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Chaotic capture of (dark) matter by binary systems Guillaume Rollin 1 , Pierre Haag 1 , Jos Lages 1 , Dima Shepelyansky 2 1 Equipe PhAs - Physique Thorique et Astrophysique Institut UTINAM - UMR CNRS 6213 Observatoire de Besanon Universit

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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 Astrophysique Institut UTINAM - UMR CNRS 6213 Observatoire de Besançon Université de Franche-Comté, France 2Laboratoire de Physique Théorique - UMR CNRS 5152 Université 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 SS barycenter.

Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

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Dark matter capture – Restricted circular three-body problem

mDMP ≪ m ≪ m⊙

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 ≪ m ≪ m⊙ ¨ r = 1 − m

  • r⊙(t) − r
  • 3
  • r⊙(t) − r
  • +

m

  • r(t) − r
  • 3
  • r(t) − r
  • G = 1,

m + m⊙ = 1,

  • ˙

r

  • ≃ 13km.s−1 = 1,
  • r
  • = 1

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 ≪ m ≪ m⊙ ¨ r = 1 − m

  • r⊙(t) − r
  • 3
  • r⊙(t) − r
  • +

m

  • r(t) − r
  • 3
  • r(t) − r
  • G = 1,

m + m⊙ = 1,

  • ˙

r

  • ≃ 13km.s−1 = 1,
  • r
  • = 1

Energy change after a passage at perihelion (wide encounter) F ∼ m m⊙

  • ˙

r

  • 2 ≃ 10−3

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 ≪ m ≪ m⊙ ¨ r = 1 − m

  • r⊙(t) − r
  • 3
  • r⊙(t) − r
  • +

m

  • r(t) − r
  • 3
  • r(t) − r
  • G = 1,

m + m⊙ = 1,

  • ˙

r

  • ≃ 13km.s−1 = 1,
  • r
  • = 1

Energy change after a passage at perihelion (wide encounter) F ∼ m m⊙

  • ˙

r

  • 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)

0.01 0.02 0.03 0.04 0.05 0.06 10 20 30 40 50

u

DMP 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 ≪ m ≪ m⊙ ¨ r = 1 − m

  • r⊙(t) − r
  • 3
  • r⊙(t) − r
  • +

m

  • r(t) − r
  • 3
  • r(t) − r
  • G = 1,

m + m⊙ = 1,

  • ˙

r

  • ≃ 13km.s−1 = 1,
  • r
  • = 1

Energy change after a passage at perihelion (wide encounter) F ∼ m m⊙

  • ˙

r

  • 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 Galactic DMPs Most of the capturable DMPs have close to parabolic ap- proaching trajectories (E ∼ 0) Direct simulation of Newton’s equations is difficult : very elongated ellipses, not many particles can be simulated, CPU time consuming (Peter 2009)

0.01 0.02 0.03 0.04 0.05 0.06 10 20 30 40 50

u

DMP 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 kick ¯ x = x + ¯ w −3/2 ← third Kepler’s law Map already used in the study of :

◮ Cometary clouds in Solar systems (Petrosky 1986) ◮ Chaotic dynamics of Halley’s comet (Chirikov & Vecheslavov 1986) ◮ 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 can simulated.

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 case

Kick functions of SS planets

  • 1
  • 0.5

0.5 1 1.5 0.25 0.5 0.75 1

xi F4 (x4 )

Mars (x10-5)

  • 1
  • 0.5

0.5 1

F3 (x3 )

Earth (x10-4)

  • 0.5

0.5

F2 (x2 )

Venus (x10-4)

  • 0.2
  • 0.1

0.1 0.2

F1 (x1 )

Mercury (x10-6)

  • 0.2

0.2 0.25 0.5 0.75 1

xi F8 (x8 )

Neptune (x10-4)

  • 0.4
  • 0.2

0.2 0.4

F7 (x7 )

Uranus (x10-4)

  • 1
  • 0.5

0.5 1 1.5

F6 (x6 )

Saturn (x10-3)

  • 0.5

0.5

F5 (x5 )

Jupiter (x10-2)

Rollin, Haag, J. L., Phys. Lett. A 379 (2015) 1017-1022 Our calculations match direct observation data 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 contributions

◮ Direct planetary Keplerian potential ◮ Rotating gravitational dipole potential

due to the Sun movement around Solar System barycenter

Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

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Halley map – Cometary case

Renormalized kick function fi(xi) = Fi(xi)/v2

i /µi

Exponential decay with q for q > 1.5ai More precisely fi ≃ 21/4π1/2 q ai −1/4 exp

  • − 23/2

3 q ai 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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.25 0.5 0.75 1

x w

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.25 0.5 0.75 1

x w

0.45 0.46 0.47 0.48 0.49 0.5 0.25 0.5 0.75 1

x w

0.45 0.46 0.47 0.48 0.49 0.5 0.25 0.5 0.75 1

x w

0.27 0.28 0.29 0.3 0.31 0.32 0.25 0.5 0.75 1

x w

1:6 1:7 2:13 2:13 3:19 3:19 3:19 3:20 3:20 3:20

0.27 0.28 0.29 0.3 0.31 0.32 0.25 0.5 0.75 1

x w

1:6 1:7 2:13 2:13 3:19 3:19 3:19 3:20 3:20 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 determination We 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 comet b : q = 1.5, ω = 0.7, i = 0 c : 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 determination We 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 given energy (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 = m m⊙

  • ˙

r

  • 2 ≃ 10−3

σp = π

  • r
  • 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 / wcap dN / dw / Np

1/|w| 1/w

2

σ/σp ≃ π

m⊙ m

wcap |w| in agreement with Khriplovich & Shepelyansky 2009

◮ Predominance of wide encounters as suggested by Peter 2009 ◮ 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 capture

Simulation of the (isotropic) injection, the capture and the escape of DMPs during the whole lifetime

  • f the Solar system.

Injection of Ntot ≃ 1.5 × 1014 DMPs with energy |w| in the range [0, ∞] with NH = 4 × 109 DMPs in the Halley’s comet energy interval [0, wH].

105 106 107 108 109 101 102 103 104 105 106 107

t Ncap

0.05 0.1 0.15 0.2 0.5 1

w x

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

  • f stability

Last invarian KAM curve ◮ Equilibrium reached after a time td ∼ 107yr similar to the diffusive escape time scale of the

Halley’s comet (Chirikov & Vecheslavov 1989) − → Equilibrium energy distribution ρ(w)

◮ 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 )

0.05 0.1 0.15 0.2 0.25 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

◮ The profile of the radial density ρ(r) ∝ dN/dr is similar to those observed for galaxies

where DMP mass is dominant. Indeed ρ(r) is almost flat (increases slowly) right after Jupiter orbit (r = 1) − → according to virial theorem the circular velocity of visible matter is consequently constant as observed e.g. in Rubin 1980 Virial 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 ∼ r1/2dr Consequently, 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 x

1 1

y x

1 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 ?

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 ∞ dv v f(v)σ(v) ≈ 35ρgtSG

  • r
  • 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 ≈ ηMAC ≈ 0.9 × 10−18M⊙ ≈ 1.5 × 1015g The captured DMP mass in the volume of the Jupiter orbit radius is M ≈ ηMAC ≈ 4.6 × 10−20M⊙ ≈ 1014g

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 ∞ dv v f(v)σ(v) ≈ 35ρgtSG

  • r
  • 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 ≈ ηMAC ≈ 0.9 × 10−18M⊙ ≈ 1.5 × 1015g The captured DMP mass in the volume of the Jupiter orbit radius is M ≈ ηMAC ≈ 4.6 × 10−20M⊙ ≈ 1014g The average volume density of captured dark matter inside the Jupiter orbit sphere is ρ = 3M 4πr3

  • ≈ 5 × 10−29g/cm3 ≈ 1.2 × 10−4ρg ≪ ρg (Galactic DMP density)

Globally, not much dark matter captured by the Solar system, but ...

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 ∞ dv v f(v)σ(v) ≈ 35ρgtSG

  • r
  • 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 ≈ ηMAC ≈ 0.9 × 10−18M⊙ ≈ 1.5 × 1015g The captured DMP mass in the volume of the Jupiter orbit radius is M ≈ ηMAC ≈ 4.6 × 10−20M⊙ ≈ 1014g The average volume density of captured dark matter inside the Jupiter orbit sphere is ρ = 3M 4πr3

  • ≈ 5 × 10−29g/cm3 ≈ 1.2 × 10−4ρg ≪ ρg (Galactic DMP density)

Globally, not much dark matter captured by the Solar system, but ... Let’s compare to the capturable DMP density ρgH = ρg √wH dv v f(v) ≈ 1.4 × 10−32g/cm3 Huge chaotic enhancement ζ = ρ/ρgH ≈ 4 × 103 of the density of actually capturable DMPs. = ⇒ The long range interaction capture mechanism is very efficient for binary systems (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

10

4

10

  • 2

10

  • 1

1 10

1

(u) u

chaotic enhancement factor

w ~w ~0.005

cap H

w > u

cap 1/2

w < 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

10

4

10

  • 2

10

  • 1

1 10

1

(u) u

chaotic enhancement factor

0.01 0.02 0.03 0.04 0.05 0.06 10 20 30 40 50

u DMP candidates for capture

w ~w ~0.005

cap H w

cap 1/2

w << u

cap

w > u

cap 1/2

w < 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

10

4

10

  • 2

10

  • 1

1 10

1

(u) u

chaotic enhancement factor

0.01 0.02 0.03 0.04 0.05 0.06 10 20 30 40 50

u DMP candidates for capture

w ~w ~0.005

cap H

5 10 15 20 25 30 35 0.02 0.04 0.06 0.08 0.1

u w

cap 1/2

w

cap 1/2

w ~ u

cap

w << u

cap

w > u

cap 1/2

w < u

cap 1/2 1/2 1/2

e.g. Black hole + Star companion V~c/40

star

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

  • 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

  • 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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Thank You !

Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016