Chaotic capture of (dark) matter by binary systems Guillaume Rollin 1 - PowerPoint PPT Presentation

Chaotic capture of (dark) matter by binary systems Guillaume Rollin 1 , Pierre Haag 1 , José Lages 1 , Dima Shepelyansky 2 1 Equipe PhAs - Physique Théorique et Astrophysique Institut UTINAM - UMR CNRS 6213 Observatoire de Besançon Université de Franche-Comté, France 2 Laboratoire 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 Institut Institut UTINAM UTINAM

(Dark) matter capture – Three-body problem Possible DMP capture (or comet capture) due to Jupiter and Sun rotations around the SS barycenter. Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Restricted circular three-body problem m DMP ≪ m � ≪ m ⊙ Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Restricted circular three-body problem Newton’s equations m DMP ≪ m � ≪ m ⊙ 1 − m � m � � � � � ¨ r = r ⊙ ( t ) − r + r � ( t ) − r � 3 � 3 � � � � � r ⊙ ( t ) − r � r � ( t ) − r � ≃ 13 km . s − 1 = 1 , � = 1 � � ˙ � � � G = 1 , m � + m ⊙ = 1 , r � � r � Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Restricted circular three-body problem Newton’s equations m DMP ≪ m � ≪ m ⊙ 1 − m � m � � � � � ¨ r = r ⊙ ( t ) − r + r � ( t ) − r � 3 � 3 � � � � � r ⊙ ( t ) − r � r � ( t ) − r � ≃ 13 km . s − 1 = 1 , � = 1 � � ˙ � � � G = 1 , m � + m ⊙ = 1 , r � � r � Energy change after a passage at perihelion (wide encounter) F ∼ m � � 2 ≃ 10 − 3 � � � ˙ r � m ⊙ Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Restricted circular three-body problem Newton’s equations m DMP ≪ m � ≪ m ⊙ 1 − m � m � � � � � r = ¨ r ⊙ ( t ) − r + r � ( t ) − r � 3 � 3 � � � � � r ⊙ ( t ) − r � r � ( t ) − r � ≃ 13 km . s − 1 = 1 , � = 1 � � ˙ � � � G = 1 , m � + m ⊙ = 1 , r � � r � Energy change after a passage at perihelion (wide encounter) F ∼ m � � 2 ≃ 10 − 3 � � � ˙ r � m ⊙ 0.06 Assuming a Maxwellian distribution of Galactic DMP veloci- 0.05 ties f ( v ) dv ∼ v 2 exp � − 3 v 2 / 2 u 2 � dv 0.04 with u ≃ 220 km . s − 1 ∼ 17 (mean DMP velocity) 0.03 0.02 0.01 0 u 10 20 30 40 50 DMP candidates for capture Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Restricted circular three-body problem Newton’s equations m DMP ≪ m � ≪ m ⊙ 1 − m � m � � � � � ¨ r = r ⊙ ( t ) − r + r � ( t ) − r � 3 � 3 � � � � � r ⊙ ( t ) − r � r � ( t ) − r � ≃ 13 km . s − 1 = 1 , � = 1 � ˙ � � � � G = 1 , m � + m ⊙ = 1 , r � � r � Energy change after a passage at perihelion (wide encounter) F ∼ m � � 2 ≃ 10 − 3 � � � ˙ r � m ⊙ 0.06 Assuming a Maxwellian distribution of Galactic DMP veloci- 0.05 ties f ( v ) dv ∼ v 2 exp � − 3 v 2 / 2 u 2 � dv 0.04 with u ≃ 220 km . s − 1 ∼ 17 (mean DMP velocity) 0.03 As F ≪ u 2 , not many candidates for capture among Galactic 0.02 DMPs Most of the capturable DMPs have close to parabolic ap- 0.01 proaching trajectories ( E ∼ 0 ) Direct simulation of Newton’s equations is difficult : very 0 elongated ellipses, not many particles can be simulated, u 10 20 30 40 50 CPU time consuming (Peter 2009) DMP candidates for capture Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Kepler map x : Jupiter’s phase when particle at pericenter ( x = ϕ/ 2 π mod 1 ) w : particle energy at apocenter ( w = − 2 E / m DMP ) Symplectic Kepler map w ¯ = w + F ( x ) = w + W sin ( 2 π x ) ← energy change after a kick w − 3 / 2 ¯ x = x + ¯ ← 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. Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Let’s make a digression ... Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Halley map – Cometary case Kick functions of SS planets Mercury (x10 -6 ) Jupiter (x10 -2 ) 0.2 F 1 (x 1 ) F 5 (x 5 ) 0.5 0.1 0 0 -0.1 -0.5 -0.2 Venus (x10 -4 ) Saturn (x10 -3 ) 1.5 F 2 (x 2 ) F 6 (x 6 ) 0.5 1 0.5 0 Rollin, Haag, J. L., Phys. Lett. A 379 (2015) 0 1017-1022 -0.5 -0.5 Our calculations match direct observation -1 data and previous numerical data (Yeomans Earth (x10 -4 ) Uranus (x10 -4 ) 1 & Kiang 1981) 0.4 F 3 (x 3 ) F 7 (x 7 ) 0.5 0.2 8 0 � F ( x 1 , . . . , x 8 ) ≃ F i ( x i ) 0 -0.5 -0.2 i = 1 � r · r i -1 -0.4 � + ∞ 1 � F i ( x i ) = − 2 µ i ∇ − . ˙ r dt Mars (x10 -5 ) Neptune (x10 -4 ) r 3 � r − r i � 1.5 −∞ F 4 (x 4 ) F 8 (x 8 ) 0.2 1 Two main contributions 0.5 ◮ Direct planetary Keplerian potential 0 0 ◮ Rotating gravitational dipole potential -0.5 -0.2 due to the Sun movement around -1 Solar System barycenter 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 x i x i Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Halley map – Cometary case Renormalized kick function f i ( x i ) = F i ( x i ) / v 2 i /µ i Exponential decay with q for q > 1 . 5 a i More precisely � q � q � � 3 / 2 � � − 1 / 4 − 2 3 / 2 f i ≃ 2 1 / 4 π 1 / 2 exp a i 3 a i (Heggie 1975, Petrosky 1986, Petrosky & Broucke 1988, Roy & Haddow 2003, Shevchenko 2011, see also Rollin’s talk yesterday) Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Halley map – Dynamical chaos Poincaré section 0.5 0.5 0.5 0.5 0.49 0.49 0.45 0.45 0.48 0.48 w w 0.4 0.4 0.47 0.47 0.46 0.46 0.35 0.35 0.45 0.45 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 Symplectic Halley map x x 0.3 0.3 w − 3 / 2 x ¯ = x + ¯ 0.25 0.25 w ¯ = w + F ( x ) w w Chaotic dynamics of 1P/Halley 0.2 0.2 “Lifetime” ∼ 10 7 years 0.32 0.32 0.15 0.15 0.31 0.31 1:6 1:6 0.1 0.1 0.3 0.3 w w 3:19 3:19 3:19 3:19 0.29 0.29 3:19 3:19 2:13 2:13 2:13 2:13 3:20 3:20 3:20 3:20 0.05 0.05 0.28 0.28 3:20 3:20 1:7 1:7 0.27 0.27 0 0 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 x x x x Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Let’s come back to (dark) matter ... Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Dark map Kick function determination We determine numerically the kick function for any pa- rabolic orbit ( q , i , ω ) F ( x ) = F q , 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 Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter capture – Dark map ~25% Kick function determination We determine numerically the kick function for any pa- rabolic orbit ( q , i , ω ) F ( x ) = F q , i ,ω ( x ) By nonlinear fit we obtain analytical functions. Chance to be captured with a given energy ( w < 0 ↔ E > 0 ) h q , i ,ω ( w ) Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016

Dark matter – Capture cross section w cap = m � � 2 ≃ 10 − 3 � � � � � 2 � ˙ σ p = π r � � r � area enclosed by Jupiter’s orbit m ⊙ 10 8 10 1 10 0 10 4 1/|w| dN / dw / N p 10 -1 σ / σ p 10 0 10 -2 1/w 2 10 -3 10 -4 10 -4 10 -8 10 -5 10 -4 10 -2 10 0 10 2 10 -4 10 -2 10 0 10 2 10 4 | | w / w cap | | w / w cap m ⊙ w cap σ/σ p ≃ π | w | in agreement with Khriplovich & Shepelyansky 2009 m � ◮ 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) Institut Institut UTINAM UTINAM Dynamics and chaos in astronomy and physics, Luchon, Sept. 22th 2016