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” Q Transformation (or direction cosine) matrix x y z x’x’ y’y’ 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 {θ d, θ e, and θ f } 1 st rotation 2 nd rotation3 rd rotation

Coordinate Transformation For example the Euler angle sequence for rotation is the rotation where you rotate by the angle α along the 3 rd axis (usually z-axis), then by β along the 1 st axis, and then by γ along the 3 rd 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 rotation where you rotate by the angle α along the 3 rd axis (usually z-axis), then by β along the 2 nd axis, and then by γ along the 1 st 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 Planets are not perfect spheres Oblateness

Perturbation to Orbits Oblateness

Perturbation to Orbits Oblateness

Perturbation to Orbits Oblateness

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 ( days) or °/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 1.Compute the orbital elements of the state 2.Find the orbit period, T, and mean motion, n 3.Find the eccentric anomaly 4.Calculate time since periapsis passage, t, using Kepler’s equation

Finding State of S/C w/Oblateness 5.Calculate new time as t f = t + Δt 6.Find the number of orbit periods elapsed since original periapsis passage 7.Find the time since periapsis passage for the final orbit 8.Find the new mean anomaly for orbit n 9.Use Newton’s method and Kepler’s equation to find the Eccentric anomaly (See slide 57)

Finding State of S/C w/Oblateness 10.Find the new true anomaly 11.Find position and velocity in the perifocal frame

Finding State of S/C w/Oblateness 12.Compute the rate of the ascending node 13.Compute the new ascending node for orbit n 14.Find the argument of periapsis rate 15.Find the new argument of periapsis

Finding State of S/C w/Oblateness 16.Compute the transformation matrix [Q] using the inclination, the UPDATED argument of periapsis, and the UPDATED longitude of ascending node 17.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 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.