Proc. Neutrino Astrophysics - MPP Theory Group

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86 Photon Dispersion in a Supernova Core Alexander Kopf Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 1, 85748 Garching, Germany In astrophysical plasmas the dispersion relation of photons is usually dominated by the electronic plasma effect. It was recently claimed [1], however, that in a supernova (SN) core the dominant contribution is caused by the photon interaction with the nucleon magnetic moments. This contribution can have the opposite sign of the plasma term so that the photon four-momentum K could become space-like, allowing the Cherenkov processes γν → ν and ν → γν. Because of a numerical error the results of Ref. [1] are far too large [2], but the correct effect is still large enough to call for a closer investigation. Therefore, we will estimate the different possible contributions to the refractive index, which is taken to be a real number, nrefr = √ ǫµ = k/ω. (1) It relates the wavenumber k and frequency ω of an electromagnetic excitation and is given in terms of the dielectric permittivity ǫ and the magnetic permeability µ [3]. We may also write the dispersion relation in the form ω2 − k2 = m2 eff , where m2eff can be positive or negative. We first assume that the medium constituents (protons, neutrons, electrons) interact with each other only electromagnetically. This leaves us with the usual plasma effect which is much larger for the electrons than for the protons. It is approximately described by n2 refr ≈ 1−ω2 p/ω2 , where ω2 p = 4παnp/mp for the protons (nondegenerate, nonrelativistic) and ω2 p = (4α/3π)µ 2 e for the electrons (degenerate, relativistic) with µe the electron chemical potential. If the nuclei are not completely dissociated, nuclear bound-free transitions might be significant. However, at least for the simple example of deuteron states this is not the case. For a medium consisting of deuterons alone we employ the usual relation between refractive index and forward-scattering amplitude f0 [4] nrefr = 1 + (2π/ω2 )nd Ref0(ω). From the values for Ref0 of Ref. [5] we conclude that the contribution from deuteron-like states never exceeds the single-proton plasma effect and is thus negligible. The main point of our work is to calculate the contribution from correlated spins. We study the simplest model, a medium consisting purely of neutrons. In the nonrelativistic case they do not respond to an applied electric field so that ǫ = 1. We write the magnetic permeability µ = 1 + χ [3] in terms of the magnetic susceptibility χ = χ ′ + iχ ′′ which implies n2 refr −1 = χ′ (ω,k). The fluctuation dissipation theorem [6] establishes a relationship between the absorptive part χ ′′ and spin fluctuations. The neutron spin fluctuations can be described by the dynamical spin-density structure function Sσ(ω,k) [7]. We use a simple model for its actual shape [8] in terms of the spin fluctuation rate Γσ which can be estimated from the nucleon interaction potential. We calculate the refractive index (or rather χ ′ ) from χ ′′ via a Kramers-Kronig relation and express it as an effective mass m2 eff = (1 − n2 refr )ω2 . In Fig. 1 we show the different contributions to m2 eff relative to the one by the electrons for typical conditions of a SN core (ρ = 7 × 1014 g cm−3 , electron to baryon ratio Ye = 0.3, T = 40MeV, Γσ = 30T). All the approximations made in our calculation work in the same direction of overestimating the magnetic-moment contribution.
2 2 / meff,e m eff 1.0 0.5 0.0 electrons protons neutrons 0.01 0.1 1 10 ω / T Figure 1: The contributions of electrons, protons and correlated neutrons for ρ = 7 × 10 14 g cm −3 , Ye = 0.3, T = 40MeV and Γσ = 30T. The contribution of the correlated spins has the opposite sign relative to the plasma terms only in a small ω range and thus cannot change the sign of the total m2 eff . Thus the refractive index is always less than 1 and Cherenkov effects remain forbidden. Acknowledgments This talk is based on work done in collaboration with G. Raffelt [8]. I thank S. Hardy for comments on the manuscript. References [1] S. Mohanty and M.K. Samal, Phys. Rev. Lett. 77, 806 (1996). [2] G.G. Raffelt, Phys. Rev. Lett. 79, 773 (1997). [3] J.D. Jackson, Classical Electrodynamics (John Wiley, New York, 1975). [4] J.J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass., 1967). [5] J. Ahrens, L.S. Ferreira, W. Weise, Nucl. Phys. A485, 621 (1988). [6] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin-Cummings, Reading, Mass., 1975). [7] H.-T. Janka, W. Keil, G. Raffelt, and D. Seckel, Phys. Rev. Lett. 76, 2621 (1996). [8] A. Kopf and G. Raffelt, Eprint astro-ph/9711196, to be published in Phys. Rev. D (1998). 87
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arXiv:astro-ph/9801320v1 30 Jan 199
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Preface This was the fourth worksho
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vi S.J. Hardy Quasilinear Diffusion
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viii
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History of Neutrino Physics: Pauli
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Brief an Oskar Klein, Stockholm, vo
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Recent observations with the Hubble
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MACHO sample of ∼100,000 light cu
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Solar Neutrinos
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14 In the depth range where an abun
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16 as follows: (1) All the detector
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18 [3] J.N. Bahcall and H.A. Bethe,
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20 the opacity has a very slowly ch
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22 Status of the Radiochemical Gall
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24 known true source activity. The
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26 Solar Neutrino Observation with
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28 Events/day/kton/bin 0.3 0.25 0.2
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30 log(Δm 2 (eV 2 log(Δm )) 2 (eV
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32 Table 1: Neutrino event rates in
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34 tank and sphere high purity wate
Page 44 and 45: 36 Measurements of Low Energy Nucle
Page 46 and 47: 38 Figure 1: The S(E) factor of 3 H
Page 48 and 49: 40 Solar Neutrinos: Where We Are an
Page 50 and 51: 42 [9] B. Pontecorvo, Chalk River R
Page 52 and 53: 44 Figure 2: The difference between
Page 55 and 56: Phenomenology of Supernova Explosio
Page 57 and 58: Figure 2: Theoretical classificatio
Page 59 and 60: Supernova Rates Bruno Leibundgut Eu
Page 61 and 62: galaxies, however, and the statisti
Page 63 and 64: Figure 1: The motions of pulsars re
Page 65 and 66: Convection in Newly Born Neutron St
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Page 69 and 70: ference). The angular variations of
Page 71 and 72: Figure 1: Neutrino luminosity and e
Page 73 and 74: [9] G.S. Bisnovatyi-Kogan, Astron.
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Page 77 and 78: Figure 1: Time variation of superno
Page 79 and 80: Figure 3: Energy spectra of the sup
Page 81 and 82: Supernova Neutrino Opacities G.G. R
Page 83 and 84: Quasilinear Diffusion of Neutrinos
Page 85 and 86: Assuming a saturated thermal distri
Page 87 and 88: Anisotropic Neutrino Propagation in
Page 89 and 90: So far the damping rate corresponds
Page 91 and 92: Cherenkov Radiation by Massless Neu
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Page 99 and 100: Gamma-Ray Burst Observations D.H. H
Page 101 and 102: Figure 1: The cumulative distributi
Page 103 and 104: [7] Lilly, S., in Critical Dialogue
Page 105 and 106: ) Phase Transitions in Neutron Star
Page 107 and 108: after the collapse began (nb., no e
Page 109 and 110: Models of Coalescing Neutron Stars
Page 111 and 112: Figure 2: Contour plots of the mass
Page 113 and 114: From our torus models we find that
Page 115: High-Energy Neutrinos
Page 118 and 119: 110 in gamma-rays. However, the acc
Page 120 and 121: 112 Atmospheric Muons and Neutrinos
Page 122 and 123: 114 Today, however, charm productio
Page 124 and 125: 116 Atmospheric Neutrinos in Super-
Page 126 and 127: 118 number of events 300 200 100 (a
Page 128 and 129: 120 Acknowledgements D.K. is suppor
Page 130 and 131: 122 Key signatures for νµ nucleon
Page 132 and 133: 124 (a.u) 0.5 0.4 0.3 0.2 0.1 0 0 1
Page 134 and 135: 126 However, in order to operate th
Page 136 and 137: 128 could be gravitationally captur
Page 138 and 139: 130 Ground-Based Observation of Gam
Page 140 and 141: 132 Figure 2: Sensitivities in unit
Page 142 and 143: 134 Crab flux Figure 4: Gamma-ray f
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136
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Helium Absorption and Cosmic Reioni
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Non-Standard Big Bang Nucleosynthes
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) Non-standard Neutrino Properties
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some heating of the plasma. In cont
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Big Bang Nucleosynthesis With Small
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Neutrinos and Structure Formation i
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Photon and Neutrino Backgrounds The
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The (photon) temperature at aeq is
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spectrum has therefore changed to 1
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Figure 4: Growth of structure in CD
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Future Prospects
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162 Figure 1: Calorimetric β − s
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164 References [1] F. Gatti: Procee
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166 The frequency of Galactic super
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168 Table 1: Comparison of proposed
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170 Neutrinos in Astrophysics José
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172 N eq 7 6.5 6 5.5 5 4.5 4 3.5 3
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174 Figure 4: SN 1987A bounds on FC
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176 [5] A. Joshipura and J. W. F. V
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178 Some Neutrino Events of the 21s
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180 2099: Relic Neutrinos And last
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182
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184 Tuesday, Oct. 21: Supernova Neu
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186 Michael Altmann Technische Univ
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188 Paolo Gondolo Max-Planck-Instit
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190 Patrizia Meunier Max-Planck-Ins
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192 Torsten Soldner Technische Univ
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194 A Experimental Particle Physics
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196 C Astrophysics and Cosmology C1
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