ARO309 - Astronautics and Spacecraft Design

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 r1and r2and Δt, what are the velocities at those two points, v1 and v2.

5. Lambert Fit • Steps to find v1 and v2: • Find the magnitude of r1 and r2 • Decide if the orbit is prograde or retrograde • Compute the following • Compute Δθ for retrograde for prograde

6. Lambert Fit • Compute the function • Find z by iterating using Newton’s method until convergence • you can start with z0 = 0 (or positive z0 if an elliptical orbit), where

7. Lambert Fit • where • 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 • Compute the function y(z) using the converged z • Compute f, g, fdot, gdot • Compute

9. Example 5.2 (Text Book)

10. Example 5.2 (Text Book)

11. Example 5.2 (Text Book)