Description of the dynamics of a parabolic problem and the mechanisms that allow to explain mass variation after galaxy close encounters Presentation to the XV Jornadas de Trabajo en Mecánica Celeste, Manresa, June 2016

1. Introduction Dynamics of the parabolic problem Numerical results Conclusions
On the dynamics of the parabolic restricted three
body problem
E. Barrab´es1
J.M. Cors 2
L. Garcia M. Oll´e2
1
Universitat de Girona
2
Universitat Polit`ecnica de Catalunya
XV Jornadas de Trabajo en Mec´anica Celeste
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 1 / 28

2. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 2 / 28

3. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Galactic encounters: bridges and tails (I)
10
NGC 2535/6 NGC 7752/3
astromania.deyave.com/ www.etsu.edu/physics/bsmith/research/sg/arp.html www.spiral-galaxies.com/Galaxies-Pegasus.html
NGC 3808
Bridges!
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 3 / 28

4. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Galactic encounters: bridges and tails (I)
9
Tails!
Arp 173
• NGC 2992/3
http://www.astrooptik.com/Bildergalerie/PolluxGallery/NGC2623.htm www.etsu.edu/physics/bsmith/research/sg/arp.html
http://www.ess.sunysb.edu/fwalter/SMARTS/findingcharts.html
NGC 2623
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 3 / 28

5. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Galactic encounters: bridges and tails (I)
Galaxies NGC 3808A (right) and NGC 3808B (left). Credits: NASA, Hubble
Space Telescope (2015)
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 3 / 28

6. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Galactic encounters: bridges and tails (II)
Toomre, A. and Toomre, J. Galactic bridges and tails, The Astrophysical
Journal, 178, 1972.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 4 / 28

7. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Galactic encounters: bridges and tails (II)
Toomre, A. and Toomre, J. Galactic bridges and tails, The Astrophysical
Journal, 178, 1972.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 4 / 28

8. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Motivations and Aims
Close approach of two galaxies: it cause significant modification of the
mass distribution or disc structure. One particle that initially stays in one
galaxy (or around one star), after the close encounter, it can jump to the
other galaxy or escape.
To study the mechanisms that explain that a particle remains or not
around each galaxy, considering a very simple model: the planar
parabolic restricted three-body problem.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 5 / 28

9. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 6 / 28

10. Introduction Dynamics of the parabolic problem Numerical results Conclusions
The Planar Parabolic Restricted Three-Body Problem
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 7 / 28

11. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Equations (I)
Parabolic problem:
d2
Z
dt2
= −(1 − µ)
Z − Z1
|Z − Z1|3
− µ
Z − Z2
|Z − Z2|3
,
Z2 = −Z1 = 1
2 (σ2
− 1, 2σ), and σ = tan(f/2)
Change to a synodic frame (primaries at fixed positions) + change of time:
z1 = (−
1
2
, 0), z2 = (
1
2
, 0),
dt
ds
=
√
2 r3/2
.
Compatification to extend the flow when the primaries are at infinity
(t, s → ±∞):
sin(θ) = tanh(s).
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 8 / 28

12. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Equations (II)
Global system



θ = cos θ,
z = w,
w = −A(θ)w + Ω(z)
where =
d
ds
and
A(θ) =
sin θ 4 cos θ
−4 cos θ sin θ
,
Ω(z) = x2
+ y2
+ 2
1 − µ
(x − µ)2 + y2
+ 2
µ
(x − µ + 1)2 + y2
.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 9 / 28

13. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Upper and Lower boundary problems
Global system Boundary problems



θ = cos θ,
z = w,
w = −A(θ)w + Ω(z)
−→
θ=±π/2
z = w,
w = w + Ω(z)
dim 5 dim 4
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 10 / 28

14. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Main properties (I)
Equilibrium points at the boundaries (as in the RTBP):
Collinear: L±
i = (xi, 0, 0, 0, ±π/2), i = 1, 2, 3
Triangular: L±
i = (µ − 1
2 , yi, 0, 0, ±π/2), i = 4, 5
Stability:
L+
1,2,3 L+
4,5
dim(Wu
) 1 2
dim(Ws
) 4 3
L−
1,2,3 L−
4,5
dim(Wu
) 4 3
dim(Ws
) 1 2
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 11 / 28

15. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Main properties (II)
Jacobi function: semi gradient property (no periodic orbits)
C = 2Ω(z) − |w|2
,
dC
ds
= 2 sin θ| w|2
Hill’s regions: {2Ω(z) − C ≥ 0} → C-criterium
−π/2 −→ θ −→ 0
-2
-1
0
1
2
-2 -1 0 1 2
y
x
-2
-1
0
1
2
-2 -1 0 1 2
y
x
-2
-1
0
1
2
-2 -1 0 1 2
y
x
π/2 ←− θ ←− 0
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16. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Main properties (III)
Homothetic solutions and connections
θ = π/2
θ = −π/2
θ = 0
L+
1
L+
4
L+
3
L−
3
L−
4
L−
1
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17. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Dynamics of the problem
In order to describe the dynamics of the parabolic problem, we will focus on
two aspects:
the final evolutions in the synodical system when time tends to infinity,
the richness in the intermediate stages due to
existence of invariant manifolds associated with the homothetic solutions
heteroclinic connections that allow the existence of orbits with passages
close to collinear and/or equilateral configurations.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 14 / 28

18. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Final evolutions
Proposition (Final evolutions)
Let γ(s) = (θ(s), z(s), w(s)), s ∈ [0, ∞), be a solution
of the global system. Then, either it is a collision orbit
(lims→∞ |z(s) − zi| = 0), or lims→∞ |z(s)| = ∞ or its ω-limit is
an equilibrium point.
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 15 / 28

19. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Connections in the the upper boundary problem
m1 m2∞
m1 ∞ m2
L+
4
L+
5
L+
1 L+
3
L+
2
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 16 / 28

20. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 17 / 28

21. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Explorations
Role of the invariant manifolds in the sets of connecting orbits between
primaries
Equal masses (µ = 0.5):
Barrab´es, Cors, Oll´e Dynamics of the parabolic restricted three-body
problem Communications in Nonlinear Science and Numerical Simulation,
29: 400–415, 2015
Different masses (µ < 0.5):
Work in progress with L. Garc´ıa and M. Oll`e
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 18 / 28

22. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Connecting orbits with passages to collinear or triangular
configurations
Connection of type mi − Lk − mj:
collision orbit with mi backwards in time
collision orbit with mj forwards in time
along its trajectory it has a close passage to Lk
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 19 / 28

23. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Connecting orbits: examples (µ = 0.5)
-1
-0.5
0
0.5
-0.5 0 0.5 1
y
x
m2 − L3 − L2 − m2/m1
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 20 / 28

24. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Connecting orbits: examples (µ = 0.5)
-4
-2
0
2
4
-4 -2 0 2 4
m1
m0
m2
-3000
-1500
0
1500
3000
-3e+06 -1.5e+06 0 1.5e+06 3e+06
Y
X
m1
m2
m0
-30
-15
0
15
30
-200 -100 0 100 200
m1
m2
m0
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 20 / 28

25. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Symmetric connecting orbits (µ = 0.5)
Connection mi − mi: crosses the section θ = 0 such that y = x = 0
I.C. (x0, 0, 0, y0)
Connection mi − mj: crosses the section θ = 0 such that x = y = 0
I.C. (0, y0, x0, 0)
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 21 / 28

26. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Symmetric connecting orbits (µ = 0.5)
mi − mi
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y’
x
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
-4
-3
-2
-1
0
1
2
3
4
-8 -6 -4 -2 0 2 4 6 8
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 22 / 28

27. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Symmetric connecting orbits (µ = 0.5)
mi − mj
-10
-8
-6
-4
-2
0
2
0 1 2 3 4 5
x’
y
capture m2
capture m1
C=C(L3)
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-4
-3
-2
-1
0
1
2
3
4
-8 -6 -4 -2 0 2 4 6 8
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 22 / 28

28. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Symmetric connecting orbits (µ = 0.5)
The invariant manifolds of L±
i , i = 1, 2, 3 separate the regions of capture
-3.33108
-3.33103
-3.33098
-3.33093
0.52952 0.5296 0.52968
x
’
y
a
c-b
d-e
f
-2
-1
0
1
-2 -1 0 1 2
y
x
a
b cd e f
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 22 / 28

29. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Symmetric connecting orbits (µ = 0.5)
The hetero/homoclinic connections of L±
i , i = 1, 2, 3 separate the regions of
capture
L−
3 → L+
3
L−
1 → L+
1
L−
2 → L+
2
L−
3 → L+
1
L−
1 → L+
3
L−
2 → L+
2
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 23 / 28

30. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Bridges and Tails?
We consider a bunch of initial conditions around m1 for θ = −π/4. We take a
snapshot of each particle at certain θ > 0 and classify them depending their
final evolution:
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
Y
X
Capture m1 Capture m2
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 24 / 28

31. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Bridges and Tails?
We consider a bunch of initial conditions around m1 for θ = −π/4. We take a
snapshot of each particle at certain θ > 0 and classify them depending their
final evolution:
-4
-2
0
2
4
-4 -2 0 2 4
Y
Y
Capture m1 Escape
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 24 / 28

32. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Evolution of sets of symmetric connecting orbits
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
µ = 0.5 µ = 0.4
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 25 / 28

33. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Evolution of sets of symmetric connecting orbits
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
µ = 0.3 µ = 0.2
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 25 / 28

34. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Evolution of sets of symmetric connecting orbits
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
-6
-4
-2
0
2
4
6
-4 -3 -2 -1 0 1 2 3 4
y
’
x
µ = 0.2 µ = 0.1
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 25 / 28

35. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 26 / 28

36. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Conclusions
Using the invariant manifolds, the symmetries of the problem and the
C-criterium it is possible to construct connecting orbits of different types
The regions of the phase space where the test particles remain or not
around each galaxy are confined by the invariant manifolds of the
collinear equilibrium points
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 27 / 28

37. Introduction Dynamics of the parabolic problem Numerical results Conclusions
Further work
Bridges: close encounters of galaxies of similar sizes; tails: one galaxy
much bigger the other.
How does the mass parameter of the parabolic problem affects?
Toomre & Toomre: explorations varying the inclination (spatial problem)
Spatial parabolic problem
Hyperbolic problem
Barrab´es, Cors, Oll´e (July 15, 2016) Parabolic problem 30/5/2016 28 / 28