Vai's Uni Notes

Tschauner-Hempel Equations

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Linearized relative motion about an elliptical reference orbit, and the Yamanaka-Ankersen solution. Generalizes the HCW equations.

Equations of Motion for an Elliptical Reference Orbit

  • TH/YA starting point, in the RSW frame of the reference orbit:
    • ̈𝑥−2̇𝑓̇𝑦−(̇𝑓2+2𝜇𝑟3)𝑥−̈𝑓𝑦=0̈𝑦+2̇𝑓̇𝑥+̈𝑓𝑥−(̇𝑓2−𝜇𝑟3)𝑦=0̈𝑧+𝜇𝑟3𝑧=0
    • 𝑓 = true anomaly of the reference orbit, 𝑟 = radius of the reference orbit, ̇𝑓 = time derivative of 𝑓
  • Time-varying coefficients (̇𝑓, 𝑟 change around the orbit); for a circular reference orbit ̇𝑓=𝑛, ̈𝑓=0, 𝜇𝑟3=𝑛2 and these reduce to HCW

Coordinate Scaling and True Anomaly

  • Apply coordinate scaling and change the independent variable from time to true anomaly 𝑓:
    • ̃𝐱=[̃𝑥̃𝑦̃𝑧̃𝑥′̃𝑦′̃𝑧′]
    • prime indicates derivative with respect to true anomaly
  • Scaling transformation:
    • ̃𝐱=𝑇(𝑓)𝐱(2)
    • 𝑇(𝑓)=[(1+𝑒cos𝑓)𝐼−𝑒sin𝑓𝐼0𝑝2ℎ(1+𝑒cos𝑓)𝐼],𝑇−1(𝑓)=[11+𝑒cos𝑓𝐼ℎ𝑒𝑝2sin𝑓𝐼0ℎ𝑝2(1+𝑒cos𝑓)𝐼](3)
  • Variables 𝑒, 𝑝, ℎ are the eccentricity, semilatus rectum, angular momentum magnitude of the reference orbit

Tschauner-Hempel Equations

  • ̃𝑥″−2̃𝑦′−31+𝑒cos𝑓̃𝑥=0̃𝑦″+2̃𝑥′=0̃𝑧″+̃𝑧=0
  • Much simpler than (1): only one coefficient still depends on 𝑓

Don't normalize

  • Don’t normalize the true anomaly here, because it represents elapsed time (see Angle Normalization)

Yamanaka-Ankersen Solution

  • Yamanaka-Ankersen (2002) has become the standard approach to solving these equations, using this integral:
    • 𝐽(𝑓,𝑓0)=∫𝑓𝑓0d𝜁(1+𝑒cos𝜁)2
    • 𝐽=ℎ𝑝2(𝑡−𝑡0)=𝑀−𝑀0(1−𝑒2)32=𝐸−𝑒sin𝐸−𝐸0+𝑒sin𝐸0(1−𝑒2)32
    • so it follows from Kepler’s equation
  • Solution (with a constant vector 𝐤):
    • ̃𝐱=Ψ(𝑓)𝐤,Ψ(𝑓)=[𝛙1𝛙2𝛙3𝛙4𝛙5𝛙6](9)
  • Columns:
    • 𝛙1=[sin𝑓(1+𝑒cos𝑓)2cos𝑓−𝑒sin2𝑓0cos𝑓+𝑒cos2𝑓−2sin𝑓(1+𝑒cos𝑓)0],𝛙2=[cos𝑓(1+𝑒cos𝑓)−2sin𝑓−𝑒sin𝑓cos𝑓0−sin𝑓−𝑒sin2𝑓𝑒−2cos𝑓(1+𝑒cos𝑓)0](10)
    • 𝛙3=[1−3𝑒2𝐾sin𝑓(1+𝑒cos𝑓)−32𝐾(1+𝑒cos𝑓)20−3𝑒2𝐾(cos𝑓+𝑒cos2𝑓)−3𝑒sin𝑓2(1+𝑒cos𝑓)3𝑒𝐾sin𝑓(1+𝑒cos𝑓)−320],𝛙4=[010000](11)
    • 𝛙5=[00sin𝑓00cos𝑓],𝛙6=[00cos𝑓00−sin𝑓](12)
  • Replace the integral 𝐽 with 𝐾:
    • 𝐾=ℎ(𝑡−𝜏)𝑝2,𝑓(𝜏)=0,𝐽=𝐾−𝐾0
    • 𝜏: time of perigee passage of the reference orbit
  • Inverse:
    • Ψ−1=[𝛔1𝛔2𝛔3𝛔4𝛔5𝛔6](13)
    • 𝛔1=[−31+𝑒2+𝑒cos𝑓(1−𝑒2)(1+𝑒cos𝑓)sin𝑓+3𝐾𝑒1−𝑒2(2+𝑒2+3𝑒cos𝑓)−3𝑒+cos𝑓1−𝑒222+𝑒2+3𝑒cos𝑓1−𝑒2−3𝑒3+𝑒2+2𝑒cos𝑓(1−𝑒2)(1+𝑒cos𝑓)sin𝑓+3𝐾1+𝑒21−𝑒2(2+𝑒2+3𝑒cos𝑓)00],𝛔2=[000100](14)
    • 𝛔3=[0000sin𝑓cos𝑓],𝛔4=[cos𝑓(1+𝑒cos𝑓)−2𝑒1−𝑒2+3𝐾𝑒21−𝑒2sin𝑓(1+𝑒cos𝑓)−sin𝑓1+𝑒cos𝑓1−𝑒22𝑒sin𝑓1+𝑒cos𝑓1−𝑒22𝑒cos𝑓(1+𝑒cos𝑓)−1−𝑒21−𝑒2+3𝐾𝑒1+𝑒21−𝑒2sin𝑓(1+𝑒cos𝑓)00](15)
    • 𝛔5=[−sin𝑓2+𝑒cos𝑓1−𝑒2+3𝐾𝑒1−𝑒2(1+𝑒cos𝑓)2−(𝑒+2cos𝑓+𝑒cos2𝑓1−𝑒2)2(1+𝑒cos𝑓)21−𝑒2−2𝑒sin𝑓2+𝑒cos𝑓1−𝑒2+3𝐾1+𝑒21−𝑒2(1+𝑒cos𝑓)200],𝛔6=[0000cos𝑓−sin𝑓](16)

Yamanaka-Ankersen State Transition Matrix

  • 𝐱(𝑡)=𝑇−1(𝑓)Ψ(𝑓)Ψ−1(𝑓0)𝑇(𝑓0)⏟STM𝐱0=Ω−1(𝑓)Ω(𝑓0)𝐱0≡Φ(𝑓,𝑓0)𝐱0
  • Eq. (17): the state transition matrix is constructed from these matrices
  • Scale to ̃𝐱, propagate the TH solution in true anomaly, scale back

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