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ARO309 - Astronautics and Spacecraft Design

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ARO309 - Astronautics and Spacecraft Design

Winter 2014

Try Lam

CalPoly Pomona Aerospace Engineering

Two-Body Dynamics: Orbits in 3D

Chapter 4

- 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

- 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

- 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

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

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

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

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

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

Transferring between pqw frame and xyz

Transformation from geocentric equatorial to perifocal frame

Transformation from perifocal to geocentric equatorial frame is then

Therefore

Oblateness

- Planets are not perfect spheres

Oblateness

Oblateness

Oblateness

Oblateness

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

- 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

- 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)

- Find the new true anomaly
- Find position and velocity in the perifocal frame

- 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

- 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

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

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

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.