Vai's Uni Notes

Perifocal Frame

3 min read

The natural frame of the orbit itself: centered at the focus, lying in the orbital plane.

Perifocal Coordinates

  • Axes (Curtis section 2.10):
    • ̂𝐏 points to perigee: ̂𝐏=̂𝐞
    • ̂𝐐 is in the orbital plane, perpendicular to ̂𝐏 (in the direction of motion at perigee)
    • ̂𝐖 is in the direction of the angular momentum: ̂𝐖=̂𝐡
  • The orbit lies entirely in the ̂𝐏-̂𝐐 plane, so positions and velocities have no ̂𝐖 component
  • We compute in the perifocal frame, then use a 3-1-3 rotation into ECI (see State Vector from Orbital Elements)

Position in Perifocal Coordinates

  • With the true anomaly 𝜃 measured from ̂𝐏 and the orbit equation:
    • 𝐫=𝑟cos𝜃̂𝐏+𝑟sin𝜃̂𝐐=𝑝cos𝜃1+𝑒cos𝜃̂𝐏+𝑝sin𝜃1+𝑒cos𝜃̂𝐐

Velocity in Perifocal Coordinates

  • Differentiate (the axes are fixed):
    • ̇𝐫=(̇𝑟cos𝜃−𝑟̇𝜃sin𝜃)̂𝐏+(̇𝑟sin𝜃+𝑟̇𝜃cos𝜃)̂𝐐
  • Radial rate from differentiating the orbit equation:
    • ̇𝑟=−𝑝(1+𝑒cos𝜃)2(−𝑒sin𝜃)̇𝜃=̇𝜃𝑝𝑒sin𝜃(1+𝑒cos𝜃)2=̇𝜃𝑟2𝑝𝑒sin𝜃
  • With ℎ=𝑟2̇𝜃 (transverse velocity) and 𝑝=ℎ2𝜇⇒ℎ=√𝜇𝑝:
    • Radial velocity:
      • 𝑣𝑟=̇𝑟=ℎ𝑟2𝑟2𝑝𝑒sin𝜃=√𝜇𝑝𝑒sin𝜃
    • Transverse velocity:
      • 𝑣⟂=𝑟̇𝜃=ℎ𝑟=ℎ𝑝𝑝𝑟=√𝜇𝑝(1+𝑒cos𝜃)
    • (their ratio is the flight-path angle tan𝛾=𝑣𝑟𝑣⟂)
  • Substitute ̇𝑟=𝑣𝑟 and 𝑟̇𝜃=𝑣⟂:
    • ̇𝐫=(√𝜇𝑝𝑒sin𝜃cos𝜃−√𝜇𝑝(1+𝑒cos𝜃)sin𝜃)̂𝐏+(√𝜇𝑝𝑒sin2𝜃+√𝜇𝑝(1+𝑒cos𝜃)cos𝜃)̂𝐐=√𝜇𝑝[−sin𝜃̂𝐏+(cos𝜃+𝑒)̂𝐐]

Motion in Perifocal Coordinates

  • 𝐫=𝑝1+𝑒cos𝜃(cos𝜃̂𝐏+sin𝜃̂𝐐)
  • ̇𝐫=√𝜇𝑝[−sin𝜃̂𝐏+(cos𝜃+𝑒)̂𝐐]
  • See Curtis equations (4.43), (4.44)

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