Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia Quarks’06, Repino 23 May 2006 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 1) Yaroslavl State University, Yaroslavl, Russia
Contents Contents I. Introduction II. Photon dispersion properties in a strongly magnetized plasma III. The absorption coefficient of Compton scattering in strongly mag- netized plasma and astrophysical applications IV. Summary 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 2) Yaroslavl State University, Yaroslavl, Russia
Introduction Introduction Critical value of magnetic field B e = m 2 e ≃ 4 . 41 × 10 13 G , e c = � = k = 1 . Magnetars SGR – soft gamma repeaters, anomalous X-ray pulsars (AXP). B ∼ 10 15 G. SGR 1806-20 – B ∼ 7 × 10 15 G (Israel et al. astro-ph/0505255). 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 3) Yaroslavl State University, Yaroslavl, Russia
Introduction Historical review Melrose, Parle 1983 - modification of the scattering amplitude Harding et al. 1986, 2000 - cross section in the limit ω ≪ 2 m Elmfors et al. 1998 - Compton scattering could compete with the photon splitting The astrophysical application Duncan, Thompson 1995 - Compton scattering in the limit T ≪ m , is suppressed in comparison with the photon splitting channel γ 1 → γ 2 γ 2 This is correct only at T ≪ 25 keV ( µ = 0 ) The limit of strongly magnetized plasma eB ≫ T 2 , µ 2 , ω 2 , E 2 The region is below cyclotron resonance: eB ≫ ( pk ) 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 4) Yaroslavl State University, Yaroslavl, Russia
Photon dispersion ... Photon dispersion properties in a strongly magnetized and charge-symmetric plasma ( µ = 0 ) The eigenvalues of the photon polarization operator in plasma could be presented in the following form − α ⊥ − q 2 Λ( B ) , P (1) ( q ) 3 π q 2 ≃ � � 4 m 2 � � − 2 eBα − q 2 Λ( B ) , P (2) ( q ) ≃ H + J ( q � ) q 2 π � − q 2 Λ( B ) , P (3) ( q ) ≃ α where Λ( B ) = 3 π [1 . 792 − ln( B/B e )] , � dp z � f E J ( q � ) = 4 q 2 � m 2 p 2 z + m 2 , , E = � ) 2 − 4( pq ) 2 ( q 2 E � f E = [exp ( E/T ) + 1] − 1 is the electron distribution function, z 1 H ( z ) = √ z − 1 arctan √ z − 1 − 1 , z ≥ 1 . 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 5) Yaroslavl State University, Yaroslavl, Russia
Photon dispersion ... Dispersion equation q 2 − P ( λ ) ( q ) = 0 ( λ = 1 , 2 , 3) . Polarization vectors ( λ = 1 , 2 ) ( ϕq ) α ( ˜ ϕq ) α ε (1) ε (2) α ( q ) ≃ � , α ( q ) ≃ � . ( qϕϕq ) ( q ˜ ϕ ˜ ϕq ) Renormalization of the mode 2 of the photon wave function � = 1 − ∂ P (2) ε (2) α → ε (2) Z − 1 Z 2 , ∂ω 2 . α 2 The four-vectors with indices ⊥ and � belong to the Euclidean { 1, 2 } -subspace and the Minkowski { 0, 3 } -subspace correspondingly in the frame were the magnetic field is ϕ αβ = 1 directed along the third axis; ϕ αβ = F αβ /B and ˜ 2 ε αβµν ϕ µν are the dimensionless field tensor and dual field tensor correspondingly. The tensors Λ αβ = ( ϕϕ ) αβ , and � ϕ ) αβ related by � Λ αβ = ( ˜ ϕ ˜ Λ αβ − Λ αβ = g αβ = diag (1 , − 1 , − 1 , − 1) are introduced. 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 6) Yaroslavl State University, Yaroslavl, Russia
Photon dispersion ... q 2 � 4 m 2 1.4 1.2 1 0.8 0.6 0.4 0.2 q 2 ⊥ 4 m 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Photon dispersion laws in the strong magnetic field B/B e = 200 and neutral plasma at the temperature: T = 1 MeV (upper solid curve), T = 0 . 5 MeV (middle solid curve), T = 0 . 25 MeV (lower solid curve). The photon dispersion without plasma is depicted by the dashed line. The dotted line corresponds to the vacuum dispersion law, q 2 � − q 2 ⊥ = 0 . 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 7) Yaroslavl State University, Yaroslavl, Russia
Photon dispersion ... 0 0 � ( q ) � ( q ) � ( q ) � ( q ) The Feynman diagrams for the Compton process in magnetic + field. 0 0 e ( p ) e ( p ) e ( p ) e ( p ) 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 8) Yaroslavl State University, Yaroslavl, Russia
Photon dispersion ... Amplitudes of the photon scattering ( qϕq ′ )( q ˜ ϕq ′ ) M 1 → 1 = − 8 παm � , eB q 2 ⊥ q ′ 2 ⊥ ( − Q 2 � ) ( q Λ q ′ )( q ′ ˜ M 1 → 2 = − 8 παm Λ Q ) � , eB q 2 ⊥ q ′ 2 � ( − Q 2 � ) ( q Λ q ′ )( q ˜ M 2 → 1 = 8 παm Λ Q ) � , eB q 2 � q ′ 2 ⊥ ( − Q 2 � ) � � q 2 � q ′ 2 ( − Q 2 � ) κ � M 2 → 2 = 16 iπαm ϕq ′ ) 2 , Λ q ′ ) 2 − κ 2 ( q ˜ ( q ˜ � � , Q 2 � = ( q − q ′ ) 2 1 − 4 m 2 /Q 2 where κ = � < 0 . 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 9) Yaroslavl State University, Yaroslavl, Russia
The absorption coefficient ... The photon scattering absorption coefficient The general expression for the photon scattering absorption coefficient by real electrons and positrons of the medium can be written in the following form � eB | M λ → λ ′ | 2 Z λ Z λ ′ × W λe ± → λ ′ e ± = 16(2 π ) 4 ω λ ′ × f E (1 − f E ′ ) (1 + f ω ′ ) δ ( ω λ ( k ) + E − ω λ ′ ( k ′ ) − E ′ ) dp z d 3 k EE ′ ω λ ′ , where f ω ′ = [exp ( ω ′ /T ) − 1] − 1 is the photon distribution function. The analytical expression for the photon scattering absorption coefficient in the case of a rare electron gas ( T ≪ m ) can be presented as W λ → λ ′ = W λe − → λ ′ e − + W λe + → λ ′ e + = n e σ λ → λ ′ , � mT 2 π 3 e − m/T . n e = eB 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 10) Yaroslavl State University, Yaroslavl, Russia
The absorption coefficient ... � ω + 2 m �� � σ 1 → 1 = α 2 π 2( ω + m ) + m 1 + ω ( eB ) 2 ω 2 ω ln , m � ω + 2 m �� � σ 2 → 1 = α 2 πZ 2 2( ω + m ) − m 1 + ω ( eB ) 2 q 2 ω ln , ⊥ m � � 4 m 2 �� � ω 2 α 2 π ( ω + 2 m ) 2 1 + 2 αeB 1 dq ′ 2 σ 1 → 2 = H × 2( eB ) 2 ω ( ω + m ) � q ′ 2 q ′ 2 π � � 0 � ω 2 − q ′ 2 � × , ( ω + 2 m ) 2 − q ′ 2 � � � � 16 m 2 α 2 π ( ω + m ) ω ( ω + 2 m ) 1 + ω σ 2 → 2 = Z 2 ( ω + m )(2 m − ω ) − ln + ω 3 ( ω + 2 m ) 2 m � 2 ω ( ω − m )(2 m + ω ) ω √ √ + 4 m 2 − ω 2 arctan . 4 m 2 − ω 2 ( ω + m )(2 m − ω ) 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 11) Yaroslavl State University, Yaroslavl, Russia
The absorption coefficient ... a 6 b 4 2 c � � b W log 10 0 W 0 a -2 -4 c -6 0.2 0.4 0.6 0.8 1 ω 2 m The dependence of the absorption coefficient of photon scattering of channels γ 2 e ± → γ 2 e ± (solid line) and γ 2 e ± → γ 1 e ± (dashed line) on the energy of the initial photon in a strong magnetic field B/B e = 200 and neutral ( µ = 0) plasma, at T = 1 MeV – a , T = 250 keV – b , T = 50 keV – c . The chain line corresponds to the probability of photon splitting, γ 2 → γ 1 γ 1 , at T = 1 MeV . Here W 0 = ( α/π ) 3 m . 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 12) Yaroslavl State University, Yaroslavl, Russia
PSfrag repla emen ts 4 a b 2 a The absorption coefficient ... 0 � � W b log 10 W 0 -2 -4 -6 1 2 3 4 5 6 7 ! 2 m The dependence of the absorption coefficient of photon scattering of channels γ 1 e ± → γ 1 e ± (firm line) and γ 1 e ± → γ 2 e ± (dashed line) on energy of initial photon in a strong magnetic field B/B e = 200 and neutral ( µ = 0) plasma, at T = 1 MeV – a , T = 250 keV – b , T = 50 keV – c . The dotted and chain lines correspond to the probability of photon splitting, γ 1 → γ 1 γ 2 and γ 1 → γ 2 γ 2 , respectively, at T = 50 keV . 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Quarks’06, Repino (page 13) Yaroslavl State University, Yaroslavl, Russia
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