1. ARO309 - Astronautics and Spacecraft Design Winter 2014 Try Lam CalPoly Pomona Aerospace Engineering

2. Lambert’s Solution Chapter 5

3. Introductions This chapter only covers the basic concept of determining an orbit from some observation In practice, this is not referred to as orbit determination Space OD is actually a statistical estimation or filtering method (example: Kalman Filter) We will only cover Lambert’s problem (Section 5.3) from this Chapter

4. Lambert’s Problem Given 2 positions on an orbit r 1 and r 2 and Δt, what are the velocities at those two points, v 1 and v 2.

5. Lambert Fit Steps to find v 1 and v 2 : 1.Find the magnitude of r 1 and r 2 2.Decide if the orbit is prograde or retrograde 3.Compute the following 4.Compute Δθ for prograde for retrograde

6. Lambert Fit 5.Compute the function 6.Find z by iterating using Newton’s method until convergence you can start with z 0 = 0 (or positive z 0 if an elliptical orbit), where

7. Lambert Fit where 7.Note: the sign of the converged z tells you the orbit type: z < 0  Hyperbolic Orbit z = 0  Parabolic Orbit z > 0  Elliptical Orbit

8. Lambert Fit 8.Compute the function y(z) using the converged z 9.Compute f, g, fdot, gdot 10.Compute

9. Example 5.2 (Text Book)