Rela%vis%c Nonthermal Par%cle Accelera%on in Magne%c Reconnec%on Dmitri Uzdensky (University of Colorado Boulder) Thanks to Univ. Colorado Theore9cal Plasma Astrophysics Group: Greg Werner Vladimir Zhdankin Mitch Begelman (support by NSF, NASA, DOE) 3 rd Purdue Workshop, May 8, 2018 D. Uzdensky 5/8/2018 1
OUTLINE: Theory of NTPA In Magne>c Reconnec>on Physical Picture and AssumpQons. Empirical (numerical) Knowledge Base. Key Ingredients of the KineQc EquaQon: AcceleraQon by reconnecqon electric field; MagneQzaQon by reconnected magneqc flux: (power-law slope) Trapping in plasmoids: (high-energy cutoff) Effects of Guide magneqc Field Conclusions (RadiaQve Turbulence) D. Uzdensky 5/8/2018 2
Physical Picture: Reconnec>on Plasmoid-dominated magneqc reconnecqon plasmoid chain, characterized by a plasmoid distribuqon funcqon F(w) and a cumulaqve distribuqon N(w): ReconnecQon may or not be relaqvisqc. ReconnecQon rate: E rec = β rec B 0 = ε V A B 0 /c = ε β A B 0 V A = cβ A = c σ h 1+σ h Hot magneqzaqon: σ h = B 02 /(4π nh); h = relaqvisqc enthalpy per parqcle. rel. limit: σ h >>1 à V A ~ c non-rel. limit σ h << 1 à β A = V A /c ~ σ h 1/2 << 1. I will also use cold σ = B 02 /(4π n b mc 2 ) But parqcles under consideraqon are ultra-relaqvisqc. D. Uzdensky 5/8/2018 3
Self-similar hierarchical plasmoid chain Cerun et al. 2013 Bhakacharjee et al. 2009 Two Phases of Reconnected MagneQc Field: - of order B1 = εb0 = 0.1 B0 in current sheets - of order B0 in plasmoids D. Uzdensky 5/8/2018 4
Dimensionality Real world is 3D. However, recent PIC simulaqons of relaqvisqc pair-plasma reconnecqon (Werner & Uzdensky 2017, also Sironi & Spitkovsky 2014, Guo et al. 2015) show that 2D and 3D give the same NTPA. (However, for non-relaqvisqc electron-ion plasma reconnecqon, 2D and 3D NTPA may be different, see Dahlin et al. 2015-2017.) Thus here we will consider 2D reconnecqon. DirecQon perpendicular to current sheet: parqcles confined to the sheet by reconnecqng magneqc field à consider only moqon along the sheet. D. Uzdensky 5/8/2018 5
Basic Picture While they are unmagneqzed, energeqc parqcles are accelerated by the main reconnecqon electric field E rec. This steady acceleraqon proceeds unql a parqcle is captured by the reconnected magneqc field. Reconnected magneqc field is bimodal: ~ B 1 ~ ε B 0 ~ 0.1 B 0 in inter-plasmoid current layers; ~ B 0 in circularized plasmoids [with size distribuqon F(w)] We will treat magneqzaqon in B 1 and trapping in plasmoids separately. D. Uzdensky 5/8/2018 6
Empirical Numerical Proper>es of Reconnec>on-driven rela>vis>c NTPA Power-law index p p scales with σ h as p(σ) 2.2 p ~ C 1 + C 2 /σ h 1/2 High-energy cutoff (Werner et al. 2016) 2.0 1.8 1.6 1.4 1.2 1.0 10 0 10 1 10 2 10 3 pair plasma: Werner et al. 2016 electron-ion plasma: Werner et al. 2018 Two high-energy cutoffs: - exp(- γ/γ c1 ); γ c1 ~ 4σ, (guide magneqc field suppresses NTPA, see below) - exp[-(γ/γ c2 ) 2 ]; γ c2 ~ 0.1 L/ρ 0 (ρ 0 = m e c 2 /e B 0 ) D. Uzdensky 5/8/2018 7
High-Energy Power-Law Cutoff (Werner, Uzdensky, CeruI, Nalewajko, Begelman, ApJL 2016) 2.2 α * (σ h ) 2.0 1.8 1.6 1.4 Two high-energy cutoffs: exp (- γ/γ c1 ); γ c1 ~ 4σ, - independent of L; γ c1 ~ 10 <ε> 1.2 σ h 1.0 10 0 10 1 10 2 10 3 exp [-(γ/γ c2 ) 2 ]; γ c2 ~ 0.1 L/ρ 0 - independent of σ. γ c1 c1 10 4 10 3 10 2 L/ 0 = 25 L/ 0 = 50 L/ 0 = 100 L/ 0 = 200 L/ 0 = 400 c1 =4 σ 10 10 1 0 10 1 10 2 10 3 available voltage drop ε max ~ e E rec L ~ 0.1 e B 0 L (Hillas, extreme accelera>on) (ρ 0 = m e c 2 /e B 0 ) Large-system regime: (γ c1 < γ c2 ): L/ρ 0 > 40 σ 10 2 10 3 10 4 10 5 L/ 0 D. Uzdensky 5/8/2018 8 γ c2 c2 10 4 10 3 10 2 10 1 = 1000 = 300 = 100 = 30 = 10 =3 γ c2 =0.1L/ 0 L/ρ 0
Why is there a γ c 4σ cutoff? Cutoff comes from small laminar elementary interplasmoid layers at the bokom of the plasmoid hierarchy (marginally stable to tearing). ParQcles are accelerated in these layers but then become trapped inside plasmoids. Cerun et al. 2013 l Zenitani & Hoshino 2001 (ρ 0 =m e c 2 /e B 0 ) σ = B 02 /(4π nmc 2 ) δ Cutoff: γ c = e E rec l /m e c 2 0.1 e B 0 l /m e c 2 = 0.1 l /ρ 0. Layers are marginally stable to tearing à l ~ 100 δ. Layer thickness: δ ρ (<γ>) = <γ> ρ 0 (σ /3) ρ 0. Thus, l /ρ 0 100 δ/ρ 0 30 σ => γ 0 = 3 σ. Further parqcle acceleraqon is possible, e.g., in plasmoid mergers, but this 2nd-stage reconnecqon acceleraqon occurs with lower sigma and smaller L. D. Uzdensky 5/8/2018 9
Rela>vis>c e-i reconnec>on: Key PIC Sims Results I Werner et al., MNRAS 2018; arxiv:1612.04493 Systema%c 2D PIC study: (no guide field, m i /m e =1836) Energy par>>oning between electrons and ions (useful prescripqon for GRMHD models of BH accreqon flows) Reconnec>on rate: v in /c = E/B 0 ions ce V A B 0 electrons V A = c σ i B 0 4πn i m i c 2 + B 0 2 = c σ i 1+σ i D. Uzdensky 5/8/2018 10 σ i In semi-rela%vis%c case ions gain 3 %mes more energy than electrons.
Rela>vis>c e-i reconnec>on: Key PIC Sims Results II Par>cle Accelera>on: - electrons: - ions? Electron power-law index p and cutoff γ c : p e σ i Observed blazar p e ~2-3 imply σ i ~1, consistent with expectaqons from MHD jet models. γ c /σ i Werner et al. 2018 D. Uzdensky 5/8/2018 11 σ i
Kine>c Equa>on Key Ingredients: AcceleraQon by main reconnecqon electric field: - independent of γ MagneQzaQon by reconnected B-field (--> power-law): Trapping by large [w > ρ L (γ)] plasmoids (à cutoff): τ tr controlled by plasmoid distribuqon funcqon (below) D. Uzdensky 5/8/2018 12
Steady State Kine>c Equa>on Since γ acc is independent of γ, we get SoluQon: where D. Uzdensky 5/8/2018 13
Magne>za>on by Reconnected Field and NTPA power law EnergeQc parqcle passes right through small plasmoids. Distance a parqcle travels before being magneqzed l mag (γ) ~ρ L (γ, B 1 ) = (B 0 /B 1 ) ρ L (γ, B 0 ) = ε -1 ρ 0 γ magneqzaqon Qme-scale Balance magneqzaqon with acceleraqon in kineqc eqn: power-law soluqon f(γ) ~ γ -p power-law index: ultra-rel (σ h >>1): p à const (cf. Zenitani & Hoshino 2001) non-rel. case (σ h >>1): p ~ σ h -1/2 (c.f., Werner et al. 2018) D. Uzdensky 5/8/2018 14
Plasmoid Chain I: single power-law EnergeQc parqcles can be trapped in large plasmoids when w =w(γ) = ρ L (γ) = ρ 0 γ High-energy cutoff is controlled by plasmoid distribuqon funcqon F(w) = - dn/dw. τ tr = λ pl (w)/c where λ pl (w) is separaqon between plasmoids of size w: λ pl (w) = L/N(w) Thus τ tr = L/c N(w) (ρ 0 = m e c 2 /e B 0 ) D. Uzdensky 5/8/2018 15
Single-Power-law plasmoid chain Consider for illustraqon: F(w) ~ w -α for w < w max CumulaQve distribuqon: N(w) ~(w/w max ) 1-α [where N(w max ) =1] where γ max = w max /ρ 0 Special case α = 2: τ trap ~ γ (same as for magneqzaqon - later) Trapping rate overtakes magneqzaqon at For w max ~ εl: γ c ~ γ max D. Uzdensky 5/8/2018 16
Establishing Cutoff Balancing acceleraqon against trapping in plasmoids: Special case α ~ 1 (ignoring log-correcqons) where (can be < γ max, if β A < 1) Special case α=2: no cutoff but another power-law: For w max ~ εl: p ~ β A -1 (~same as without plasmoids) Combined with magneqzaqon: à steeper (by x2) power law. D. Uzdensky 5/8/2018 17
Plasmoid Chain II: Realis>c double-power-law Real simulaqons show double-power-law plasmoid distribuqons (Loureiro et al. 2012, Huang et al. 2013, Sironi et al. 2016, Petropoulou et al. 2018) For α 1 1: ~ γ c = w c /ρ 0 ~ γ c Most likely: α 1 1, α 2 2 Important QuesQon: What controls w c? If α 2 2: possible 2 nd (steeper) power law above spectral break at γ c D. Uzdensky 5/8/2018 18
Important Ques>on: What controls w c? Consider large-system regime: L >> δ ~ <ρ> ~ σ ρ 0 The plasmoid distribuqon break size w c may be anywhere between microscopic ~ σ ρ 0 and macroscopic ~ L. If w c ~ σ ρ 0, then γ c ~ σ, e.g., γ c 4σ (Werner et al. 2016) If w c ~ L, then γ c ~ L/ρ 0 -- extreme (Hillas) acceleraqon limit. D. Uzdensky 5/8/2018 19
Effect of Guide Magne>c Field (Werner & Uzdensky 2017 ApJL) Par>cle spectra for different B z /B 0 : L z /L x = 1 γ 2 f(γ) γ Par>cle accelera>on is nega>vely affected by strong guide field. Explana9on: Guide field s iner%a: Guide-field needs to be advected out of the layer with the plasma, so guide field s inerqa contributes B gz2 /4π to the enthalpy in the denominator of σ h : σ h,eff = B 02 /(B gz 2 + 4π nh) -- reduces in-plane V A, hence recn. ouƒlow speed, hence E rec D. Uzdensky 5/8/2018 20
Conclusions RelaQvisQc Nonthermal ParQcle AcceleraQon (NTPA) in reconnecqon is an interplay of: steady acceleraqon by reconnecqon electric field; checked by escape from acceleraqon zone: - (1) magneqzaqon by general reconnected magneqc field B 1 ~ 0.1 B 0 à power-law index p ~ (E rec /B 1 ) -1 ~ 1/ β A ~ [1+σ h )/σ h ] 1/2 - (2) capture/trapping by plasmoids with w ~ ρ L (γ) =γ ρ 0. à high-energy cutoff. - Cutoff depends on plasmoid-distribuqon funcqon, e.g., for α=1 it is simple exponenqal with γ c = w max /ρ 0. - RealisQc double-power-law F(w): cutoff at break size w c and - possible steeper power law (if α 2 2) above w c Guide field s inerqa B gz2 /4π adds to enthalpy, reduces effecqve σ h D. Uzdensky 5/8/2018 21