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

Introductions Class Materials at Course: ARO 309: Astronautics and Spacecraft Design (3 units) Description: Space mission and trajectory design. Kepler’s laws. Orbits, hyperbolic escape trajectories, interplanetary transfers, gravity assists. Special orbits including geostationary, Molniya, sunsynchronous. [Kepler's equation, orbit determination, attitude dynamics and control.] Prerequisite: C or better in ME215 (dynamics) Section 01: 5:30 PM – 6:45 PM MW (15900) Room Section 02: 7:00 PM – 8:15 PM MW (15901) Room Holidays: 1/20 Text Book: H. Curtis, Orbital Mechanics for Engineering Students, Butterworth-Heinemann (preference: 2 nd Edition) Grades: 10% Homework, 25% Midterm, 25% Final, 40% Quizzes (4 x 10% each)

Introductions Things you should know (or willing to learn) to be successful in this class – Basic Math – Dynamics – Basic programing/scripting

What are we studying?

Earth Orbiters

Pork Chop Plot

High Thrust Interplanetary Transfer

Low-Thrust Interplanetary Transfer

Low-Thrust Europa End Game

Stable for > 100 days Orbit Stability Enceladus Orbit

Juno

Other Missions

Lecture 01 and 02: Two-Body Dynamics: Conics Chapter 2

Equations of Motion

Fundamental Equations of Motion for 2-Body Motion

Conic Equation From 2-body equation to conic equation

Angular Momentum Other Useful Equations

Energy NOTE: ε = 0 (parabolic), ε > 0 (escape), ε < 0 (capture: elliptical and circular)

Conics

Circular Orbits

Elliptical Orbits

Parabolic Orbits Parabolic orbits are borderline case between an open hyperbolic and a closed elliptical orbit NOTE: as v  180°, then r  ∞

Hyperbolic Orbits

Hyperbolic excess speed

Properties of Conics 0 < e < 1

Conic Properties

Vis-Viva Equation Vis-viva equation Mean Motion

Perifocal Frame “natural frame” for an orbit centered at the focus with x-axis to periapsis and z- axis toward the angular momentum vector

Perifocal Frame FROM THEN

Lagrange Coefficients Future estimated state as a function of current state Solving unit vector based on initial conditions and Where

Lagrange Coefficients Steps finding state at a future Δθ using Lagrange Coefficients 1.Find r 0 and v 0 from the given position and velocity vector 2.Find v r0 (last slide) 3.Find the constant angular momentum, h 4.Find r (last slide) 5.Find f, g, fdot, gdot 6.Find r and v

Lagrange Coefficients Example (from book)

Lagrange Coefficients Example (from book)

Lagrange Coefficients Example (from book)

Lagrange Coefficients ALSO Since V r0 is < 0 we know that S/C is approaching periapsis (so 180°<θ<360°)

CR3BP Circular Restricted Three Body Problem (CR3BP)

CR3BP Kinematics (LHS):

CR3BP Kinematics (RHS):

CR3BP CR3BP Plots are in the rotating frame Tadpole Orbit Horseshoe Orbit Lyapunov Orbit DRO

CR3BP: Equilibrium Points Equilibrium points or Libration points or Lagrange points L1L2L3 L4 L5 Jacobi Constant