Orbital Mechanics I

Discussion

circular orbits

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

Fc = Fg  ⇒  m2v2  =  Gm1m2  ⇒  v = √  Gm
r r2 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

lagrange points

the three body problem, lagrange libration points are the simplest solutions

still just a balance between centripetal and gravitational forces, but now much more complicated

[slide]

The five Lagrange points of the earth-sun system. Satellites in orbit at these locations remain fixed with respect to the earth and sun. This figure is not drawn to scale.

L1 and L2 are approximately four times farther from the earth than the moon. L3 is a very near the "anti-earth" point.

L4 and L5 are at the vertex of an equilateral triangle formed with the earth and sun. L4 leads the earth and L5 follows.

Objects can settle in an orbit around a Lagrange point. Orbits around the three collinear points, L1, L2, and L3, are unstable. They last but days before the object will break away. L1 and L2 last about 23 days. Objects orbiting around L4 and L5 are stable because the Coriolis force keeps them spinning around the Lagrange point.

noncircular orbits

qualitative description of noncircular orbits

centripetal-gravitational forces don't balance

uses