO). The moon, earth's only natural satellite, is approximately nine times farther from the earth than the ring of geosynchronous satellites. Source: NASA.

binary systems

circular motion about the center of mass

still just a balance between centripetal and gravitational force, but slightly more complicated