Orbital motion in space from the conservation laws: the shape of the orbit falls out of the Eccentricity Vector without integrating anything.
Orbital Motion From Conservation Laws
- Rearrange the eccentricity vector:
- Dot with and use the scalar triple product:
- With the angle from to , :
- is the semi-latus rectum
- is the eccentricity
Terminology
- “anomaly”, “argument” = angle
- is the true anomaly: the angle from the eccentricity vector (perigee) to the satellite
The Orbit Equation
Orbit Equation
- Curtis (2.45)
- Relates , the geocentric distance, to , the true anomaly
- Closest approach (perigee) at , farthest (apogee) at when
- Writing it in Cartesian coordinates shows the orbit is a conic section (Kepler’s first law)