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Chaotic capture of (dark) matter by binary systems
Guillaume Rollin1, Pierre Haag1, José Lages1, Dima Shepelyansky2
1Equipe PhAs - Physique Théorique et AstrophysiqueInstitut UTINAM - UMR CNRS 6213
Observatoire de BesançonUniversité de Franche-Comté, France
2Laboratoire de Physique Théorique - UMR CNRS 5152Université Paul Sabatier, Toulouse, France
– Dynamics and chaos in astronomy and physics, Luchon 2016 –
Papers :J. L., D. Shepelyansky, Dark matter chaos in the Solar system, MNRAS Letters 430, L25-L29 (2013)G. Rollin, J. L., D. Shepelyansky, Chaotic enhancement of dark matter density in binary systems, A&A 576, A40 (2015)P. Haag, G. Rollin, J. L., Symplectic map description of Halley’s comet dynamics, Physics Letters A 379 (2015) 1017-1022
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(Dark) matter capture – Three-body problem
Possible DMP capture (or comet capture) due to Jupiter and Sun rotations around the SSbarycenter.
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter capture – Restricted circular three-body problem
Newton’s equations
mDMP � mX � m�
r =1− mX∥∥r�(t)− r
∥∥3
(r�(t)− r
)+
mX∥∥rX(t)− r∥∥3
(rX(t)− r
)G = 1, mX + m� = 1,
∥∥rX∥∥ ' 13km.s−1
= 1,∥∥rX
∥∥ = 1
Energy change after a passage at perihelion (wide encounter)
F ∼mXm�
∥∥rX∥∥2 ' 10−3
Assuming a Maxwellian distribution of Galactic DMP veloci-ties
f (v)dv ∼ v2 exp(−3v2
/2u2)
dv
with u ' 220km.s−1 ∼ 17 (mean DMP velocity)
As F � u2, not many candidates for capture among GalacticDMPsMost of the capturable DMPs have close to parabolic ap-proaching trajectories (E ∼ 0)Direct simulation of Newton’s equations is difficult : veryelongated ellipses, not many particles can be simulated,CPU time consuming (Peter 2009)
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50uDMP candidates
for capture
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter capture – Restricted circular three-body problem
Newton’s equations mDMP � mX � m�
r =1− mX∥∥r�(t)− r
∥∥3
(r�(t)− r
)+
mX∥∥rX(t)− r∥∥3
(rX(t)− r
)G = 1, mX + m� = 1,
∥∥rX∥∥ ' 13km.s−1
= 1,∥∥rX
∥∥ = 1
Energy change after a passage at perihelion (wide encounter)
F ∼mXm�
∥∥rX∥∥2 ' 10−3
Assuming a Maxwellian distribution of Galactic DMP veloci-ties
f (v)dv ∼ v2 exp(−3v2
/2u2)
dv
with u ' 220km.s−1 ∼ 17 (mean DMP velocity)
As F � u2, not many candidates for capture among GalacticDMPsMost of the capturable DMPs have close to parabolic ap-proaching trajectories (E ∼ 0)Direct simulation of Newton’s equations is difficult : veryelongated ellipses, not many particles can be simulated,CPU time consuming (Peter 2009)
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50uDMP candidates
for capture
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter capture – Restricted circular three-body problem
Newton’s equations mDMP � mX � m�
r =1− mX∥∥r�(t)− r
∥∥3
(r�(t)− r
)+
mX∥∥rX(t)− r∥∥3
(rX(t)− r
)G = 1, mX + m� = 1,
∥∥rX∥∥ ' 13km.s−1
= 1,∥∥rX
∥∥ = 1
Energy change after a passage at perihelion (wide encounter)
F ∼mXm�
∥∥rX∥∥2 ' 10−3
Assuming a Maxwellian distribution of Galactic DMP veloci-ties
f (v)dv ∼ v2 exp(−3v2
/2u2)
dv
with u ' 220km.s−1 ∼ 17 (mean DMP velocity)
As F � u2, not many candidates for capture among GalacticDMPsMost of the capturable DMPs have close to parabolic ap-proaching trajectories (E ∼ 0)Direct simulation of Newton’s equations is difficult : veryelongated ellipses, not many particles can be simulated,CPU time consuming (Peter 2009)
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50uDMP candidates
for capture
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
UTINAMInstitut
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Dark matter capture – Restricted circular three-body problem
Newton’s equations mDMP � mX � m�
r =1− mX∥∥r�(t)− r
∥∥3
(r�(t)− r
)+
mX∥∥rX(t)− r∥∥3
(rX(t)− r
)G = 1, mX + m� = 1,
∥∥rX∥∥ ' 13km.s−1
= 1,∥∥rX
∥∥ = 1
Energy change after a passage at perihelion (wide encounter)
F ∼mXm�
∥∥rX∥∥2 ' 10−3
Assuming a Maxwellian distribution of Galactic DMP veloci-ties
f (v)dv ∼ v2 exp(−3v2
/2u2)
dv
with u ' 220km.s−1 ∼ 17 (mean DMP velocity)
As F � u2, not many candidates for capture among GalacticDMPsMost of the capturable DMPs have close to parabolic ap-proaching trajectories (E ∼ 0)Direct simulation of Newton’s equations is difficult : veryelongated ellipses, not many particles can be simulated,CPU time consuming (Peter 2009)
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50uDMP candidates
for capture
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
UTINAMInstitut
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Dark matter capture – Restricted circular three-body problem
Newton’s equations mDMP � mX � m�
r =1− mX∥∥r�(t)− r
∥∥3
(r�(t)− r
)+
mX∥∥rX(t)− r∥∥3
(rX(t)− r
)G = 1, mX + m� = 1,
∥∥rX∥∥ ' 13km.s−1
= 1,∥∥rX
∥∥ = 1
Energy change after a passage at perihelion (wide encounter)
F ∼mXm�
∥∥rX∥∥2 ' 10−3
Assuming a Maxwellian distribution of Galactic DMP veloci-ties
f (v)dv ∼ v2 exp(−3v2
/2u2)
dv
with u ' 220km.s−1 ∼ 17 (mean DMP velocity)
As F � u2, not many candidates for capture among GalacticDMPsMost of the capturable DMPs have close to parabolic ap-proaching trajectories (E ∼ 0)Direct simulation of Newton’s equations is difficult : veryelongated ellipses, not many particles can be simulated,CPU time consuming (Peter 2009)
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50uDMP candidates
for capture
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Kepler map
x : Jupiter’s phase when particle at pericenter (x = ϕ/2π mod 1)w : particle energy at apocenter (w = −2E/mDMP)
Symplectic Kepler map
w = w + F(x) = w + W sin(2πx) ← energy change after a kickx = x + w−3/2 ← third Kepler’s law
Map already used in the study of :I Cometary clouds in Solar systems (Petrosky 1986)I Chaotic dynamics of Halley’s comet (Chirikov & Vecheslavov 1986)I Microwave ionization of hydrogen atoms (see e.g. Shepelyansky, scholarpedia)
Advantage : if the kick function F(x) is known the dynamics of a huge number of particles cansimulated.
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Let’s make a digression ...
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Halley map – Cometary caseKick functions of SS planets
-1
-0.5
0
0.5
1
1.5
0 0.25 0.5 0.75 1xi
F4 (
x 4 )
Mars (x10-5
)
-1
-0.5
0
0.5
1
F3 (
x 3 )
Earth (x10-4
)
-0.5
0
0.5
F2 (
x 2 )
Venus (x10-4
)
-0.2
-0.1
0
0.1
0.2F
1 (
x 1 )
Mercury (x10-6
)
-0.2
0
0.2
0 0.25 0.5 0.75 1xi
F8 (
x 8 )
Neptune (x10-4
)
-0.4
-0.2
0
0.2
0.4
F7 (
x 7 )
Uranus (x10-4
)
-1
-0.5
0
0.5
1
1.5
F6 (
x 6 )
Saturn (x10-3
)
-0.5
0
0.5
F5 (
x 5 )
Jupiter (x10-2
)
Rollin, Haag, J. L., Phys. Lett. A 379 (2015)1017-1022Our calculations match direct observationdata and previous numerical data (Yeomans& Kiang 1981)
F(x1, . . . , x8) '8∑
i=1
Fi(xi)
Fi(xi) = −2µi
∫ +∞
−∞∇(
r · ri
r3−
1‖r− ri‖
).r dt
Two main contributionsI Direct planetary Keplerian potentialI Rotating gravitational dipole potential
due to the Sun movement aroundSolar System barycenter
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Halley map – Cometary caseRenormalized kick function
fi(xi) = Fi(xi)/v2i /µi
Exponential decay with q for q > 1.5aiMore precisely
fi ' 21/4π
1/2(
qai
)−1/4
exp
(−
23/2
3
(qai
)3/2)
(Heggie 1975, Petrosky 1986, Petrosky & Broucke 1988, Roy & Haddow 2003, Shevchenko 2011,see also Rollin’s talk yesterday)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Halley map – Dynamical chaos
Poincaré section
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.25 0.5 0.75 1
x
w
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.25 0.5 0.75 1
x
w
0.45
0.46
0.47
0.48
0.49
0.5
0 0.25 0.5 0.75 1
x
w
0.45
0.46
0.47
0.48
0.49
0.5
0 0.25 0.5 0.75 1
x
w
0.27
0.28
0.29
0.3
0.31
0.32
0 0.25 0.5 0.75 1
x
w
1:6
1:7
2:132:133:19
3:19 3:19
3:203:20
3:20
0.27
0.28
0.29
0.3
0.31
0.32
0 0.25 0.5 0.75 1
x
w
1:6
1:7
2:132:133:19
3:19 3:19
3:203:20
3:20
Symplectic Halley map
x = x + w−3/2
w = w + F(x)
Chaotic dynamics of 1P/Halley
“Lifetime” ∼ 107 years
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Let’s come back to (dark) matter ...
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter capture – Dark map
Kick function determinationWe determine numerically the kick function for any pa-rabolic orbit (q, i, ω)
F(x) = Fq,i,ω(x)
By nonlinear fit we obtain analytical functions.
a : Halley’s cometb : q = 1.5, ω = 0.7, i = 0c : q = 0.5, ω = 0., i = π/2
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter capture – Dark map
Kick function determinationWe determine numerically the kick function for any pa-rabolic orbit (q, i, ω)
F(x) = Fq,i,ω(x)
By nonlinear fit we obtain analytical functions.
~25%
Chance to be captured with a givenenergy (w < 0↔ E > 0)
hq,i,ω(w)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark matter – Capture cross section
wcap =mXm�
∥∥rX∥∥2 ' 10−3
σp = π∥∥rX
∥∥2 area enclosed by Jupiter’s orbit
10-8
10-4
100
104
108
10-4
10-2
100
102
w / wcap
σ/
σp
| | | |
10-5
10-4
10-3
10-2
10-1
100
101
10-4
10-2
100
102
104
w / wcapd
N /
dw
/ N
p1/|w|
1/w2
σ/σp ' πm�mX
wcap
|w|in agreement with Khriplovich & Shepelyansky 2009
I Predominance of wide encounters as suggested by Peter 2009I Very small contribution from close encounters invalidating previous numerical results (Gould &
Alam 2001 and Lundberg & Edsjö 2004)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Dark map – Dark matter captureSimulation of the (isotropic) injection, the capture and the escape of DMPs during the whole lifetimeof the Solar system.Injection of Ntot ' 1.5× 1014 DMPs with energy |w| in the range [0,∞] with NH = 4× 109 DMPs inthe Halley’s comet energy interval [0,wH ].
105
106
107
108
109
101
102
103
104
105
106
107
t
Ncap
0 0.05 0.1 0.15 0.20
0.5
1
wx
10-6
10-5
10-4
10-3
10-2
10-1
1
(w)
10-4
10-3
10-2
10-1
10-6
10-5
10-4
10-3
10-2
10-1
1
w
(w)
w-3/2
(w)~alpha centauri
r<100au
Slow
Chaotization
Chaotic
component
Islands
of stability
Last invarian
KAM curve
I Equilibrium reached after a time td ∼ 107yr similar to the diffusive escape time scale of theHalley’s comet (Chirikov & Vecheslavov 1989) −→ Equilibrium energy distribution ρ(w)
I The dynamics of dark matter particles in the Solar system is essentially chaotic
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Back to real space – Density distribution of captured DMPs
Nowadays equilibrium density distribution ( tS = 4.5× 109yr )
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6
r
ρ(r)
0.001
0.01
0.1
0.2 0.5 5 1 10
r
ρ(r)/r2
I The profile of the radial density ρ(r) ∝ dN/dr is similar to those observed for galaxieswhere DMP mass is dominant. Indeed ρ(r) is almost flat (increases slowly) right afterJupiter orbit (r = 1) −→ according to virial theorem the circular velocity of visible matter isconsequently constant as observed e.g. in Rubin 1980Virial theorem : v2
m ∼∫ r
0 dr′ρ(r′)/r ∼ ρ(r) ∼ r2 (ρ(r)/r2) ∼here
r2r−1.53 ∼ r1/2
Ergodicity along radial dynamics : dµ ∼ dN ∼ ρ(r)dr ∼ dt ∼ dr/vr ∼ r1/2drConsequently, vm ∝ r0.25 (Dark map) to compare to vm ∝ r0.35 (Rubin 1980)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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Back to real space – Density distribution of captured DMPs
Surface density
ρs(z, R) ∝ dN/dzdR
where
R =√
x2 + y2
Volume density
ρv(x, y, z) ∝ dN/dxdydz
Ecliptic
Ecliptic
Ecliptic
Ecliptic
z
x1
1
y
x1
1
y
x
1
1
z
x
1
1
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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How much dark matter is present in the Solar system ?
The total mass of DMP passed through the System solar during its lifetime tS = 4.5× 109yr is
Mtot = ρgtS
∫ ∞0
dv v f (v)σ(v) ≈ 35ρgtSG∥∥rX
∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀
At time tS the mass of captured DMPs in the Solar system is
MAC ≈ ηACMtot ≈ 2× 10−15M� within r < 0.5 distanceSun-αCentauri
M100au ≈ η100auMtot ≈ 1.3× 10−17M� within r < 100au
The captured DMP mass in the volume of the Neptune orbit radius is
M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g
The captured DMP mass in the volume of the Jupiter orbit radius is
MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g
The average volume density of captured dark matter inside the Jupiter orbit sphere is
ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4
ρg � ρg (Galactic DMP density)
Globally, not much dark matter captured by the Solarsystem, but ...
Let’s compare to the capturable DMP density
ρgH = ρg
∫ √wH
0dv v f (v) ≈ 1.4× 10−32g/cm3
Huge chaotic enhancement ζ = ρX/ρgH ≈ 4× 103 ofthe density of actually capturable DMPs.
=⇒
The long rangeinteraction capturemechanism is veryefficient for binarysystems (1+2) with
m1 � m2
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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How much dark matter is present in the Solar system ?The total mass of DMP passed through the System solar during its lifetime tS = 4.5× 109yr is
Mtot = ρgtS
∫ ∞0
dv v f (v)σ(v) ≈ 35ρgtSG∥∥rX
∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀
At time tS the mass of captured DMPs in the Solar system is
MAC ≈ ηACMtot ≈ 2× 10−15M� within r < 0.5 distanceSun-αCentauri
M100au ≈ η100auMtot ≈ 1.3× 10−17M� within r < 100au
The captured DMP mass in the volume of the Neptune orbit radius is
M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g
The captured DMP mass in the volume of the Jupiter orbit radius is
MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g
The average volume density of captured dark matter inside the Jupiter orbit sphere is
ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4
ρg � ρg (Galactic DMP density)
Globally, not much dark matter captured by the Solarsystem, but ...
Let’s compare to the capturable DMP density
ρgH = ρg
∫ √wH
0dv v f (v) ≈ 1.4× 10−32g/cm3
Huge chaotic enhancement ζ = ρX/ρgH ≈ 4× 103 ofthe density of actually capturable DMPs.
=⇒
The long rangeinteraction capturemechanism is veryefficient for binarysystems (1+2) with
m1 � m2
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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How much dark matter is present in the Solar system ?The total mass of DMP passed through the System solar during its lifetime tS = 4.5× 109yr is
Mtot = ρgtS
∫ ∞0
dv v f (v)σ(v) ≈ 35ρgtSG∥∥rX
∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀
At time tS the mass of captured DMPs in the Solar system is
MAC ≈ ηACMtot ≈ 2× 10−15M� within r < 0.5 distanceSun-αCentauri
M100au ≈ η100auMtot ≈ 1.3× 10−17M� within r < 100au
The captured DMP mass in the volume of the Neptune orbit radius is
M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g
The captured DMP mass in the volume of the Jupiter orbit radius is
MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g
The average volume density of captured dark matter inside the Jupiter orbit sphere is
ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4
ρg � ρg (Galactic DMP density)
Globally, not much dark matter captured by the Solarsystem, but ...
Let’s compare to the capturable DMP density
ρgH = ρg
∫ √wH
0dv v f (v) ≈ 1.4× 10−32g/cm3
Huge chaotic enhancement ζ = ρX/ρgH ≈ 4× 103 ofthe density of actually capturable DMPs.
=⇒
The long rangeinteraction capturemechanism is veryefficient for binarysystems (1+2) with
m1 � m2
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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How much dark matter is present in the Solar system ?The total mass of DMP passed through the System solar during its lifetime tS = 4.5× 109yr is
Mtot = ρgtS
∫ ∞0
dv v f (v)σ(v) ≈ 35ρgtSG∥∥rX
∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀
At time tS the mass of captured DMPs in the Solar system is
MAC ≈ ηACMtot ≈ 2× 10−15M� within r < 0.5 distanceSun-αCentauri
M100au ≈ η100auMtot ≈ 1.3× 10−17M� within r < 100au
The captured DMP mass in the volume of the Neptune orbit radius is
M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g
The captured DMP mass in the volume of the Jupiter orbit radius is
MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g
The average volume density of captured dark matter inside the Jupiter orbit sphere is
ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4
ρg � ρg (Galactic DMP density)
Globally, not much dark matter captured by the Solarsystem, but ...
Let’s compare to the capturable DMP density
ρgH = ρg
∫ √wH
0dv v f (v) ≈ 1.4× 10−32g/cm3
Huge chaotic enhancement ζ = ρX/ρgH ≈ 4× 103 ofthe density of actually capturable DMPs.
=⇒
The long rangeinteraction capturemechanism is veryefficient for binarysystems (1+2) with
m1 � m2
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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(Dark) matter capture in binary systems
104
10-2
10-1 1 10
1
(u)
u
chaoti
c e
nhancem
en
t fa
cto
r
w ~w ~0.005cap H
w > ucap
1/2w < u
cap
1/2
u=220km/s
G. Rollin, J. L., D. Shepelyansky, A&A 576, A40 (2015)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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(Dark) matter capture in binary systems
104
10-2
10-1 1 10
1
(u)
u
chaoti
c e
nhancem
en
t fa
cto
r
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50u
DMP candidates
for capture
w ~w ~0.005cap H
wcap
1/2
w << ucap
w > ucap
1/2w < u
cap
1/2
1/2
u=220km/s
G. Rollin, J. L., D. Shepelyansky, A&A 576, A40 (2015)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
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(Dark) matter capture in binary systems
104
10-2
10-1 1 10
1
(u)
u
chaoti
c e
nhancem
en
t fa
cto
r
0
0.01
0.02
0.03
0.04
0.05
0.06
10 20 30 40 50u
DMP candidates
for capture
w ~w ~0.005cap H
0
5
10
15
20
25
30
35
0 0.02 0.04 0.06 0.08 0.1u wcap
1/2
wcap
1/2
w ~ ucap
w << ucap
w > ucap
1/2w < u
cap
1/2
1/2
1/2
e.g.
Black hole
+
Star companion
V~c/40star
u=220km/s
G. Rollin, J. L., D. Shepelyansky, A&A 576, A40 (2015)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
UTINAMInstitut
UTINAMInstitut
(Dark) matter capture in binary systems
G. Rollin, J. L., D. Shepelyansky, A&A 576, A40 (2015)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
UTINAMInstitut
UTINAMInstitut
(Dark) matter capture in binary systems
G. Rollin, J. L., D. Shepelyansky, A&A 576, A40 (2015)
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016
UTINAMInstitut
UTINAMInstitut
Thank You !
Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016