Some Mass Transfer Questions in the Solar System

Transcription

1 Some Mass Transfer Questions in the Solar System INDAM Workshop Mathematical Models and Methods for Planet Earth Gerard Gómez Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona & IEEC Roma, May 28th 2013 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

2 In collaboration with... Josep J. Masdemont (Univ. Politècnica de Catalunya & IEEC), Elisa M. Alessi (Istituto di Fisica Applicata "Nello Carrara" - CNR), Esther Barrabés (Univ. de Girona), Elena Fantino (Univ. Politècnica de Catalunya), Josep M. Mondelo (Univ. Autònoma de Barcelona), Zubin Olikara (IEEC), Mercè Ollé (Univ. Politècnica de Catalunya), Daniel Pérez (Univ. de Barcelona), Yuan Ren (York Univ.),... G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

3 Purpose and Outline Preliminaries and models Purpose of the talk: To stress the role of invariant manifolds in the dynamics in (models of) the Solar System Outline 1 Preliminaries and models 2 Spacecraft applications of the stable/unstable manifolds 3 Astronomical applications of the stable/unstable manifolds G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

4 Applications of the stable/unstable manifolds 1 Preliminaries and models G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

5 Applications of the stable/unstable manifolds Models with saddle critical points A model with saddle centre critical points: The planar restricted 3 body problem (R3BP) L 4 Rock L 3 L 1 L 2 Sun Planet L 5 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

6 Applications of the stable/unstable manifolds Models with saddle critical points Homoclinic and heteroclinic connections between saddle points Heteroclinic connection Homoclinic connection Heteroclinic intersections Homoclinic intersections G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

7 Applications of the stable/unstable manifolds Homo and heteroclinic connections in the R3BP Examples of homo and heteroclinic connections in the R3BP q z q q y x Homoclinic connection at L 2 q y q x Heteroclinic connection between periodic orbits around L 1 and L 2 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

8 Applications of the stable/unstable manifolds Models with critical points and periodic orbits with saddle behaviour Other simple models: Two patched R3BPs y SE y EM x EM L 2 EM Sun L 1 SE Earth L 1 EM L 2 SE Moon xse G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

9 Applications of the stable/unstable manifolds Other simple models: Bicircular model Models with critical points and periodic orbits with saddle behaviour L L 1 2 Sun w M t Earth Mars G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

10 Applications of the stable/unstable manifolds Models with critical points and periodic orbits with saddle behaviour Other simple models: Coherent bicircular model L L 1 2 Sun Earth Mars This model considers the gravitational effect of Sun, Earth and Mars on an infinitesimal particle. The motion of the Sun, Earth and Mars verifies the equations of the three-body problem, and their orbits are given by their (truncated) Fourier expansions G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

11 Applications of the stable/unstable manifolds 2 Spacecraft applications of the stable/unstable manifolds Computation of Earth-Moon transfers Protecting the Earth s environment from space debris G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

12 Applications of the stable/unstable manifolds Solar and astronomical observatories around the L 1 and L 2 equilibrium points ISEE 3 acrobatics Herschel spacecraft transfer and nominal orbit G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

13 Applications of the stable/unstable manifolds Computation of Earth-Moon transfers Transfer from the Earth to a halo orbit of the Sun-Earth system using W s (halo) G. Gómez, A. Jorba, J.J. Masdemont, C. Simó: Study of the Transfer from the Earth to a Halo Orbit Around the Equilibrium Point L1, Cel. Mech. & Dyn. Astron. 56 (4) (1993) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

14 Applications of the stable/unstable manifolds Computation of Earth-Moon transfers Effect of the Moon on the distance from the orbits of the W s (halo) to the Earth G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

15 Applications of the stable/unstable manifolds Computation of Earth-Moon transfers Transfers to the Moon using two patched R3BPs W.S. Koon, M.W. Lo, J.E. Marsden, S.D. Ross: Low Energy Transfer to the Moon, Cel. Mech. & Dyn. Astron. 81 (1) (2001) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

16 Applications of the stable/unstable manifolds Computation of Earth-Moon transfers Transfers between two Jupiter moons using two patched R3BPs G. Gómez, W.S. Koon, M.W. Lo, J.E. Marsden, J.J. Masdemont, S.D. Ross: Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity 17 (5) (2004) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

17 Applications of the stable/unstable manifolds Protecting the Earth s environment from space debris End-of-life mission design at Sun-Earth libration point orbits Primary options from L 1 / L 2 libration point orbits: Perform maneuver to close the zero-velocity curve/surface Use natural dynamics to put the spacecraft into long-term orbit avoiding bottleneck Put the spacecraft into highly-elliptic Earth orbit Intentional Earth reentry or Moon collision There are some end-of-life design limitations From Herschel/Planck work, wanted to perform all maneuvers within 3 months after the end of the mission Collision orbits (Earth or Moon) are regarded as potentially complex and risky Z. Olikara, G. Gómez, J.J. Masdemont: Ongoing research G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

18 Applications of the stable/unstable manifolds Protecting the Earth s environment from space debris End-of-life mission design at Sun-Earth LPO s Leaving L 2 orbit through L 1 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

19 Applications of the stable/unstable manifolds Protecting the Earth s environment from space debris End-of-life mission design at Sun-Earth LPO s Cost to close ZVC at L1 For specified initial L 2 orbit, must wait at least 1 year before performing maneuver. The pink regions correspond to orbits that pass within GEO radiu G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

20 Astronomical applications 3 Astronomical applications of the stable/ unstable manifolds G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

21 Astronomical applications Astronomical applications Astronomical applications: Formation of spiral arms in barred galaxies M. Romero, E. Athanassoula, J.J. Masdemont, C. García.: The Formation of Spiral Arms and Rings in Barred Galaxies, Astronomy & Astrophysics 472 (1) (2007) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

22 Astronomical applications Astronomical applications Astronomical applications: Planet formation hal/talks.html H. Levison. Modeling the Formation of Giant Planet Cores G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

23 Astronomical applications Astronomical applications: Distribution of low-energy lunar impact craters Distribution of low-energy lunar impact craters Nearside Farside E.M. Alessi, G. Gómez, J.J. Masdemont: A motivating exploration on lunar craters and low-energy dynamics in the Earth-Moon system, Cel. Mech. & Dyn. Astron. 107 (1-2) (2010) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

24 Astronomical applications Astronomical applications: Distribution of low-energy lunar impact craters Some questions and the approach Questions 1 Is there a reason for the alleged asymmetry of impact between nearside and farside? 2 Does the Moon act as a shield for the Earth? 3 Which is the role of the Sun in this process? Approach 1 Main channel to get to the Moon W s (W c L 2 ) 2 Use transit trajectories inside W s (W c L 2 ) for C 3 < C J < C 2 3 Use different distributions of initial conditions 4 Use different values of the Earth-Moon distance: d EM G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

25 Astronomical applications W s (W c L 2 ): The main channel Astronomical applications: Distribution of low-energy lunar impact craters y 0 y x x G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

26 Astronomical applications Astronomical applications: Distribution of low-energy lunar impact craters Distribution of impacts for i.c. uniformly distributed in W s (W c L 2 ) latitude e e e-08 3e e e e e-08 2e e-08 latitude e e e-08 3e e e e e-08 2e e-08 longitude longitude d EM = km d EM = km G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

27 Astronomical applications Astronomical applications: Distribution of low-energy lunar impact craters The effect of the Sun: The Bicircular Restricted 4 Body Problem The infinitesimal mass affected by the gravitational attractions of Earth, Moon and Sun moving in circular orbits. y P r 2 r s m 1 = 1-μ x m 2 = μ θ 0 a s m s We use the same initial conditions as the ones uniformly distributed inside W s (W c L 2 ) for the CR3BP G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

28 Astronomical applications Astronomical applications: Distribution of low-energy lunar impact craters Distributions of impacts d EM = km e e e e e e e e e e-08 latitude 0 2.8e-08 latitude 0 2.8e e e e e e e e e e e-08 longitude longitude θ 0 = 108 θ 0 = 252 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

29 From Mars to the Earth Mass transport from Mars to the Earth (martian meteorites) Y. Ren, J.J. Masdemont, G. Gómez, E. Fantino: Two mechanisms of natural transport in the Solar System. To appear in Communications in Nonlinear Science and Numerical Simulation, G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

30 Motivation From Mars to the Earth B.J. Gladman, J.A. Burns, M. Duncan, P. Lee, H.F. Levison: The Exchange of Impact Ejecta Between Terrestrial Planets, Science 271 (5254) (1996) G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

31 From Mars to the Earth Minimum and maximum heliocentric distances (in AU) PCR3BP [ R W u(s) L 1 ] [ ] R W u(s) L 2 min max min max Sun-Mercury Sun-Venus Sun-Earth Sun-Mars Sun-Jupiter * Sun-Saturn * Sun-Uranus * Sun-Neptune * The above "connections" do not explain exchange of material between terrestrial planets G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

32 From Mars to the Earth Perihelion and aphelion distances in the Sun-Jupiter CR3BP For a fixed value of C and y = 0, we determine, for each value of (x, ẋ), the first perihelion and aphelion distances With these results we distinguish several regions in the (x, ẋ) plane G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

33 From Mars to the Earth (x, ẋ) regions for a fixed value of C 1.5 Forbidden Region Region 1 Region 2 Region Forbidden Region Region 1: I.C. of inertial orbits that do not intersect the orbits of the Earth and Mars Region 2: I.C. of orbits which only intersect the orbit of Mars Region 3: I.C. of orbits which intersect the orbit of the Earth G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

34 From Mars to the Earth Poincaré map behaviour for C = 3.14, 3.09, 3.06, 3.03 (x E 0.2, x M 0.3) Transport is possible when the chaotic sea reaches regions 2 and 3 G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

35 From Mars to the Earth Heteroclinic intersections 1 The manifolds of one of the order-4 saddle points have intersections with the orbit of the Earth 2 The manifolds of the order-5 saddle point have intersections with the orbit of Mars 3 These manifolds have intersections with each other 4 The transport between regions of the phase space can be completely described by the dynamical evolution of the lobe determined by the region enclosed by the above stable and unstable manifolds. G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

36 From Mars to the Earth Lobe evolution providing natural transport between regions 2 and 3 Tï6 T2 1 T5 Tï3 T Tï Vx T4 0 Tï4 ï0.5 Tï5 T ï2 T 21 T1 ï1 T T 6 ï0.7 ï0.6 ï0.5 x ï0.4 ï0.3 3 T ï1 ï0.2 T represents the initial states near the intersection of the manifolds of the order-4 and order-5 saddle points. After 21 iterations in the forward direction, some of these states move from region 2 to region 3. G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

37 From Mars to the Earth Example of transport orbit in the CR3BP for C = 3.03 (synodic and inertial) Orbit of Mars Orbit of Earth Y 0 Jupiter Sun Y Orbit of Jupiter X X Integration time = 1283 years G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40

38 From Mars to the Earth Thank you for your attention G. Gómez (MAiA-UB & IEEC) Some Mass Transfer Questions in the Solar System Roma, May 28th / 40