The third conservation law of the Two-Body Problem; it fixes the shape and orientation of the orbit within its plane.
Derivation
- Notation: ,
- Cross the equation of motion with :
- is constant (Specific Angular Momentum), so the left side is a total
derivative:
- Expand with the triple product and
:
- Compare with the derivative of the unit vector :
- So the right side of step 1 is also a total derivative:
Eccentricity Vector
- Its magnitude is the eccentricity
- Scaled version: (the Laplace / Runge-Lenz vector)
Properties
- lies in the orbital plane and points toward perigee
- the axis of the conic section and the axis of the Perifocal Frame are aligned with it
- Dotting with gives the Orbit Equation
- Dotting with itself gives the relation of physical and geometric quantities
- , so only two of its components are independent (see Conserved Quantities)