Transport in the Solar System

Introduction General Setting Methodology and Results Work in progress

Unstable orbits and transport in the Solar System

E. Barrabes

(University of Girona)

(Dynamical Systems Seminar, BU) September 16, 2013 1 / 28


Work in collaboration with

Gerard Gomez Josep M. Mondelo Merce OlleU. de Barcelona U. Autonoma de Barcelona U. Politecnica de Catalunya





Outline

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 verydistant locations

NATURAL TRANSPORT

Aim: give a dynamical mechanism to explain natural transport inthe Solar System



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.



Planar Circular Restricted Three Body Problem (CRTBP)

L 1L 2

L 5

L 4

L 355 5

5

5

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



Planar Circular Restricted Three Body Problem (CRTBP)

The transport phenomena in the CRTBP can be obtained from the analysis ofthe behavior of the stable and unstable invariant manifolds associated toperiodic solutions, specifically those around the collinear equilibrium points.

(from Koon et al, (2002))



Trajectories with prescribed itineraries and resonant transitions

Comet Oterma: resonant transitions

(from Koon et al, Chaos (2000))



Dynamical channels in the Solar SystemThe invariant manifolds associated to L1 and L2 for each of the giant outerplanets. There are intersections between manifolds of collinear points ofadjacent Sun-planet CRTBP.

(from Koon et al, Chaos (2000))



Short-time natural transport

(2012) Y. Ren, J.J. Masdemont, G. Gomez, 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 connectionsbetween libration point orbits of a pair of consecutive Sun-planet PC3BPs.



Short-time natural transport

This short-time transport concept cannot explain the exchange of naturalmaterial throughout the inner Solar System.



Models

Model 1: a chain of Bicircular Problems (BCP) with the Sun, Jupiter, aplanet 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



The Solar SystemTransport 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

SJ

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


The Restricted Bicircular ProblemA restricted BiCircular Problem (BCP): Sun, Jupiter, Planet, a particle(S-J-Planet-particle) We suppose that both, Jupiter and the planet are incircular 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


The Restricted Bicircular ProblemA restricted BiCircular Problem (BCP): Sun, Jupiter, Planet, a particle(S-J-Planet-particle) We suppose that both, Jupiter and the planet are incircular 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


Equations of the restricted BCP

The equations may be written as a Hamiltonian system

H =1

2(p2x + p2y + p2z) + ypx − xpy −

1− µρ1− µ

ρ2︸︷︷︸RTBP Sun+Jupiter

−µPρP− µPa2P

(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, Jupiterand the Planet and

µ =mJ

mJ +mS, µP =

mP

mJ +mS



Equations of the restricted BCP

The restricted BCP may be regarded as a periodic perturbation of theCRTBP

H = HCRTBP + µPHP

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 7→ φθ0t (q0) := φ(t; t0, q0, θ0).



Dynamical substitutes of the equilibrium points

The equilibrium points Li give rise to hyperbolic periodic orbits with the sameperiod Tp than the planet: dynamical substitutes

Look for q0 s.t. φθ0Tp(q0) = q0

Initial seeds: the equilibrium points of the CRTBP

They are good enough only for ip = 7, 8. Multiple shooting strategy isneeded for other dynamical substitutes.




Two dynamical substitutes: periodic orbits L1 and L2




-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



Computation of the invariant manifolds

Let q0 be an initial condition of the periodic orbit of period T = 2π/w, Λ aneigenvalue of the monodromy matrix Dφθ0T (q0) and v0 and associatedeigenvalue.

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−θ02π Dφθ0(θ−θ0)/ω(q0)v0

are parameterizations of the periodic orbit and the eigenvector, andξ ' 10−6, 10−7.It satisfies

φθt(ψ(θ, ξ)

)= ψ(θ + tω,Λt/T ξ) +O(ξ2).



Computation of the invariant manifolds

Let q0 be an initial condition of the periodic orbit of period T = 2π/w, Λ aneigenvalue of the monodromy matrix Dφθ0T (q0) and v0 and associatedeigenvalue.

It is enough to start at different distances of the initial point

qi = φθ0ti(q0 + ξΛ−ti/T v0

)= φθ0ti (q0) + ξΛ−ti/TDφθ0ti (q0)v0︸︷︷︸

linear approx

+O(ξ2).

q0

v0

q1q2

q3

0

consider θi = θ0 + tiw ∈ [0, 2π],i = 1, . . . , N and the set of orbits

{φθ0t (q0 + ξΛ

−tiT v0)

}

for t ≤ tmax + ti.



Transport between consecutive BCP

Considering the BCPi and BCPi−1

Compute the unstable manifold (inwards branch) of the dynamicalsubstitute of L1, Wu(L1) in the BCPi

Compute the stable manifold (outwards branch) of the dynamicalsubstitute of L2, W s(L2) in the BCPi−1

Look for intersections of invariant manifolds at an intermediate Poincaresection Σ = {r = ctant} or until t ≤ Tmax

!

OP

OP

!

"

#

$

%

&"

'(

)



Transport between Neptune and Uranus

Semiaxis vs eccentricity of orbits on Wu(L1, ip = 8) (red) and W s(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

ecce

ntr

icity

semiaxis

Wu-(L1,ip8),Ws+(L2,ip7)




Orbital distance to the Sunr(t) of orbits on Wu(L1, ip = 8) (left) and W s(L2, ip = 7) (right)




Minimum distance between Wu(L1, ip = 8) ∩ Σ and W s(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. positio

ns

θ

minimum distance in positions





1e-05

0.0001

0.001

0.01

0.1

0 1 2 3 4 5 6

Dif. velo

citie

s

θ

minimum distance in velocity





0.0001

0.001

0.01

0.1

0 1 2 3 4 5 6

Dif. velo

citie

s

θ

minimum distance in velocity for those points at a distance ≤ 10−5



Transport between Uranus and Saturn

Semiaxis vs eccentricity of orbits on Wu(L1, ip = 7) (red) and W s(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

eccentr

icity

semiaxis



Transport between Uranus and Saturn

Orbital distance to the Sunr(t) of orbits on Wu(L1, ip = 7) (left) and W s(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 substitutesL1 and L2 of Saturn are “more chaotic”



Transport towards the inner planets

The dynamical substitutes L1 and L2 are highly unstable (it is evendifficult 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, 12.473Saturn 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



Still working on it...

The behavior of the manifolds of the Lyapunov periodic orbits in a chainof BCP can give a first indicator of transport in the Solar System withinthe exterior Solar System

Can we obtain similar results in the inner Solar System with a goodaccuracy?

Computation of an approximation of a parametrization of the invariantmanifolds up to a higher order (≥ 2)



Model 2: n body problem

n bodies (Sun and planets are taken into account) interacting betweemthem

a particle attracted by them

For a given initial time, we take the REAL positions and velocities (fromthe JPL ephemerides)

Aim:

We pursue an statistical result: starting from initial conditions on i.m. ofa hyperbolic invariant object, it should be more likely to see transporttowards the inner Solar System than starting from a random set of initialconditions



2do, drawbacks, difficulties, ...

The initial configuration of the planets it is important if we do notcompute the dynamical substitutes of L1 and L2 and we just consider theinitial data those from the BCP

Computation of dynamical substitutes of L1 and L2 (quasi-periodicorbits)

Computation of their invariant manifolds and possible connectionsbetween them



Thank you for your attention

Suggestions, comments,... are very welcome!
