Two-body orbit propagation without leaving Cartesian coordinates: “the and functions”.
Two-Body Initial and Boundary Value Problems
- Initial value problem: propagation
- Boundary value problem: the Lambert problem
- Orbit propagation: given an orbital state vector at one time, what is it at a future time?
- : the epoch time
The f and g Functions
- Method of Lagrange coefficients (Curtis section 2.11, but here using the change in eccentric anomaly rather than the change in true anomaly)
- The motion stays in the plane of and , so the new state is a linear combination of them:
- At the epoch:
- Solve for by crossing with :
- In in-plane (perifocal) components:
- Solve for by crossing with :
The f and g in Terms of Delta E
- Perifocal coordinates in terms of the eccentric anomaly:
- Rate of the eccentric anomaly, from Kepler’s equation and :
- Compute :
- Use and :
- using
Lagrange Coefficients
- comes from solving Kepler’s equation for the time step
- Worked example: Curtis Example 2.13
In Universal Variables
- With the universal anomaly and the Stumpff functions , :
Universal Lagrange Coefficients
- Note the (the radius at the new time) in the denominator of (circled in the notes)
Propagation with Lagrange Coefficients
- Block-matrix form of the propagation:
- : the identity matrix
- Careful: this is not the state transition matrix, because , themselves depend on the initial state (see State Transition Matrix)