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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. Introductions. So far we have focus on the orbital mechanics of a spacecraft in 2D.

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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


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


Orbital Elements


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


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 (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

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.


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