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Compton scattering in strongly magnetized plasma Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia Quarks06, Repino 23 May 2006 23 May 2006 Compton scattering in strongly magnetized plasma Rumyantsev D.A.,

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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 Quarks’06, Repino Compton scattering in strongly magnetized plasma

(page 1)

Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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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 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Introduction

Introduction

Critical value of magnetic field Be = m2

e

e ≃ 4.41 × 1013 G, c = = k = 1. Magnetars SGR – soft gamma repeaters, anomalous X-ray pulsars (AXP). B ∼ 1015 G. SGR 1806-20 – B ∼ 7 × 1015 G (Israel et al. astro-ph/0505255).

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Introduction

Historical review

Melrose, Parle 1983 - modification of the scattering amplitude Harding et al. 1986, 2000 - cross section in the limit ω ≪ 2m 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 γ1 → γ2γ2 This is correct only at T ≪ 25 keV (µ = 0) The limit of strongly magnetized plasma eB ≫ T 2, µ2, ω2, E2 The region is below cyclotron resonance: eB ≫ (pk)

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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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 P(1)(q) ≃ − α 3π q2

⊥ − q2 Λ(B),

P(2)(q) ≃ −2eBα π

  • H

4m2 q2

  • + J (q)
  • − q2 Λ(B),

P(3)(q) ≃ − q2 Λ(B), where Λ(B) =

α 3π [1.792 − ln(B/Be)] ,

J (q) = 4q2

m2

dpz E fE (q2

)2 − 4(pq)2

  • ,

E =

  • p2

z + m2,

fE = [exp (E/T) + 1]−1 is the electron distribution function, H(z) = z √z − 1 arctan 1 √z − 1 − 1, z ≥ 1.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Photon dispersion ...

Dispersion equation q2 − P(λ)(q) = 0 (λ = 1, 2, 3). Polarization vectors (λ = 1, 2) ε(1)

α (q) ≃

(ϕq)α

  • (qϕϕq)

, ε(2)

α (q) ≃

( ˜ ϕq)α

  • (q ˜

ϕ ˜ ϕq) . Renormalization of the mode 2 of the photon wave function ε(2)

α → ε(2) α

  • Z2,

Z−1

2

= 1 − ∂P(2) ∂ω2 . 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 directed along the third axis; ϕαβ = Fαβ/B and ˜ ϕαβ = 1

2εαβµνϕµν are the dimensionless

field tensor and dual field tensor correspondingly. The tensors Λαβ = (ϕϕ)αβ, and

  • Λαβ = ( ˜

ϕ ˜ ϕ)αβ related by Λαβ − Λαβ = gαβ = diag(1, −1, −1, −1) are introduced.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Photon dispersion ...

0.2 0.4 0.6 0.8 1 1 1.2 1.4 0.25 0.75 1.25 1.5 1.75 2 0.5

q2

  • 4m2

q2

⊥

4m2

Photon dispersion laws in the strong magnetic field B/Be = 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, q2

− q2 ⊥ = 0. 23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Photon dispersion ...

The Feynman diagrams for the Compton process in magnetic field.

e(p)
  • (q
) e(p )
  • (q
) + e(p)
  • (q
) e(p )
  • (q
)

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Photon dispersion ...

Amplitudes of the photon scattering

M1→1 = −8παm eB (qϕq′)(q ˜ ϕq′)

  • q2

⊥q′2 ⊥(−Q2 )

, M1→2 = −8παm eB (qΛq′)(q′˜ ΛQ)

  • q2

⊥q′2 (−Q2 )

, M2→1 = 8παm eB (qΛq′)(q˜ ΛQ)

  • q2

q′2 ⊥(−Q2 )

, M2→2 = 16iπαm q2

q′2

  • (−Q2

) κ

(q˜ Λq′)2 − κ2(q ˜ ϕq′)2, where κ =

  • 1 − 4m2/Q2

, Q2 = (q − q′)2 < 0. 23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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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 Wλe±→λ′e± = eB 16(2π)4ωλ

  • | Mλ→λ′ |2 ZλZλ′ ×

×fE (1 − fE′) (1 + fω′)δ(ωλ(k) + E − ωλ′(k′) − E′)dpz d3k

′

EE′ωλ′ , where fω′ = [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 Wλ→λ′ = Wλe−→λ′e− + Wλe+→λ′e+ = neσλ→λ′, ne = eB

  • mT

2π3 e−m/T.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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The absorption coefficient ...

σ1→1 = α2π (eB)2 ω2 ω + 2m 2(ω + m) + m ω ln

  • 1 + ω

m

  • ,

σ2→1 = α2πZ2 (eB)2 q2

⊥

ω + 2m 2(ω + m) − m ω ln

  • 1 + ω

m

  • ,

σ1→2 = α2π(ω + 2m)2 2(eB)2ω(ω + m)

ω2

  • dq′2
  • 1 + 2αeB

π 1 q′2

  • H

4m2 q′2

  • ×

×

  • ω2 − q′2
  • (ω + 2m)2 − q′2
  • ,

σ2→2 = 16m2α2π(ω + m) ω3(ω + 2m)2 Z2

  • ω(ω + 2m)

(ω + m)(2m − ω) − ln

  • 1 + ω

m

  • +

+ 2ω(ω − m)(2m + ω) (ω + m)(2m − ω) √ 4m2 − ω2 arctan ω √ 4m2 − ω2

  • .

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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The absorption coefficient ...

0.2 1 0.8 0.4 0.6 2 4 6

  • 2
  • 4
  • 6

a a b b c c log10

  • W

W0

  • ω

2m

The dependence of the absorption coefficient of photon scattering of channels γ2e± → γ2e± (solid line) and γ2e± → γ1e± (dashed line) on the energy of the initial photon in a strong magnetic field B/Be = 200 and neutral (µ = 0) plasma, at T = 1MeV – a, T = 250keV – b, T = 50keV – c. The chain line corresponds to the probability of photon splitting, γ2 → γ1γ1, at T = 1MeV . Here W0 = (α/π)3 m.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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The absorption coefficient ...

PSfrag repla emen ts 1 2 2 3 4 4 5 6 7
  • 2
  • 4
  • 6
a a b b log 10
  • W
W
  • !
2m

The dependence of the absorption coefficient of photon scattering of channels γ1e± → γ1e± (firm line) and γ1e± → γ2e± (dashed line) on energy of initial photon in a strong magnetic field B/Be = 200 and neutral (µ = 0) plasma, at T = 1MeV – a, T = 250keV – b, T = 50keV – c. The dotted and chain lines correspond to the probability of photon splitting, γ1 → γ1γ2 and γ1 → γ2γ2, respectively, at T = 50keV .

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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The absorption coefficient ...

The channels γ2e± → γ2e± and γ2e± → γ1e± are dominated over the photon splitting channel γ2 → γ1γ1. The absorption coefficient of scattering γ1e± → γ1e± is compared with photon splitting at ω < 2m and T ≤ 25 keV Low limit of the photon energy can be obtained from equation

R

  • R∗

drWλ→λ′(r, ωsc) = 1. T ≥ 16 keV = ⇒ the photon of the second mode can’t leave the region filled by the strong magnetic field and plasma. The result for the channel γ1e± → γ1e± is ωsc ≃ 85 keV (T = 25 keV). Duncan and Thompson obtained that ωsp ≃ 37.5 keV in the channel γ1 → γ2γ2.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia

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Summary

Summary

  • The process of Compton scattering, γe± → γe±, in a strongly magnetized medium of

an arbitrary temperature and zeroth chemical potential is considered.

  • The analytical expressions for the partial cross section in the small number of density

limit of the electron-positron plasma are obtained.

  • The numerical estimations for partial probabilities of this process are presented by

taking into account the photon dispersion in a strong magnetic field and a charge- symmetric plasma of an arbitrary temperature.

  • The comparison of the scattering probability with photon splitting in a plasma of

arbitrary temperatures was performed. The results show, that the photon scattering and photon splitting channels are comparable at T ≥ 25 keV and magnetic field strength B = 200Be.

  • The estimation of the low limit of photon energy, at which the optical depth is equal

to one was obtained. This work supported in part by the Council on Grants by the President of Russian Federation for the Support of Young Russian Scientists and Leading Scientific Schools of Russian Federation under the Grant No. NSh-6376.2006.2, and by the Russian Foundation for Basic Research under the Grant No. 04-02-16253.

23 May 2006 Quarks’06, Repino Compton scattering in strongly magnetized plasma

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Rumyantsev D.A., Chistyakov M.V. Yaroslavl State University, Yaroslavl, Russia