Chaotic capture of (dark) matter by binary systems · UTINAM IInnssttiitut Chaotic capture of (dark) matter by binary systems Guillaume Rollin 1, Pierre Haag , José Lages1, Dima

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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


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Newton’s equations mDMP � mX � m�

r =1− mX∥∥r�(t)− r

∥∥3

(r�(t)− r

)+


(rX(t)− r

)G = 1, mX + m� = 1,

∥∥rX∥∥ ' 13km.s−1

= 1,∥∥rX

∥∥ = 1


F ∼mXm�

∥∥rX∥∥2 ' 10−3



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


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r =1− mX∥∥r�(t)− r

∥∥3

(r�(t)− r

)+


(rX(t)− r

)G = 1, mX + m� = 1,

∥∥rX∥∥ ' 13km.s−1

= 1,∥∥rX

∥∥ = 1


F ∼mXm�

∥∥rX∥∥2 ' 10−3



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


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r =1− mX∥∥r�(t)− r

∥∥3

(r�(t)− r

)+


(rX(t)− r

)G = 1, mX + m� = 1,

∥∥rX∥∥ ' 13km.s−1

= 1,∥∥rX

∥∥ = 1


F ∼mXm�

∥∥rX∥∥2 ' 10−3



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


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r =1− mX∥∥r�(t)− r

∥∥3

(r�(t)− r

)+


(rX(t)− r

)G = 1, mX + m� = 1,

∥∥rX∥∥ ' 13km.s−1

= 1,∥∥rX

∥∥ = 1


F ∼mXm�

∥∥rX∥∥2 ' 10−3



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


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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.


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Let’s make a digression ...


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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


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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)


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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


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Let’s come back to (dark) matter ...


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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


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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)


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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)


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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


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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)


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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


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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


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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


∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀





M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g


MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g


ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4




ρgH = ρg

∫ √wH

0dv v f (v) ≈ 1.4× 10−32g/cm3


=⇒


m1 � m2


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Mtot = ρgtS

∫ ∞0


∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀





M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g


MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g


ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4




ρgH = ρg

∫ √wH

0dv v f (v) ≈ 1.4× 10−32g/cm3


=⇒


m1 � m2


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Mtot = ρgtS

∫ ∞0


∥∥M�/u ≈ 0.9× 10−6M� ∼ M♀





M[ ≈ η[MAC ≈ 0.9× 10−18M� ≈ 1.5× 1015g


MX ≈ ηXMAC ≈ 4.6× 10−20M� ≈ 1014g


ρX =3MX4πr3X≈ 5× 10−29g/cm3 ≈ 1.2× 10−4




ρgH = ρg

∫ √wH

0dv v f (v) ≈ 1.4× 10−32g/cm3


=⇒


m1 � m2


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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)


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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



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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



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Thank You !


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Chaotic capture of (dark) matter by binary systems · UTINAM IInnssttiitut Chaotic capture of (dark) matter by binary systems Guillaume Rollin 1, Pierre Haag , José Lages1, Dima