Lunar perturbation of the metric associated to the averaged orbital - - PowerPoint PPT Presentation

Lunar perturbation of the metric associated to the averaged orbital transfer

Bernard Bonnard IHP - 28th of November, 2014 Institut de Math´ ematiques de Bourgogne

bernard.bonnard@u-bourgogne.fr

Riemannian metric for the unperturbed problem

Controlled Kepler equation (planar case)

dx dt =

2

∑

i=1

uiFi(x,l) dl dt = w0(x,l)+g(x,l,u). x = (n,ρ,θ) are the slow orbital elements of

the satellite.

n is the mean motion, ρ the eccentricity, θ the argument of periapsis, l is the fast angle pointing out the satellite’s

position on his orbit. Averaged Hamiltonian for the energy cost

H = 1 4n

5 3

• 18n2p2

n +5(1−ρ2)p2 ρ +(5−4ρ2)p2 θ

ρ2

• .

The Riemannian metric associated

g = 1 9n

1 3

dn2 + 2n

5 3

5(1−ρ2)dρ2 + 2n

5 3

5−4ρ2dθ 2.

Model for the lunar perturbation

The perturbed dynamic can be written as

dx dt = F0(x,l,l′)+

2

∑

i=1

uiFi(x,l) dl dt = w0(x,l)+g(x,l,u) dl′ dt = w1(l′)

adding l′ as a fast angle pointing out the satellite’s position on his orbit. The double averaged drift vector field corresponding to the lunar perturbation is

F0(n, p,θ) = 3 4 n′2 n

• 1−ρ2 ∂

∂θ (n, p,θ).

Zermelo navigation problem

A Zermelo navigation problem on a n-dimensional Riemannian manifold (X ,g) is a time minimal problem associated to the system

dx dt = X0(x)+

n

∑

i=1

uiXi(x)

where Fi form an orthonormal frame for the metric g and |u| ≤ 1. The double averaged Zermelo Hamiltonian associated to our controlled system is

Hperb = p,F0(x, p)+ε

• H(x, p)

= pθ 3 4 n′2 n

• 1−ρ2
• +ε
• 1

4n

5 3

• 18n2p2

n +5(1−ρ2)p2 ρ +(5−4ρ2) p2 θ

ρ2

• .

.

Geometric concept of conjugate point

Let −

→ H pert the Hamiltonian vector field associated to Hpert.

• z = (x, p) is a reference extremal, solution of −

→ H pert on [0,t f].

Jacobi equation

˙ δz(t) = d− → H pert(z(t))δz(t)

A Jacobi field is a non trivial solution δz = (δx,δ p), vertical at time t if δx(t) = 0.

• tc : the first conjugate time such that the map p0 → expx0(t, p0) = x(t,x0, p0) is

not an immersion at t = tc.

• Rank test condition to com-

pute tc for a given trajectory

x(t,x0, p0) det[δx1(tc),...,δxn−1(tc), ˙ x(tc)] = 0.

50 100 150 200 250 300 350 −0.05 0.05 0.1 0.15 0.2 det(δx1(t) δx2(t) ˙ x(t)) t

Time evolution of trajectories

50 100 150 200 250 300 350 6 7 8 9 t (days) rad/day n 50 100 150 200 250 300 350 1.4 1.45 1.5 ψ rad t (days) 50 100 150 200 250 300 350 0.5 1 1.5 θ t (days) rad final point conjugate point

Evolution of the state vector

(n(t),ψ(t) = π

2 −asin(ρ(t)),θ(t)).

50 100 150 200 250 300 350 −0.12 −0.11 −0.1 −0.09 t (days) pn 50 100 150 200 250 300 350 −0.5 0.5 pρ t (days) 50 100 150 200 250 300 350 400 −0.05 0.05 pθ t (days) final point conjugate point

Evolution of the adjoint vector

(pn(t), pρ(t), pθ(t)).

solide line : unperturbed case, dash-dot line : perturbed case.

Trajectories in (θ,ψ) coordinates

pi/2 pi 3pi/2 2pi pi/4 pi/2 (θ , ψ) θ ψ conjuguate points

Trajectories from the unperturbed case.

pi/2 pi 3pi/2 2pi 5pi/2 pi/4 pi/2 (θ , ψ) θ ψ conjuguate points

Trajectories from the perturbed case.