application of cas to celestial mechanics: two examples

The Tide-Generating PotentialAnalytical models for the motion of a LT spacecraft

Application of CAS to Celestial Mechanics:two examples

Francesco Biscani

Advanced Concepts TeamEuropean Space Agency (ESTEC)

Course on Differential Equations and Computer AlgebraEstella, Spain – October 29-30, 2010

Francesco Biscani Application of CAS to Celestial Mechanics


Harmonic Development of the TGPTheories of MotionTransforming The Theories

Outline

1 The Tide-Generating PotentialHarmonic Development of the TGPTheories of MotionTransforming The Theories

2 Analytical models for the motion of a LT spacecraftMotivationLow-thrust dynamicsConvergent perturbative theoriesDivergent perturbative theory?




Tide-Generating Potential (TGP): a brief history

A (short) history of the development of the Tide-GeneratingPotential (TGP)

LAPLACE (1798): separation into species (LP, D, SD)

DARWIN, G.H. (1880): analytical, not truly harmonic, Hansen LT

DOODSON (1921): fully harmonic, Hill-Brown LT, Newcomb ST

CARTWRIGHT+TAYLER+EDDEN (1971-73): numerical spectralanalysis, Hill-Brown-Eckert, Newewcomb ST

BULLESFELD (1985): numerical, H-B-E LT, Newcomb

XI (1987): analytical computer algebra manipulations

TAMURA (1987): numerical, 1200 lines

HARTMANN+WENZEL (1995): numerical s.a., 12935 lines, numericalephemerides (Sun, Moon, Mercury-Saturn)

ROOSBEEK (1996): analytical, ∼7000 lines, uses algebraic manipulation,accounts for indirect lunar perturbations on the Sun




TGP: definition

FTGP = F (rP)− F (rE )

=∂

∂r[V (r)−W (r)]

∣∣∣∣r=rP

=∂UTGP (rP)

∂rP

Definition:

UTGP (r) =GML

d

∞∑n=2

(aed

)nPn (cosψ)




TGP: The Laplace Decomposition

Addition theorem:

UL (r) =GML

ae

∞∑n=2

n∑m=0

(2− δ0m) ·4πan+1

e

2n + 1Ymn

(ϕp , λp

)·(

1

d

)n+1

Ym∗n

(δL, αL

)eimθg︸︷︷︸

Time-dependent




TGP: Decomposition

Separation of geographical and astronomical components

Doodson (1922):

UL (r) = g∞∑n=2

n∑m=0

B∗nm(t)Ymn

(ϕp, λp

)Bnm(t) = (2− δ0m)

4π

2n + 1

MLaeMT

(aed

)n+1

Ymn

(δL, αL

)e−imθg

Goal: spectral decomposition

Bnm(t) =∑k

Hk cos (Θk)




Theories of Motion as Poisson Series

Lunar Theory ELP2000:

(r , b, l) ∼ ω1δV +∑

i1,...,ip

A(n)i1,...,ip

sincos

(i1λ1 + i2λ2 + . . .+ ipλp + φ

(n)i1,...,ip

)

Nutation Theory IAU2000:

∆ψ ∼∑i

Si sinAi ∆ε ∼∑i

Ci cosAi

Planetary Theory VSOP87:

(x , y , z)(r ,B, L)

(a, λ, k, h, q, p)∼∑i

(Si sinϕi + Ki cosϕi )




Alternative Procedures

Using the theories:

Analytical theories are typically not expressed in the appropriatereference frame:

Bnm(t) = (2− δ0m)4π

2n + 1

MLaeMT

(aed

)n+1Ymn

(δL, αL

)e−imθg

Two possibilities:

Transformation of the theories (as done by Doodson,Roosbeek, etc.)

Transformation of functions of the theories: rotation andtranslation of Spherical Harmonics




SH Transformations

Invoke Wigner’s rotation theorem

Ymn (θ′, φ′) =

n∑k=−n

Dnkm(α, β, γ)Y k

n (θ, φ)




SH Transformations

Invoke Wigner’s rotation theorem

Ymn (θ′, φ′) =

n∑k=−n

Dnkm(α, β, γ)Y k

n (θ, φ)

Where . . .

Dnkm(α, β, γ) = e−ik(α−π

2 ) dnkm(β) e−im(γ+ π

2 ),

dnkm(β) =

t2∑t=t1

(−1)t(n − k)!(n + m)!

t!(n + k − t)!(n −m − t)!(m − k + t)!·

(cos

β

2

)2n−(m−k+2t)

·(

sinβ

2

)m−k+2t

,

t1 = max(0, k −m),

t2 = min(n −m, n + k).




SH Transformations

Translation theorem

Imn (r) =∞∑k=0

jmax∑t=jmin

(−1)t+n+1

(k + n −m + t

n −m

)Im−tk+n (r2)R l

k (r1)




SH Transformations

Translation theorem

Imn (r) =∞∑k=0

jmax∑t=jmin

(−1)t+n+1

(k + n −m + t

n −m

)Im−tk+n (r2)R l

k (r1)

Where . . .

Imn (r) = r−n−1Y mn (θ, φ)

Rmn (r) = rnY m

n (θ, φ)

jmin = m − n − k

jmax = m + n + k




Rotations

TGP: Mean Ecliptic → Mean Equator (Simplified Model)

Ymn

(δL, αL

)=

n∑k=−n

Dnkm (0,−ε, 0)Y k

n (l , b)




Rotations

TGP: Mean Ecliptic → True Equator (Accounting for Nutation)

Ymn

(δL, αL

)=

n∑k=−n

Dnkm (−∆ψ,−ε−∆ε, 0)Y k

n (l , b)




Rotations

TGP: Mean Ecliptic → True Equator (Accounting for Nutation)

Ymn

(δL, αL

)=

n∑k=−n


n (l , b)

Where . . .

Dnkm (−∆ψ,−ε−∆ε, 0) = eik∆ψdn

km (−ε−∆ε) ei π2

(k−m)

dnkm (−ε−∆ε) =

u2∑u=u1

wnukm

(cos

ε+ ∆ε

2

)2n−(m−k+2u)

·(− sin

ε+ ∆ε

2

)m−k+2u




Nutation effects

Equation of equinoxes: effects on Earth orientation

θg = ∆ψ cos (ε+ ∆ε) + θg




After the transformations

Bnm is ready for substitutions from the theories

Bnm(t) = (2− δ0m)4π

2n + 1

MLaeMT

(aeρ

)n+1

e−im[∆ψ cos(−ε−∆ε)+θg ]

n∑k=−n


n (l , b)

Automatic manipulations are now required

Complex exponential (i.e., trigonometric functions) of Poissonseries

Negative powers of Poisson series




A brief example . . .

Complex exponential of the lunar longitude

b =∑

m

[Cm cos (m ·D)]

eikb = eik∑

m[Cm cos(m·D)]

=∏m

∞∑s=0

(2− δ0s) J2s (kCm) cos [2s m ·D] +

+ 2i∞∑s=0

J2s+1 (kCm) sin [(2s + 1)m ·D]




The results



MotivationLow-thrust dynamicsConvergent perturbative theoriesDivergent perturbative theory?

Outline

1 The Tide-Generating PotentialHarmonic Development of the TGPTheories of MotionTransforming The Theories

2 Analytical models for the motion of a LT spacecraftMotivationLow-thrust dynamicsConvergent perturbative theoriesDivergent perturbative theory?




Optimisation of chemical trajectories

chemical thrust: very high acceleration, very short time

modelled as instantaneous ∆V changes

keplerian ballistic arcs

deep-space maneuvers (DSM)

planetary flybys

optimisation: adjust launch/DSM/flyby epochs, initialvelocity, . . . and, e.g., maximise final mass

nonlinear programming problem (NLP)




Chemical trajectories – an example




Optimisation of low-thrust (LT) trajectories

low thrust: very low acceleration, very long time

non-keplerian trajectory arcs

direct methods:

Sims-Flanagan modeloptimal control approximated with a piecewise constantparametrizationtrajectory divided in small segmentsthe optimisation becomes an NLPwithin each segment, dynamics is still assumed to be keplerianfeasible/optimal trajectories in this transcription might not befeasible/optimal with more realistic dynamics




LT trajectories – an example




Plan & objectives

use perturbation theory to model the low-thrust segments

devise qualitative and quantitative models for fixed low-thrustpropulsion

increase the fidelity of the dynamical model

be faster than numerical integration

be well-suited for NLP




Equations of motion

Equation of motion:

a = − r

r3+ εux ,y ,z

Hamiltonian:

H =1

2v2 − 1

r− ε x , y , z

= H0 + εH1,

with

H0 =1

2v2 − 1

r, H1 = −x , y , z




Transformation in modified Delaunay variables

Λ =√a, λ = M +$,

P =√a(

1−√

1− e2), p = −$,

Q = 2√

a (1− e2) sin2 i

2, q = −Ω.

Hamiltonian:

H0 = − 1

2Λ2,

H1 = H1 (Λ,P,Q, λ, p, q) ,

where H1 is a Poisson seriesFrancesco Biscani Application of CAS to Celestial Mechanics



The planar case

Thrust direction along the x axis. Perturbative Hamiltonian (order1 in P, or order 2 in e):

H1 = H1 (Λ,P,−, λ, p,−) =

3

2Λ

32C

12

2 P12 cos (p) +

(1

2ΛC2P − Λ2

)cos (λ)− 1

2Λ

32C

12

2 P12 cos (2λ+ p)− 3

8ΛC2P

cos (3λ+ 2p)− 1

8ΛC2P cos (λ+ 2p)

We can eliminate λ with Lie series transformations.




The planar case: normal form

normal form calculated via integration over λ of the terms ofH1 depending on λ

general expression of the normal form up to order n in ε:

H =n∑

i=0

εncn (Λ,P) cos (np) ,

with cn (Λ,P) Laurent series in Λ12 , P

12




The planar case: 1st order

Hamiltonian of the averaged system:

H′ = − 1

2 (Λ′)2+ εc1

(Λ′,P ′

)cos p′

Equations of motion:

dΛ′

dt= −∂H

′

∂λ′= 0,

dλ′

dt=∂H′

∂Λ′=

1

(Λ′)3+ ε

∂c1

∂Λ′cos p′,

dP ′

dt= −∂H

′

∂p′= εc1 sin p′,

dp′

dt=∂H′

∂P ′= ε

∂c1

∂P ′cos p′.




The planar case: observations

mean semi-major axis is conserved

time variation of P ′ has the sign of sin p′ = − sinω′

time variation of p′ has the sign of cos p′ = cosω′

an initial condition of p′ = ±π2 results in a “frozen orbit” in

which eccentricity increases/decreases with the rest of themean elements evolving keplerianly




The planar case: lines of constant H

0.05 0.10 0.15 0.20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

P

cosp




The spatial case

Thrust direction along the z axis. Perturbative Hamiltonian (order1 in P,Q):

H1 = H1 (Λ,P,Q, λ, p,Q) = −2Λ32C− 1

22 Q

12 sin (λ+ q)−

ΛP12Q

12 sin (2λ+ p + q)− 3ΛP

12Q

12 sin (p − q)

We can eliminate λ with Lie series transformations.




The spatial case: normal form

normal form calculated via integration over λ of the terms ofH1 depending on λ

general expression of the normal form up to order n in ε:

H =n∑

i=0

εncn (Λ,P,Q)cossin

[n (p − q)] ,

with cn (Λ,P,Q) Laurent series in Λ12 , P

12 , Q

12

H can be reduced to 1 DOF




The spatial case: normal form

Canonical transformation:

Λt = Λ, λt = λ,

Pt = P + Q, pt = p,

Qt = Q, qt = q − p = ω.

The inverse transformation P = Pt − Qt can be expandedwith the binomial theorem




The spatial case: 1st order

Hamiltonian of the averaged system:

H′ = − 1

2 (Λ′)2+ εc1

(Λ′,P ′t ,Q

′) sinω′

Equations of motion:

dΛ′

dt= −∂H

′

∂λ′= 0,

dλ′

dt=∂H′

∂Λ′=

1

(Λ′)3+ ε

∂c1

∂Λ′sinω′,

dP ′tdt

= −∂H′

∂p′= 0,

dp′

dt=∂H′

∂P ′t= ε

∂c1

∂P ′tsinω′,

dQ ′

dt= −∂H

′

∂ω′= −εc1 cosω′,

dω′

dt=∂H′

∂Q ′= ε

∂c1

∂Q ′sinω′.




Divergent perturbative theory?

Issues with convergent theories:

explicit solution still requires the integration of non-trivialfunctions

not much better than ODE integration for the practical usesoutlined previously

singularities


application of cas to celestial mechanics: two examples

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