We study the natural transport from the outer to the inner Solar System by modelling the Solar System as a sequence of coupled bicircular models.

1. Introduction General Setting Methodology and Results Work in progress
Unstable orbits and transport in the Solar System
E. Barrab´es
(University of Girona)
(Dynamical Systems Seminar, BU) September 16, 2013 1 / 28

2. Introduction General Setting Methodology and Results Work in progress
Work in collaboration with
Gerard G´omez Josep M. Mondelo Merc´e Oll`e
U. de Barcelona U. Aut`onoma de Barcelona U. Polit`ecnica de Catalunya
(Dynamical Systems Seminar, BU) September 16, 2013 2 / 28

3. Introduction General Setting Methodology and Results Work in progress
(Dynamical Systems Seminar, BU) September 16, 2013 3 / 28

4. Introduction General Setting Methodology and Results Work in progress
Outline
Introduction
General Setting
Methodology and Results
Work in progress
(Dynamical Systems Seminar, BU) September 16, 2013 4 / 28

5. Introduction General Setting Methodology and Results Work in progress
Motivation
Comets, asteroids and small particles in the Solar System are ca-
pable of performing transfers from their original location to very
distant locations
NATURAL TRANSPORT
Aim: give a dynamical mechanism to explain natural transport in
the Solar System
(Dynamical Systems Seminar, BU) September 16, 2013 5 / 28

6. Introduction General Setting Methodology and Results Work in progress
Meteorits from Mars
(1996) B.J. Gladman, J.A. Burns, M. Duncan, P. Lee, and H.F. Levison.
The exchange of impact ejecta between terrestrial planets.
Science, 271(5254):1387–1392.
Evolution of 200 particles launched from Mars. Those with 1 < a(1 + e) = Q
(aphelion) and q = a(1 − e) < 1 (perihelion) can cross the Earth’s orbit.
(Dynamical Systems Seminar, BU) September 16, 2013 6 / 28

7. Introduction General Setting Methodology and Results Work in progress
Planar Circular Restricted Three Body Problem (CRTBP)
L1
L2
L5
L4
L3
SE 0.5
0.5
−0.5
−0.5
Primaries:
masses 1 − µ, µ
circular orbits
fixed in a rotating frame
Equilibrium points:
L1, L2, L3 collinear points
L4, L5 triangular points
(Dynamical Systems Seminar, BU) September 16, 2013 7 / 28

8. Introduction General Setting Methodology and Results Work in progress
Planar Circular Restricted Three Body Problem (CRTBP)
The transport phenomena in the CRTBP can be obtained from the analysis of
the behavior of the stable and unstable invariant manifolds associated to
periodic solutions, specifically those around the collinear equilibrium points.
(from Koon et al, (2002))
(Dynamical Systems Seminar, BU) September 16, 2013 7 / 28

9. Introduction General Setting Methodology and Results Work in progress
Trajectories with prescribed itineraries and resonant transitions
Comet Oterma: resonant transitions
(from Koon et al, Chaos (2000))
(Dynamical Systems Seminar, BU) September 16, 2013 8 / 28

10. Introduction General Setting Methodology and Results Work in progress
Dynamical channels in the Solar System
The invariant manifolds associated to L1 and L2 for each of the giant outer
planets. There are intersections between manifolds of collinear points of
adjacent Sun-planet CRTBP.
(from Koon et al, Chaos (2000))
(Dynamical Systems Seminar, BU) September 16, 2013 9 / 28

11. Introduction General Setting Methodology and Results Work in progress
Short-time natural transport
(2012) Y. Ren, J.J. Masdemont, G. G´omez, and E. Fantino.
Two mechanisms of natural transport in the solar system.
Communications in Nonlinear Science and Numerical Simulation,
The short-time mechanism is based on the existence of heteroclinic connections
between libration point orbits of a pair of consecutive Sun-planet PC3BPs.
(Dynamical Systems Seminar, BU) September 16, 2013 10 / 28

12. Introduction General Setting Methodology and Results Work in progress
Short-time natural transport
This short-time transport concept cannot explain the exchange of natural
material throughout the inner Solar System.
(Dynamical Systems Seminar, BU) September 16, 2013 10 / 28

13. Introduction General Setting Methodology and Results Work in progress
Models
Model 1: a chain of Bicircular Problems (BCP) with the Sun, Jupiter, a
planet and a massless particle
Model 2: n-body problem and a massless particle
Tools: Dynamical systems theory
unstable invariant objects and their manifolds
equilibrium points, periodic and quasi-periodic orbits
(Dynamical Systems Seminar, BU) September 16, 2013 11 / 28

14. Introduction General Setting Methodology and Results Work in progress
The Solar System
Transport from the exterior region to the interior one
-40
-30
-20
-10
0
10
20
30
40
-40 -30 -20 -10 0 10 20 30 40
N
U
S
J
Planets: Mercury (ip = 1), Venus (ip = 2),...., Saturn(ip = 6), Uranus (ip = 7),
Neptune (ip = 8)
(Dynamical Systems Seminar, BU) September 16, 2013 12 / 28

15. Introduction General Setting Methodology and Results Work in progress
The Restricted Bicircular Problem
A restricted BiCircular Problem (BCP): Sun, Jupiter, Planet, a particle
(S-J-Planet-particle) We suppose that both, Jupiter and the planet are in
circular orbits with respect the Sun.
-2
0
2
4
6
-2 0 2 4 6
S
J
Planet
particle
Inertial frame
(Dynamical Systems Seminar, BU) September 16, 2013 13 / 28

16. Introduction General Setting Methodology and Results Work in progress
The Restricted Bicircular Problem
A restricted BiCircular Problem (BCP): Sun, Jupiter, Planet, a particle
(S-J-Planet-particle) We suppose that both, Jupiter and the planet are in
circular orbits with respect the Sun.
-6
-4
-2
0
2
4
6
-2 0 2 4 6
SJ
Planet
particle
Rotating frame
(Dynamical Systems Seminar, BU) September 16, 2013 13 / 28

17. Introduction General Setting Methodology and Results Work in progress
Equations of the restricted BCP
The equations may be written as a Hamiltonian system
H =
1
2
(p2
x + p2
y + p2
z) + ypx − xpy −
1 − µ
ρ1
−
µ
ρ2
RTBP Sun+Jupiter
−
µP
ρP
−
µP
a2
P
(y sin(θ0 + t(ωP − 1)) − x cos(θ0 + t(ωP − 1)))
planet perturbation
where ρ1, ρ2 and ρP are the distances from the particle to the Sun, Jupiter
and the Planet and
µ =
mJ
mJ + mS
, µP =
mP
mJ + mS
(Dynamical Systems Seminar, BU) September 16, 2013 14 / 28

18. Introduction General Setting Methodology and Results Work in progress
Equations of the restricted BCP
The restricted BCP may be regarded as a periodic perturbation of the
CRTBP
H = HCRT BP + µP HP
Mercury 0.1658 × 10−6
Jupiter 0.9539 × 10−3
Venus 0.2445 × 10−5
Saturn 0.2856 × 10−3
Earth 0.3037 × 10−5
Uranus 0.4362 × 10−4
Mars 0.3224 × 10−6
Neptune 0.5146 × 10−4
It is a non-autonomous, periodic system.
˙q = f(q, θ0 + t(ωP − 1))
Associated flow:
t → φθ0
t (q0) := φ(t; t0, q0, θ0).
(Dynamical Systems Seminar, BU) September 16, 2013 15 / 28

19. Introduction General Setting Methodology and Results Work in progress
Dynamical substitutes of the equilibrium points
The equilibrium points Li give rise to hyperbolic periodic orbits with the same
period Tp than the planet: dynamical substitutes
Look for q0 s.t. φθ0
Tp
(q0) = q0
Initial seeds: the equilibrium points of the CRTBP
They are good enough only for ip = 7, 8. Multiple shooting strategy is
needed for other dynamical substitutes.
(Dynamical Systems Seminar, BU) September 16, 2013 16 / 28

20. Introduction General Setting Methodology and Results Work in progress
Dynamical substitutes of the equilibrium points
Two dynamical substitutes: periodic orbits L1 and L2
(Dynamical Systems Seminar, BU) September 16, 2013 17 / 28

21. Introduction General Setting Methodology and Results Work in progress
Dynamical substitutes of the equilibrium points
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
SJ
L1
L2
L1
L2
L1
L2
A chain of dynamical substitutes
(Dynamical Systems Seminar, BU) September 16, 2013 17 / 28

22. Introduction General Setting Methodology and Results Work in progress
Computation of the invariant manifolds
Let q0 be an initial condition of the periodic orbit of period T = 2π/w, Λ an
eigenvalue of the monodromy matrix Dφθ0
T (q0) and v0 and associated
eigenvalue.
The linear approximation of the invariant manifold can be given by
¯ψ(θ, ξ) := ϕ(θ) + ξv(θ),
where
ϕ(θ) = φθ0
(θ−θ0)/ω(q0), v(θ) = Λ−
θ−θ0
2π Dφθ0
(θ−θ0)/ω(q0)v0
are parameterizations of the periodic orbit and the eigenvector, and
ξ 10−6
, 10−7
.
It satisfies
φθ
t
¯ψ(θ, ξ) = ¯ψ(θ + tω, Λt/T
ξ) + O(ξ2
).
(Dynamical Systems Seminar, BU) September 16, 2013 18 / 28

23. Introduction General Setting Methodology and Results Work in progress
Computation of the invariant manifolds
Let q0 be an initial condition of the periodic orbit of period T = 2π/w, Λ an
eigenvalue of the monodromy matrix Dφθ0
T (q0) and v0 and associated
eigenvalue.
It is enough to start at different distances of the initial point
qi = φθ0
ti
q0 + ξΛ−ti/T
v0 = φθ0
ti
(q0) + ξΛ−ti/T
Dφθ0
ti
(q0)v0
linear approx
+ O(ξ2
).
q0
v0
q1
q2
q3
0
consider θi = θ0 + tiw ∈ [0, 2π],
i = 1, . . . , N and the set of orbits
φθ0
t (q0 + ξΛ
−ti
T v0)
for t ≤ tmax + ti.
(Dynamical Systems Seminar, BU) September 16, 2013 18 / 28

24. Introduction General Setting Methodology and Results Work in progress
Transport between consecutive BCP
Considering the BCPi and BCPi−1
Compute the unstable manifold (inwards branch) of the dynamical
substitute of L1, Wu
(L1) in the BCPi
Compute the stable manifold (outwards branch) of the dynamical
substitute of L2, Ws
(L2) in the BCPi−1
Look for intersections of invariant manifolds at an intermediate Poincar´e
section Σ = {r = ctant} or until t ≤ Tmax
OP
OP
I
*
R
?
-
6
(Dynamical Systems Seminar, BU) September 16, 2013 19 / 28

25. Introduction General Setting Methodology and Results Work in progress
Transport between Neptune and Uranus
Semiaxis vs eccentricity of orbits on Wu
(L1, ip = 8) (red) and Ws
(L2, ip = 7)
(blue)
0.05
0.1
0.15
0.2
0.25
0.3
4.2 4.4 4.6 4.8 5 5.2 5.4 5.6
0.05
0.1
0.15
0.2
0.25
0.3
eccentricity
semiaxis
Wu-(L1,ip8),Ws+(L2,ip7)
(Dynamical Systems Seminar, BU) September 16, 2013 20 / 28

26. Introduction General Setting Methodology and Results Work in progress
Transport between Neptune and Uranus
Orbital distance to the Sun
r(t) of orbits on Wu
(L1, ip = 8) (left) and Ws
(L2, ip = 7) (right)
(Dynamical Systems Seminar, BU) September 16, 2013 20 / 28

27. Introduction General Setting Methodology and Results Work in progress
Transport between Neptune and Uranus
Minimum distance between Wu
(L1, ip = 8) ∩ Σ and Ws
(L2, ip = 7) ∩ Σ
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0 1 2 3 4 5 6
Dif.positions
θ
minimum distance in positions
(Dynamical Systems Seminar, BU) September 16, 2013 21 / 28

28. Introduction General Setting Methodology and Results Work in progress
Transport between Neptune and Uranus
Minimum distance between Wu
(L1, ip = 8) ∩ Σ and Ws
(L2, ip = 7) ∩ Σ
1e-05
0.0001
0.001
0.01
0.1
0 1 2 3 4 5 6
Dif.velocities
θ
minimum distance in velocity
(Dynamical Systems Seminar, BU) September 16, 2013 21 / 28

29. Introduction General Setting Methodology and Results Work in progress
Transport between Neptune and Uranus
Minimum distance between Wu
(L1, ip = 8) ∩ Σ and Ws
(L2, ip = 7) ∩ Σ
0.0001
0.001
0.01
0.1
0 1 2 3 4 5 6
Dif.velocities
θ
minimum distance in velocity for those points at a distance ≤ 10−5
(Dynamical Systems Seminar, BU) September 16, 2013 21 / 28

30. Introduction General Setting Methodology and Results Work in progress
Transport between Uranus and Saturn
Semiaxis vs eccentricity of orbits on Wu
(L1, ip = 7) (red) and Ws
(L2, ip = 6)
(blue)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2 3 4 5 6 7
eccentricity
semiaxis
(Dynamical Systems Seminar, BU) September 16, 2013 22 / 28

31. Introduction General Setting Methodology and Results Work in progress
Transport between Uranus and Saturn
Orbital distance to the Sun
r(t) of orbits on Wu
(L1, ip = 7) (left) and Ws
(L2, ip = 6) (right)
2.6
2.8
3
3.2
3.4
3.6
0 2000 4000 6000 8000 10000
r
t
2
2.5
3
3.5
4
4.5
-10000 -8000 -6000 -4000 -2000 0
r
t
The orbits on the invariant manifolds associated to the dynamical substitutes
L1 and L2 of Saturn are “more chaotic”
(Dynamical Systems Seminar, BU) September 16, 2013 22 / 28

32. Introduction General Setting Methodology and Results Work in progress
Transport towards the inner planets
The dynamical substitutes L1 and L2 are highly unstable (it is even
difficult to calculate them numerically)
The linear approximation is invariant under the flow up to order (Λξ)2
.
Values of the eigenvalue Λ 1 for each BCP and dynamical substitute L1
and L2.
Neptune 3.492, 3.286 Uranus 14.105, 12.473
Saturn 6.5× 104
, 2.5× 104
Mars 9× 107
, 2.5× 108
Earth 2.8× 107
, 3.4× 107
Venus 1.5× 107
, 1× 107
For t 10000 the matrix Dφt has big (1012
) components
(Dynamical Systems Seminar, BU) September 16, 2013 23 / 28

33. Introduction General Setting Methodology and Results Work in progress
Still working on it...
The behavior of the manifolds of the Lyapunov periodic orbits in a chain
of BCP can give a first indicator of transport in the Solar System within
the exterior Solar System
Can we obtain similar results in the inner Solar System with a good
accuracy?
Computation of an approximation of a parametrization of the invariant
manifolds up to a higher order (≥ 2)
(Dynamical Systems Seminar, BU) September 16, 2013 24 / 28

34. Introduction General Setting Methodology and Results Work in progress
Model 2: n body problem
n bodies (Sun and planets are taken into account) interacting betweem
them
a particle attracted by them
For a given initial time, we take the REAL positions and velocities (from
the JPL ephemerides)
Aim:
We pursue an statistical result: starting from initial conditions on i.m. of
a hyperbolic invariant object, it should be more likely to see transport
towards the inner Solar System than starting from a random set of initial
conditions
(Dynamical Systems Seminar, BU) September 16, 2013 25 / 28

35. Introduction General Setting Methodology and Results Work in progress
2do, drawbacks, difficulties, ...
The initial configuration of the planets it is important if we do not
compute the dynamical substitutes of L1 and L2 and we just consider the
initial data those from the BCP
Computation of dynamical substitutes of L1 and L2 (quasi-periodic
orbits)
Computation of their invariant manifolds and possible connections
between them
(Dynamical Systems Seminar, BU) September 16, 2013 26 / 28

36. Introduction General Setting Methodology and Results Work in progress
Thank you for your attention
Suggestions, comments,... are very welcome!
(Dynamical Systems Seminar, BU) September 16, 2013 27 / 28