Chaotic dark matter in the Solar system and galaxiesperso.utinam.cnrs.fr/~lages/presentations/DMPIHPnov2013.pdf · J. L., D. Shepelyansky, Dark matter chaos in the Solar system, MNRASL

UTINAMInstitut

UTINAMInstitut

Chaotic dark matter in the Solar system and galaxies

Guillaume Rollin1, 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

Workshop « Dynamics & Kinetic theory of self-gravitating systems », GRAVASCO, Institut Henri Poincaré, November 2013

References :J. L., D. Shepelyansky, Dark matter chaos in the Solar system, MNRASL 430, L25-L29 (2013), arXiv :1211.0903G. Rollin, J. L., D. Shepelyansky, Chaotic enhancement of dark matter density in binary systems and galaxies, to be submitted

UTINAMInstitut

UTINAMInstitut

Dark matter capture – Three-body problem

DMP

Ecliptic

p

Possible DMP capture due to Jupiter rotation around the Sun

Chaotic dark matter in the Solar system and galaxies, J. Lages, GRAVASCO IHP, Nov 2013

UTINAMInstitut

UTINAMInstitut

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 (in absence of close encounter)

F ∼mXm�

∥∥rX∥∥2 ' 10−3 (Petrosky 86’)

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 09’)

0

0.01

0.02

0.03

0.04

0.05

0.06

10 20 30 40 50uDMP candidates

for capture


UTINAMInstitut

UTINAMInstitut


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 (Petrosky 86’)



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


UTINAMInstitut

UTINAMInstitut



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 (Petrosky 86’)



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


UTINAMInstitut

UTINAMInstitut



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 (Petrosky 86’)



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


UTINAMInstitut

UTINAMInstitut



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 (Petrosky 86’)



/2u2)

dv



0

0.01

0.02

0.03

0.04

0.05

0.06


for capture


UTINAMInstitut

UTINAMInstitut

Dark matter capture – Kicked model

x : Jupiter’s phase when DMP at perihelion (x = ϕ/2π mod 1)w : DMP energy (w = −2E/mDMP)

Symplectic Kepler-Petrosky map

x = x + w−3/2 third Kepler’s laww = w + F(x) energy change after a kick

Map already used in the study of :I Cometary clouds in Solar systems (Petrosky 86’)I Chaotic dynamics of Halley’s comet (Chirikov, Vecheslavov 89’)I Microwave ionization of hydrogen atoms (see e.g. Shepelyansky, scholarpedia)

Advantage : providing the fact the kick function F(x) is known the dynamics of a huge number ofparticles can simulated.


UTINAMInstitut

UTINAMInstitut








UTINAMInstitut

UTINAMInstitut








UTINAMInstitut

UTINAMInstitut

Dark matter capture – Dark map

Kick function determination

F(x)→ F`,θ,φ(x) ∼ Fq,θ,φ(x)

The kick function is different for each approaching con-figuration (`, θ, φ) ∼ (q, θ, φ)For close to 1 eccentricities the perihelion is such as

q ∼`2

2

For each approaching configuration (q, θ, φ), the inte-gration of the three-body problem gives the kick func-tion.

θ

φ

q


UTINAMInstitut

UTINAMInstitut

Dark matter capture – Dark map

Kick function determination



q ∼`2

2


-0.001

0

0.001

0 0.25 0.5 0.75 1

x

F (

x) c

-0.002

0

0.002

F (

x) b

-0.006

0

0.006

F (

x) a

a : Halley’s cometb : q = 1.5, n = 4, θ = 0.7, φ = 0c : q = 0.5, n = 4, θ = 0., φ = π/2


UTINAMInstitut

UTINAMInstitut

Dark matter capture – Dark mapKick function determination



q ∼`2

2


10-6

10-4

10-2

100

0 0.5 1 1.5 2 2.5 3 3.5 4

q / rp

Fm

ax

c

a

b

Exponential decay in agreement with Petrosky 86’

-0.001

0

0.001

0 0.25 0.5 0.75 1

x

F (

x) c

-0.002

0

0.002

F (

x) b

-0.006

0

0.006

F (

x) a

a : Halley’s cometb : q = 1.5, n = 4, θ = 0.7, φ = 0c : q = 0.5, n = 4, θ = 0., φ = π/2


UTINAMInstitut

UTINAMInstitut

Dark matter – Capture cross section

wcap =mXm�

∥∥rX∥∥2 ' 10−3

σp = π∥∥rX

∥∥2 Jupiter orbit area

10-8

10-4

100

104

108

10-4

10-2

100

102

w / wcap

σ/ σ

p

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

win agreement with Khriplovich & Shepelyansky 09’

I Predominance of long range interaction as suggested by Peter 09’I Very small contribution from close encounters invalidating previous numerical results (Gould &

Alam 01’ and Lundberg & Edsjö 04’)


UTINAMInstitut

UTINAMInstitut

Dark matter evolution – Chaotic dynamicsSimulation 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 ].

I Equilibrium reached after a time td ∼ 107yr similar to the diffusive escape time scale of theHalley’s comet (Chirikov & Vecheslavov 89’) −→ Equilibrium energy distribution ρ(w)

I The dynamics of dark matter particles in the Solar system is essentially chaotic


UTINAMInstitut

UTINAMInstitut

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 80’More precisely, vm ∝ r0.25 (Dark map) quite close to vm ∝ r0.35 (Rubin 80’)


UTINAMInstitut

UTINAMInstitut

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


UTINAMInstitut

UTINAMInstitut

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


UTINAMInstitut

UTINAMInstitut

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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut

Conclusion : Dark matter capture in binary systems (preliminary results)

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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut


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


UTINAMInstitut

UTINAMInstitut



UTINAMInstitut

UTINAMInstitut



UTINAMInstitut

UTINAMInstitut

Thank You !


Documents

Chaotic dark matter in the Solar system and galaxiesperso.utinam.cnrs.fr/~lages/presentations/DMPIHPnov2013.pdf · J. L., D. Shepelyansky, Dark matter chaos in the Solar system, MNRASL