practical stability boundaries around l4,5 in the spatial r3bp

Practical Stability Boundariesaround L4,5 in the spatial R3BP

Carles Simo* Priscilla A. Sousa Silva** Maisa O. Terra**

* Universitat de Barcelona - MAiA** Instituto Tecnologico de Aeronautica (SP/Brasil)

San Martino al Cimino - Viterbo, Italy03/09/2013

Introduction

Introduction

Domains of Practical* Stability occur in non-integrable dynam-ical systems with N > 2 D.O.F.For N > 2

N -dimensional invariant tori are not able to confine trajectoriesin the 2N phase-spaceArnold diffusion effects are expected (slow rates of diffusion)

*Practical (or Effective) = trajectories remainbounded for a very long time span (not time!)

Questions:Up to which distance escape is slow, and why crossing some quasi-boundaries it becomes relatively fast?Which objects are responsible for the quasi-confinement?

P.A. Sousa Silva (ITA) Practical Stability Boundaries 03/09/13 2 / 19

Introduction

Motivation

Jupiter co-orbitals3893 Greeks around L4 and 1993 Trojans around L5

user:Mdf/Wikimedia Commons/Public

Domain

On 19/08/2013, there were 5901 Trojan minor planets listed in the Solar System:Earth (1), Mars (5), Neptune (9), Jupiter (5886).

Data from the Minor Planet Center - IAU (www.minorplanetcenter.org)


Introduction The Restricted Three-body Problem

The Restricted Three-body Problem

3 D.O.F. HamiltonianParticle P3 of negligible mass moving under the gravitational influenceof P1 and P2 of masses m1 and m2.The primaries describe circular coplanar orbits around the barycenterof P1-P2 and are fixed in the synodic reference frame (which rotatesw.r.t. an inertial frame).Distance between the primaries, the sum of their masses and theirangular velocity around the barycenter are normalized to one. = m2/(m1 +m2), m1 > m2 is the only parameter.

Equations of Motion:

x 2y = x, y + 2x = y , z = z ,(x, y, z) =

1

2(x2 + y2) +

1 r1

+

r2+(1 )

2,

r1 =

(x )2 + y2 + z2, r2 =

(x+ 1 )2 + y2 + z2

Integral of motion: J(x, y, z, x, y, z) = 2(x, y, z) (x2 + y2 + z2) = C,5D manifold M(,C) = {(x, y, z, x, y, z) R6|J(x, y, z, x, y, z) = const.}

Hill regions:

Projection of M onto configuration spaceAreas accessible to trajectories for each CBounded by the Zero Velocity Surfaces

Symmetries:

(x, y, z, x, y, z, t) (x,y, z,x, y,z,t)(x, y, z, x, y, z, t) (x, y,z, x, y,z, t)-


Introduction The Restricted Three-body Problem

Equilibrium points

Collinear points: L1,2,3, on the x-axiscenter-center-saddle4D central manifold: vertical and horizontal Lyapunov orbits, Halo orbits, invarianttori, other periodic orbits, chaotic regions.

Triangular points: L4,5, at x = 12 , y =

32, z = 0

Planar case: nonlinear stability for (0, 1)\{2, 3}1 =

(969)18 , 2 =

(451833)90 , 3 =

(15213)30 .

Spatial case: nonlinear stability (on a ball of small -dependent radius around L4,5)for (0, 1)\{2, 3}, except a set of initial conditions of small Lebesgue measurefor fixed .

Normal forms give rise to Nekhorosev-like estimates of diffusion: possible escape isextremely slow

Markeev, A.P., On the stability of the triangular libration points in the circular bounded three-body problem,Applied Math. Mech. 33 (1969), 105-110.

Markeev, A.P., Stability of the triangular Lagrangian solutions of the restricted three-body problem in thethree-dimensional circular case, Soviet Astronomy 15 (1972), 682-686.

Leontovich, A.M., On the stability of the Lagrange periodic solutions of the restricted problem of threebodies, Soviet Math. Dokl., 3 (1962), 425-428.

Deprit, A. and Deprit-Bartholome, A., Stability of the triangular Lagrangian points, The AstronomicalJournal 72, (1967), 173-179.


Introduction Some previous results

Brief History of the problem

Rough investigation of the planar R3BP for the Earth-Moon system (480 dimensionless units of time). McKenzie,R. and Szebehely, V., Nonlinear stability motion around the triangular

libration points, Celestial Mechanics 23 (1981), 223-229.

Numerical evidence that the boundary of the stable do-main in the planar case is related to the central manifoldof L3. Gomez, G. and Jorba, A. and Simo, C. and Masdemont, J., Dy-namics and Mission Design Near Libration Points, Volume IV: Advanced

Methods for Triangular Points, World Scientific (2001).

Prediction of practical stability domain in the spatial casebased on Nekhorosev type estimates. Giorgilli, A., Delshams,A., Fontich, E., Galgani, L., Simo, C. Effective Stability for a Hamilto-

nian System near an Elliptic Equilibrium Point, with an application to the

Restricted Three Body Problem, J. of Diff. Eq. 77 (1989), 167-370.

Numerical simulations show that the stable do-main in the 3D case is larger than the one in the2D case

Simo, C., Effective Computations in Celestial Mechanics and Astrodynamics, in Modern Methods of An-alytical Mechanics and their Applications, Ed. Rumyantsev, V. and Karapetian, A. CISM Courses andLectures Vol. 387, 55102, Springer (1998)Simo, C., Slides of the talk Boundaries of Stability, given at UB (June 3, 2006)available at www.maia.ub.es/dsg/2006/


Detecting effective stability domains in the spatial R3BP

Selecting initial conditions for numerical explorations

The Zero Velocity Surface (3D subspace with null initial velocity)

x = + (1 + %) cos(2pi),

y = (1 + %) sin(2pi)

z = z0 > 0,x = y = z = 0,

The values for % R are found by = 1 0.5z20 + 0.09375z40 +O(z50) which arethe first terms of the solution of z20 = 4(1 + R

2)2 R2, with (z0) = R andR = 1 + %.

[0, 0.5]L5 is at % = 0, = 1/3, z0 = 0

IC set expected to provide seeds to detect invariantobjects at the stability boundary.

For L4 results are obtained from the time reversibility of the R3BP:

S : (x, y, z, x, y, z, t) (x,y, z,x, y,z,t).P.A. Sousa Silva (ITA) Practical Stability Boundaries 03/09/13 7 / 19


Extensive detection of the stability region and its boundaryfor different values of the mass parameter

= 2 104 SJ EM

(, , z = 0)

= 2 104 ( of Sun-Saturn); Sun-Jupiter: SJ = 9.538754 104Earth-Moon: EM = 1.21506683 102

Simo, C., Sousa-Silva, P., Terra, M., Practical Stability Domains near L4,5 in the Restricted Three-BodyProblem: some preliminary facts, in Progress and challenges in dynamical systems, Springer Proceedingsin Mathematics and Statistics, (2013)More cases shown in Simo, C., Slides of the talk Boundaries of Stability, given at UB (June 3, 2006)available at www.maia.ub.es/dsg/2006/



The shape of the effective stability region

= 2 104, = 107, t = 106 (rev)(, , z) (x, y, z)

(z, C)

, , t (rev)

z0 = 0.1 z0 = 0.8

Showing escape times only for t = 104 (rev)



Quasi-confined (top) and unconfined orbits (bottom)

xy projection of Poincare iterates through z = 0, z > 0

z0 = 0.1 z0 = 0.8



Escape patterns

xy projection of the Poincare Section of unconfinedtrajectories through of z = 0, z > 0

= 1/3, z0 = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.70.8.0, 0.85, 0.867



The escape process: dynamics near Hyperbolic Structures

In the planar case there is one periodic orbit around L3 for a givenC < C3. In the Poincare section each is a saddle. In the completephase space W s and Wu are 2D. So...

The central manifold of L3 is the 2D foliation of the unipara-metric family of P.Os and has 3D (codimension-1) hyperbolicstable and unstable manifolds which account for the quasi-confinement!

In the 3D case the libration point is centre centre saddle. So...The central manifold of L3 is a 4D sphere (3D for fixed C < C3) with the familyof planar Lyapunov orbits, the family of vertical Lyapunov orbits, a Cantorianfamily of invariant tori connecting them for every energy level, later on Haloorbits appear... + there are other periodic orbits (associated to resonances) andalso small chaotic zones

Codimension-1 is now 5D!

For fixed energy, we will be looking for 2D Cantor sets of 2D tori with 3Dhyperbolic manifolds.Moreover...

Different escape patterns suggest that we can find different objects playing arole in the quasi-confinement of trajectories around L5.






In the 3D case the libration point is centre centre saddle. So...

The central manifold of L3 is a 4D sphere (3D for fixed C < C3) with the familyof planar Lyapunov orbits, the family of vertical Lyapunov orbits, a Cantorianfamily of invariant tori connecting them for every energy level, later on Haloorbits appear... + there are other periodic orbits (associated to resonances) andalso small chaotic zones











For fixed energy, we will be looking for 2D Cantor sets of 2D tori with 3Dhyperbolic manifolds.

Moreover...



The central manifold of a family of periodic orbits near L5

The case of large z0

Quasi-confinedtrajectories visitrecurrently two2D tori, seen asapproximatedinvariant curveson the PoincareSection z = 0,

z > 0.

z0 = 0.46, = 0.35, C = 2.79507059



z0 = 0.8, = 1/3, C = 2.43554203

The approximated invariant curves from the integrated data are refined usingstandard Fourier methods. Castella`, E. and Jorba, A., On the vertical families of two-dimensionaltori near the triangular points of the Bicirbular Problem, Celestial Mechanics and Dynamical Astronomy

76 (2000), 35-54.P.A. Sousa Silva (ITA) Practical Stability Boundaries 03/09/13 14 / 19


Continuation of tori varying rotation number ()

(xmax, )

The tori found at the boundary are continued until two symmetric (w.r.t z = 0)periodic orbits, 1 and 2, are reached.



A relevant family of periodic orbits

v5C > C: center-center-centerC < C: center-center-saddle

1,2center-center-saddle

C = 1.94769998

v5 : red

1: green

C = C: blue(2 not shown)


The central manifold of L3

The case of small z0

z0 = 0.1, = 0.16,C = 2.99024550

-

(xy, z = 0, z > 0)-

Green: 0 6 t 6 0.1tescRed: 0.9tesc 6 t 6 tesc

To correctly identify tori that play a role in quasi-confinement, one needs to deal with:- Slow rotational dynamics under the Poincare Map;- Large instability;- Recurrent visits to the L3 region related to hetero-clinic connections for a given C.

(xy projection)


The central manifold of L3

Correspondence between tori approximated from the timeseries of trajectories and tori continued from v3

C = 2.99033255--

--

L3 Vertical LyapunovL3 Planar Lyapunov

Hyperbolic 2D torus (dots)Poincare section of 2D torus (line)


Conclusions

We have seen that...There are relatively large practical stability domains with sharp boundariesfor small values of .Different objects play a role in the the quasi-confinement of trajectories.How? By means of their invariant manifolds.Codimension-1 manifolds are important to achieve a complete descriptionof the dynamics.

Ongoing work...Systematic computation of the hyperbolic invariant manifolds of the objectssitting at the boundary.Computation and representation of relevant codimension-1 manifolds: sectionsof sections to visualize 5D objects.The splitting of these manifolds.

Next...Perturbation methods for very small .Other mass parameters (other objects may be important due to higher orderresonances): Sun-Jupiter and Earth-Moon.More realistic models.

Thanks for your attention!

: 2013/07174-4 and 2012/21023-6

: PDE-201932/2010-5


Conclusions

We have seen that...There are relatively large practical stability domains with sharp boundariesfor small values of .Different objects play a role in the the quasi-confinement of trajectories.How? By means of their invariant manifolds.Codimension-1 manifolds are important to achieve a complete descriptionof the dynamics.

Ongoing work...Systematic computation of the hyperbolic invariant manifolds of the objectssitting at the boundary.Computation and representation of relevant codimension-1 manifolds: sectionsof sections to visualize 5D objects.The splitting of these manifolds.

Next...Perturbation methods for very small .Other mass parameters (other objects may be important due to higher orderresonances): Sun-Jupiter and Earth-Moon.More realistic models.

Thanks for your attention!

: 2013/07174-4 and 2012/21023-6

: PDE-201932/2010-5


IntroductionThe Restricted Three-body ProblemSome previous results

Detecting effective stability domains in the spatial R3BPThe central manifold of a family of periodic orbits near L5The central manifold of L3Conclusions

practical stability boundaries around l4,5 in the spatial r3bp

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