ARO309 - Astronautics and Spacecraft Design

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

Two-Body Dynamics: Orbits in 3D Chapter 4

Introductions • So far we have focus on the orbital mechanics of a spacecraft in 2D. • In this Chapter we will now move to 3D and express orbits using all 6 orbital elements

Geocentric Equatorial Frame

Orbital Elements • Classical Orbital Elements are: a = semi-major axis (or h or ε) e = eccentricity i = inclination Ω = longitude of ascending node ω = argument of periapsis θ = true anomaly

Orbital Elements

Coordinate Transformation • Answers the question of “what are the parameters in another coordinate frame” y y’ x’ Q x Transformation (or direction cosine) matrix z z’ Q is a orthogonal transformation matrix

Coordinate Transformation Where And Where is made up of rotations about the axis {a, b, or c} by the angle {θd, θe, and θf} 3rd rotation 2nd rotation 1st rotation

Coordinate Transformation For example the Euler angle sequence for rotation is the 3-1-3 rotation where you rotate by the angle α along the 3rd axis (usually z-axis), then by β along the 1st axis, and then by γ along the 3rd axis. Thus, the angles can be found from elements of Q

Coordinate Transformation Classic Euler Sequence from xyz to x’y’z’

Coordinate Transformation For example the Yaw-Pitch-Roll sequence for rotation is the 1-2-3 rotation where you rotate by the angle α along the 3rd axis (usually z-axis), then by β along the 2nd axis, and then by γ along the 1st axis. Thus, the angles can be found from elements of Q

Coordinate Transformation Yaw, Pitch, and Roll Sequence from xyz to x’y’z’

Transformation between Geocentric Equatorial and Perifocal Frame Transferring between pqw frame and xyz Transformation from geocentric equatorial to perifocal frame

Transformation between Geocentric Equatorial and Perifocal Frame Transformation from perifocal to geocentric equatorial frame is then Therefore

Perturbation to Orbits Oblateness • Planets are not perfect spheres

Perturbation to Orbits Oblateness

Sun-Synchronous Orbits Orbits where the orbit plane is at a fix angle α from the Sun-planet line Thus the orbit plane must rotate 360° per year (365.25 days) or 0.9856°/day

Finding State of S/C w/Oblateness • Given: Initial State Vector • Find: State after Δt assuming oblateness (J2) • Steps finding updated state at a future Δt assuming perturbation • Compute the orbital elements of the state • Find the orbit period, T, and mean motion, n • Find the eccentric anomaly • Calculate time since periapsis passage, t, using Kepler’s equation

Finding State of S/C w/Oblateness • Calculate new time as tf = t + Δt • Find the number of orbit periods elapsed since original periapsis passage • Find the time since periapsis passage for the final orbit • Find the new mean anomaly for orbit n • Use Newton’s method and Kepler’s equation to find the Eccentric anomaly (See slide 57)

Finding State of S/C w/Oblateness • Find the new true anomaly • Find position and velocity in the perifocal frame

Finding State of S/C w/Oblateness • Compute the rate of the ascending node • Compute the new ascending node for orbit n • Find the argument of periapsis rate • Find the new argument of periapsis

Finding State of S/C w/Oblateness • Compute the transformation matrix [Q] using the inclination, the UPDATED argument of periapsis, and the UPDATED longitude of ascending node • Find the r and v in the geocentric frame

Ground Tracks Projection of a satellite’s orbit on the planet’s surface

Ground Tracks Projection of a satellite’s orbit on the planet’s surface Ground Tracks reveal the orbit period Ground Tracks reveal the orbit inclination If the argument of perispais, ω, is zero, then the shape below and above the equator are the same.

ARO309 - Astronautics and Spacecraft Design