Neutrino acceleration: analogy with Fermi acceleration and - PowerPoint PPT Presentation

Neutrino acceleration: analogy with Fermi acceleration and Comptonization Yudai Suwa 1,2 1 Yukawa Institute for Theoretical Physics, Kyoto University 2 Max Planck Institute for Astrophysics, Garching 21/08/2015, MICRA2015@AlbaNova University Center

Neutrinos Supernovae, collapsars, mergers High temperature (~10 MeV), high density ( >10 12 g cm -3 ) Copious amount of neutrinos generated ( ~10 53 erg) Even neutrinos become optically thick F ν “neutrinospheres” Thermal distribution (0-th approx.) ε ν Cross section ( σ ν ∝ ε ν 2 ) Small change of distribution function can lead to signi fj cant di fg erence of interaction rates

Supernova shock heating layer cooling layer ν absorption ν emission PNS see talks by Fischer, Takiwaki, Kuroda, Messer, Sumiyoshi, O’Connor, Pan

Neutrino-driven jet McFadyen & Woosley 99 see talks by Just, Richers

Problems? ★ For supernovae, explosion energy in simulation ( E exp =10 49-50 erg) is much smaller than observation ( E exp ~10 51 erg) ★ For collapsars, neutrino annihilation might not produce enough strong jet for GRBs ★ Is there something missing? ★ Let’s reconsider about neutrino spectrum in more detail, beyond thermal spectrum

Courtesy of M. Liebendörfer Number Sphere Energy Sphere Transport Sphere high ρ low ρ

Analogy ★ Number and energy spheres can be called in di fg erent way ‣ chemical equilibrium : => thermal equilibrium => inside number sphere ‣ kinetic equilibrium : => does not change particle number => between number and energy spheres

Courtesy of M. Liebendörfer Number Sphere Energy Sphere Transport Sphere chemical kinetic equilibrium equilibrium Thermal Non-thermal?

Non-thermal neutrinos Energy Sphere Transport Sphere Matter fm ow Matter fm ow ν Thermal What distribution? ν G a i n e n e r g y b y s c a t t e r i n g b o d i e s ’ k i n e t i c e n e r g y � ∆ E � � � · u ”Fermi acceleration” of ν E κ Non-thermal neutrinos

Fermi acceleration (c)M. Scholer e.g., Axford+ (1977), Blandford & Ostriker (1978), Bell (1977)

Bulk Comptonization ★ The application of Fermi acceleration to photons ★ Compressional fm ow ( ∇ .V<0 ) leads to acceleration of photons ★ Compression is naturally realized for accretion fm ows onto black holes / neutron stars ( WITHOUT shock!) ★ Non-thermal components are generated from thermal components Blandford & Payne (1981), Payne & Blandford (1981)

Let’s go to neutrinos

Boltzmann eq. w/ di fg usion approx. Blandford & Payne 1981, Titarchuk+ 1997, Psaltis 1997 � ∂ f � k µ ∂ µ n ( k ) = ∂ t coll = 3 k B T � u 2 � + V 2 n (l , ν ) = ¯ n ( ν ) + 3l · f( ν ) � u � = V m di fg usion approx. bulk velocity thermal & turbulent vel. � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n � mc 2 � 4 � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n � 2 ν 3 �� ν �� ν �� ν ν

Transfer equation Boltzmann equation with di fg usion approx., up to O((u/c) 2 ) � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n � mc 2 � 4 � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n � 2 ν 3 �� ν �� ν �� ν ν di fg usion term bulk term recoil term source term thermal & turbulent terms n : ν ’s number density ε ν : ν energy V : velocity of matter κ : opacity T : temperature of matter

First order term By neglecting O((u/c) 2 ) terms and recoil term, we get � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n � mc 2 � 4 � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n � 2 ν 3 �� ν �� ν �� ν ν This is exactly the same equation we are solving with MGFLD or IDSA MGFLD Bruenn (1985) IDSA Liebendörfer+ (2009) Original ν -Boltzmann eq. (Lindquist 1966, Castor 1972) spherically symmetric � � d ln � cdt + 3 u �� � � d ln � cdt + 3 u � � cdt + µ � f d f (1 − µ 2 ) � f − u � f µ 2 � r + � µ + � ν µ cr cr cr �� ν up to O(u/c) E 2 � � � � Rf � dµ � − f R (1 − f � ) dµ � = j (1 − f ) − � f + (1 − f ) d ln ρ /dt= ∇ . V c ( hc ) 3

Order of approx. � ∂ f � k µ ∂ µ n ( k ) = ∂ t coll O(u/c) diffusion approx. � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n cdt + µ � f � � d ln � cdt + 3 u �� (1 − µ 2 ) � f � � d ln � cdt + 3 u � � � f d f − u � mc 2 � 4 � t + V · � n = � · 3 � � n 3( � · V ) � ν + j ( r , � ν ) µ 2 � r + � µ + µ � ν � 2 ν 3 �� ν �� ν �� ν cr cr cr �� ν ν E 2 � � � � Rf � dµ � − f R (1 − f � ) dµ � = j (1 − f ) − � f + (1 − f ) c ( hc ) 3 diffusion approx. O(u/c) � c + 1 � n � n � � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n �� ν

Transfer equation Boltzmann equation with di fg usion approx., up to O((u/c) 2 ) � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n � mc 2 � 4 � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n � 2 ν 3 �� ν �� ν �� ν ν di fg usion term bulk term recoil term source term thermal & turbulent terms n : ν ’s number density Solve this equation with adequate ε ν : ν energy boundary condition. The background matter is assumed to V : velocity of matter be free fall and stationary solution κ : opacity ( ∂ / ∂ t=0 ) is obtained. T : temperature of matter

Analytic solutions Nondimensional equation f ν ( τ ,x)=R( τ ) τ 5/2 x - α (separation of variables) Boundary conditions 1. fm ux ∝ τ 2 ( τ→ 0 ) 2. remain fj nite for τ >>1 ∞ � c n L 5 / 2 R ( τ ) = n (2 τ ) ( n=0,1,2… ) n =0 spectral energy fm ux YS, MNRAS (2013)

Numerical solution ★ Solved the transfer equation using relaxation method ★ At τ = τ 0 (@energy sphere) , thermal distribution is imposed non-thermal stronger non-thermal component with thermal deeper injection YS, MNRAS (2013)

Neutrino annihilation Energy injection rate by neutrino pair annihilation 〈 ε ν 〉 / 〈 ε ν 〉 thermal 〈 ε ν 2 〉 / 〈 ε ν 2 〉 thermal τ 0 Ampli fj cation Goodman+ 87, Setiawan+06 0.1 1.01 1.02 1.03 � � � � ⎛ ⎞ ε 2 ε 2 ⟨ ε ¯ ν ⟩ + ⟨ ε ν ⟩ ¯ ν ν ˙ 0.2 1.03 1.05 1.08 E ν ¯ ν = C F 3 , ν F 3 , ¯ ⎝ ⎠ ⎝ ν ⎠ ⟨ ε ν ⟩⟨ ε ¯ ν ⟩ 0.5 1.07 1.16 1.24 � f ν ε i and ⟨ ε 2 � where F i, ν = ν d ε ν , , ⟨ ε ν ⟩ = F 3, ν / F 2, ν ν ⟩ = F 4 , ν /F 2 , ν . where d , F F and 1.0 1.16 1.37 1.59 factor C includes the the weak interaction c fficients and infor- � � F 2 ε 2 2.0 1.37 1.99 2.73 3 , ν ν ˙ ν ∝ ⟨ ε ν ⟩⟨ ε 2 ν ⟩ . E ν ¯ ν ∝ . ¯ 3.0 1.60 2.83 4.52 ⟨ ε ν ⟩ 2 . and 5.0 1.95 4.49 12.5 We can evaluate t 10.0 2.43 7.12 17.3 Annihilation rate can be ampli fj ed by a factor of ~10 for the case of τ 0 =10

Does it work for supernova? ★ Unfortunately, no ★ To accelerate radiations ∇ .V need to be large at optically thick regime, but ∇ .V is small in the vicinity of PNS ★ For a black-hole forming collapse, this mechanism naively works (competition of acceleration and advection times)

Higher order e fg ects? ★ Bulk Comptonization is O(u/c) e fg ect WITH compressional fm ow ★ Is there any e fg ects from higher order? Let’s learn from photon case again ‣ Thermal Comptonization ‣ Turbulent Comptonization

Turbulent Comptonization ★ When there are turbulent fm ows, stochastic scattering can accelerate particles, like second order Fermi acceleration ★ Compressional fm ow is unnecessary, i.e., even when ∇ . V =0 , particle acceleration is possible e.g., Zel’dovich, Illarinov, Sunyaev (1972), Thompson (1994), Socrates (2004)

Neutrino transfer Boltzmann solv oltzmann solver max(v/c) in max(v/c) PNS PNS O(u/c) O((u/c) 2 ) spherical sometimes ~<10 -3 symmetry included included (1D) multi sometimes ~0.1? dimension not included included (2D/3D)

Turbulent velocity log( ρ ) v/c from neutrino-radiation hydro. simulation by Suwa+ (2014)

Summary � � � c n + ( k B T + mV 2 + 1 + 1 � �� � n � n � ) � n � mc 2 � 4 � t + V · � n = � · 3( � · V ) � ν + j ( r , � ν ) 3 � � n � 2 ν 3 �� ν �� ν �� ν ν di fg usion term bulk term recoil term source term thermal & turbulent terms ★ Based on analogy of photons, neutrino acceleration is investigated ★ O(u/c) : bulk Comptonization for γ => non-thermal ν from collapsars ★ O((u/c) 2 ) : thermal/turbulent Comptonization for γ => non-thermal ν from supernovae ★ Non-thermal ν can amplify neutrino interaction rate due to its high-energy tail