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20 the opacity has a very slowly changing gradient too high values for the damping parameter lead to an interpolated run of opacity with T and/or R which shows step-like gradient variations. This is illustrated in Fig. 1 where the full line shows the interpolated κ(R) using a high damping parameter and the dashed line the run of κ without damping. In this case a simple cubic spline seems to describe a better fit. The greatest deviations of κ between two tabulated values is only about 3%, which is smaller than the quoted uncertainty for the opacities of about 10%. We would like to note here that neither of the chosen interpolation schemes can claim to reproduce the true values, as interpolation is always an estimation. To illustrate the influence of the opacity interpolation, two solar models were calculated, GARSOM3 with no and GARSOM2 with strong spline damping. The run of sound speed of both models is compared with a seismic mode from [8] in Fig. 2. The deviation of GARSOM2 (dashed line) from the seismic model just below the convective zone is about twice as large as compared with GARSOM3 (full line). At r = 0.67 R⊙ the temperature is approximately 2 million Kelvin, log10R ≈ −1.7. Regarding Fig. 1 one notices that at this radius the model is in T-R-regions of the opacity tables where the different interpolations schemes differ most. Figure 2: The run of quadratic sound speed of various models compared to seismic model from [7]. Full line: GARSOM2, dashed line: GARSOM3, dotted line: reference model from which the seismic model is inferred. As the possible error in the tabulated opacities (10%) is bigger than the difference due to interpolation (3%), neither of the models can be favoured from the theoretical point of view. Although we must admit that κ(R,T) inferred from cubic splines without damping does look more reliable, we can not really rule out GARSOM2. It is therefore necessary to improve the grid density in the tables and the input physics in the opacities. To show that GARSOM3 is compatible with solar models from other groups, we have plotted in Fig. 2 also the run of sound speed of the reference model which was used to infer the seismic model of [8]. The remaining discrepancies between the reference model and GARSOM3 may be due to slightly different ages or nuclear reaction rates.
Acknowledgments We would like to thank Jørgen Christensen-Dalsgaard who helped to detect the sensitivity of opacity to interpolation. Additionally we acknowledge Sarbani Basu who provided us her seismic model. References [1] H. Schlattl, A. Weiss and H.-G. Ludwig, A&A 322 (1997) 646. [2] F.J. Rogers, F.J. Swenson and C.A. Iglesias, ApJ 456 (1996) 902. [3] C.A. Iglesias and F.J. Rogers, ApJ 464 (1996) 943. [4] D.R. Alexander and J.W. Fergusson, ApJ 437 (1994) 879. [5] A.A. Thoul, J.N. Bahcall and A. Loeb, ApJ 421 (1994) 828. [6] V. Castellani et al., Phys.Rev.D 50 (1994) 4749. [7] H. Späth, Spline-Algorithmen zur Konstruktion glatter Kurven und Flächen, Oldenbourg, München, 1973. [8] S. Basu, Mon. Not. R. Astr. Soc. (1997) in press. 21
Page 1 and 2: arXiv:astro-ph/9801320v1 30 Jan 199
Page 3: Preface This was the fourth worksho
Page 6 and 7: vi S.J. Hardy Quasilinear Diffusion
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Page 11 and 12: History of Neutrino Physics: Pauli
Page 13 and 14: Brief an Oskar Klein, Stockholm, vo
Page 15 and 16: Recent observations with the Hubble
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Page 19: Solar Neutrinos
Page 22 and 23: 14 In the depth range where an abun
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Page 26 and 27: 18 [3] J.N. Bahcall and H.A. Bethe,
Page 30 and 31: 22 Status of the Radiochemical Gall
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Page 40 and 41: 32 Table 1: Neutrino event rates in
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Figure 3: Energy spectra of the sup
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Supernova Neutrino Opacities G.G. R
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Quasilinear Diffusion of Neutrinos
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Assuming a saturated thermal distri
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Anisotropic Neutrino Propagation in
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So far the damping rate corresponds
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Cherenkov Radiation by Massless Neu
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on ω so that they are well approxi
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2 2 / meff,e m eff 1.0 0.5 0.0 elec
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Gamma-Ray Burst Observations D.H. H
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Figure 1: The cumulative distributi
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[7] Lilly, S., in Critical Dialogue
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) Phase Transitions in Neutron Star
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after the collapse began (nb., no e
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Models of Coalescing Neutron Stars
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Figure 2: Contour plots of the mass
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From our torus models we find that
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High-Energy Neutrinos
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110 in gamma-rays. However, the acc
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112 Atmospheric Muons and Neutrinos
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114 Today, however, charm productio
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116 Atmospheric Neutrinos in Super-
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118 number of events 300 200 100 (a
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120 Acknowledgements D.K. is suppor
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122 Key signatures for νµ nucleon
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124 (a.u) 0.5 0.4 0.3 0.2 0.1 0 0 1
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126 However, in order to operate th
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128 could be gravitationally captur
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130 Ground-Based Observation of Gam
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132 Figure 2: Sensitivities in unit
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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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