Modelling galaxy close encounters by a parabolic problem
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Transcript of Modelling galaxy close encounters by a parabolic problem
Introduction Dynamics of the parabolic problem Numerical results Conclusions
On the dynamics of the parabolic restricted threebody problem
E. Barrabes1 J.M. Cors 2 L. Garcia M. Olle2
1Universitat de Girona
2Universitat Politecnica de Catalunya
XV Jornadas de Trabajo en Mecanica Celeste
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 1 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 2 / 28
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!
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 3 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 3 / 28
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, HubbleSpace Telescope (2015)
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 3 / 28
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 AstrophysicalJournal, 178, 1972.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 4 / 28
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 AstrophysicalJournal, 178, 1972.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 4 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Motivations and Aims
Close approach of two galaxies: it cause significant modification of themass distribution or disc structure. One particle that initially stays in onegalaxy (or around one star), after the close encounter, it can jump to theother galaxy or escape.
To study the mechanisms that explain that a particle remains or notaround each galaxy, considering a very simple model: the planarparabolic restricted three-body problem.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 5 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 6 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
The Planar Parabolic Restricted Three-Body Problem
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 7 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Equations (I)
Parabolic problem:
d2Z
dt2= −(1− µ)
Z− Z1
|Z− Z1|3− µ Z− Z2
|Z− Z2|3,
Z2 = −Z1 = 12 (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).
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 8 / 28
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
dsand
A(θ) =
(sin θ 4 cos θ−4 cos θ sin θ
),
Ω(z) = x2 + y2 + 21− µ√
(x− µ)2 + y2+ 2
µ√(x− µ+ 1)2 + y2
.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 9 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 10 / 28
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− 12 , yi, 0, 0,±π/2), i = 4, 5
Stability:
L+1,2,3 L+
4,5
dim(Wu) 1 2dim(W s) 4 3
L−1,2,3 L−4,5dim(Wu) 4 3dim(W s) 1 2
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 11 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 12 / 28
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−3L−
4
L−1
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 13 / 28
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 ontwo 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 passagesclose to collinear and/or equilateral configurations.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 14 / 28
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 solutionof the global system. Then, either it is a collision orbit(lims→∞ |z(s)− zi| = 0), or lims→∞ |z(s)| =∞ or its ω-limit isan equilibrium point.
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 15 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 16 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 17 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Explorations
Role of the invariant manifolds in the sets of connecting orbits betweenprimaries
Equal masses (µ = 0.5):Barrabes, Cors, Olle Dynamics of the parabolic restricted three-bodyproblem Communications in Nonlinear Science and Numerical Simulation,29: 400–415, 2015
Different masses (µ < 0.5):Work in progress with L. Garcıa and M. Olle
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 18 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Connecting orbits with passages to collinear or triangularconfigurations
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 19 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 20 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 20 / 28
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, y′0)
Connection mi −mj : crosses the section θ = 0 such that x = y′ = 0
I.C. (0, y0, x′0, 0)
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 21 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 22 / 28
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 m2capture m1C=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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 22 / 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
ab
cd e f
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 22 / 28
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 ofcapture
L−3 → L+
3
L−1 → L+
1
L−2 → L+
2
L−3 → L+
1
L−1 → L+
3
L−2 → L+
2
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 23 / 28
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 asnapshot of each particle at certain θ > 0 and classify them depending theirfinal 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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 24 / 28
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 asnapshot of each particle at certain θ > 0 and classify them depending theirfinal evolution:
-4
-2
0
2
4
-4 -2 0 2 4
Y
Y
Capture m1 Escape
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 24 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 25 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 25 / 28
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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 25 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Outline
Introduction
Dynamics of the parabolic problem
Numerical results
Conclusions
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 26 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Conclusions
Using the invariant manifolds, the symmetries of the problem and theC-criterium it is possible to construct connecting orbits of different types
The regions of the phase space where the test particles remain or notaround each galaxy are confined by the invariant manifolds of thecollinear equilibrium points
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 27 / 28
Introduction Dynamics of the parabolic problem Numerical results Conclusions
Further work
Bridges: close encounters of galaxies of similar sizes; tails: one galaxymuch 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
Barrabes, Cors, Olle (July 15, 2016) Parabolic problem 30/5/2016 28 / 28