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Neutrino-Cooled Accretion Models for Gamma-Ray Bursts

Workshop on AGN and BH, TIARA, Hsinchu. Neutrino-Cooled Accretion Models for Gamma-Ray Bursts. Tong Liu, Wei-Min Gu, Li Xue, & Ju-Fu Lu Institute of Theoretical Physics and Astrophysics, Xiamen University. Workshop on AGN and BH, TIARA, Hsinchu. Introduction

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Neutrino-Cooled Accretion Models for Gamma-Ray Bursts

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  1. Workshop on AGN and BH, TIARA, Hsinchu Neutrino-Cooled Accretion Models for Gamma-Ray Bursts Tong Liu, Wei-Min Gu, Li Xue, & Ju-Fu Lu Institute of Theoretical Physics and Astrophysics, Xiamen University

  2. Workshop on AGN and BH, TIARA, Hsinchu • Introduction • Physics of Neutrino-Cooled Accretion Disks • 2.1 Hydrodynamics • 2.2 Thermodynamics • 2.3 Microphysics • 2.3.1 Neutrino Optical Depth • 2.3.2 Electron Fraction • 2.3.3 Electron Chemical Potential • Numerical Results for the Disk Structure • Neutrino Radiation and Annihilation Luminosities ITPA,Xiamen University

  3. Gamma-ray bursts (GRBs): ~ 1050 – 1052 ergs (depending on whether emission is isotropic or beamed) Theoretical models are in two categories: • Fireballs: relativistic outflows, shocks, gamma rays, afterglows • Central engine of fireballs Most popular models in the latter category are in common invoking a stellar-mass black hole accreting at a hypercritical rate, of the order of 1M⊙ s –1 ITPA,Xiamen University Introduction Workshop on AGN and BH, TIARA, Hsinchu

  4. Introduction Workshop on AGN and BH, TIARA, Hsinchu Main problem: How to convert the gravitational energy released by the accreted matter into a relativistic outflow? Two mechanisms have been proposed: ▪ Neutrino emission and annihilation (in inner region of a hyperaccretion diskρ~ 1010g cm-3, T ~ 1010K, photons are trapped) ▪ Energy extraction from disk and/or black hole via MHD processes The former mechanism is easier to understand and can be calculated more accurately ITPA,Xiamen University

  5. Introduction Workshop on AGN and BH, TIARA, Hsinchu Key question from observational point of view: Whether neutrino annihilation can provide sufficient energy for GRBs? • Popham et al. 1999: Yes, but assumed neutrinos transparent • Di Matteo et al. 2002: No, considered neutrino opacity, but used Newtonian potential • Gu et al. 2006: Yes when general relativistic effects are considered and contribution from optically thick region is included ITPA,Xiamen University

  6. Introduction Workshop on AGN and BH, TIARA, Hsinchu Other factors may influence structure and luminosity of a neutrino-cooled disk: • Kohri & Mineshige 2002: Electron degeneracy • Kohri et al. 2005: Electron fraction Ye=np/(np+nn)<0.5 Both are likely to suppress neutrino emission Full knowledge of disk structure is required to evaluate neutrino luminosity ITPA,Xiamen University

  7. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu • Hydrodynamics: similar to that of normal accretion disksinX-ray binaries Paczynski-Wiita Potential: where M is the black hole mass, R is the radius, and is the Schwarzschild radius. ITPA,Xiamen University

  8. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu Accretion in the disk is driven by viscous stress, and the kinematic viscosity coefficient is expressed as , where H is the half thickness of the disk; is the isothermal sound speed, with P and being the pressure and mass density, respectively; and is a dimensionless constant parameter that absorbs all the detailed microphysics of viscous processes. ITPA,Xiamen University

  9. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The angular velocity is approximately Keplerian,i.e., The disk is in the vertical hydrostatic equilibrium, and this gives . With these simplifications the problem is reduced to be one-dimensional, i.e., all physical quantities depend on R only. ITPA,Xiamen University

  10. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The constant mass accretion rate is expressed from the continuity equation as where is the radial velocity that can be read from the angular momentum equation as where j is an integration constant determined by the zero- torque boundary condition at the last stable orbit, and it represents the specific angular momentum (per unit mass) of the matter accreted into the black hole. ITPA,Xiamen University

  11. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu • Thermodynamics The energy equation is generally written as the balance between the viscous heating and the cooling rates, the viscous heating rate Qvis is similar to that of normal accretion disks, but the cooling rate Q - is crucially different, it has three contributions: ITPA,Xiamen University

  12. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The cooling rate by photodisintegration of -particles Qphotodis is where Xnuc is the mass fraction of free nucleons (e.g., Kohri et al. 2005). The advective cooling rate Qadv is where s is the specific entropy, is taken to be equal to 1,i.e., ds/dR is approximated as s/R (Kohri & Mineshige 2002),and mu is the mean mass of a nucleon. The entropy of degenerate particles is small and can be neglected. ITPA,Xiamen University

  13. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu is the energy density of neutrinos, for which we adopt a bridging formula valid in both the optically thin and thick regimes (Popham & Narayan 1995; Di Matteo et al. 2002), where is the total optical depth for neutrinos, is the absorption optical depth for neutrinos, and the subscript i runs for the three species of neutrinos. The cooling rate due to neutrino loss Qνis expressed as (Kohri et al. 2005), ITPA,Xiamen University

  14. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The equation of state, The gas pressure from free nucleons and -particles Pgas is The photon radiation pressure Prad is ITPA,Xiamen University

  15. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The electron pressure Pe The neutrino pressure Pν is ITPA,Xiamen University

  16. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu • Microphysics The total optical depth for neutrinos is The optical depth for neutrinos through scattering off free nucleons, -particles, and electrons is given by ITPA,Xiamen University

  17. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu where is the mean free path, and are the cross sections of scattering on protons, neutrons, -particles, and electrons; and are the number densities of free protons, free neutrons, -particles, electrons, and positrons, respectively. The four cross sections are given by (Burrows & Thompson 2002): ITPA,Xiamen University

  18. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  19. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu where , is the mean energy of neutrinos in units of (mec2), gA ≈ 1.26, ITPA,Xiamen University

  20. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu Since and , the free proton fraction Yp, the free neutron fraction Yn, and the -particle fraction Yα are related to Ye and Xnuc as ITPA,Xiamen University

  21. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu and are given by the Fermi-Dirac integration, ITPA,Xiamen University

  22. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The absorption depth for neutrinos is defined by where is the total neutrino cooling rate (per unit volume) and is the sum of four terms, ITPA,Xiamen University

  23. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The neutrino cooling rate due to the URCA processes qURCA relates only to ve and is represented by the sum of three terms ITPA,Xiamen University

  24. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu where , , Q = (mn – mp)c2 ,and is the Fermi-Dirac function. ITPA,Xiamen University

  25. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The electron-positron pair annihilation rate into neutrinos is ITPA,Xiamen University

  26. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The nucleon-nucleon bremsstrahlung rate qbrem through the processes is the same for the three species of neutrinos, ITPA,Xiamen University

  27. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu As to the plasmon decay rate qplasmon, only that through the process needs to be considered, where plasmons are photons interacting with electrons, (Ruffert et al. 1996). ITPA,Xiamen University

  28. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu • Electron Fraction By assuming that the chemical potential of neutrinos can be ignored and that electrons and positrons have equal chemical potentials, the β-equilibrium condition is if the disk material is opaque to neutrinos, where and are chemical potentials of neutrons and protons, respectively. ITPA,Xiamen University

  29. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu On the other hand,if the material is transparent to neutrinos, Yuan (2005) showed that the β-equilibrium condition reads ITPA,Xiamen University

  30. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu The neutron-to-proton ratio in β-equilibrium is given by which results in for the neutrino opaque limit, and for the neutrino transparent limit. ITPA,Xiamen University

  31. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu In order to allow for a transition from the optically thin to optically thick regime, we adopt a treatment similar to that in Lee et al. (2005), i.e. introducing a weight factor and writing in a combined form, ITPA,Xiamen University

  32. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu We finally arrive at Another equation is from nuclear statistical equilibrium (Chen & Beloborodov 2006) ITPA,Xiamen University

  33. Physics of neutrino-cooled accretion disks Workshop on AGN and BH, TIARA, Hsinchu • Electron Chemical Potential The electron chemical potential is determined by the condition of charge neutrality among protons ,electrons , and positrons, ITPA,Xiamen University

  34. Summary of Equations Workshop on AGN and BH, TIARA, Hsinchu with given ρ and T (from hydrodynamics and thermodynamics) Equations relating Ye, Xnuc, and ηe: • β-equilibrium • Nuclear statistical equilibrium • Charge neutrality

  35. Workshop on AGN and BH, TIARA, Hsinchu Numerical Results for the Disk Structure ITPA,Xiamen University

  36. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  37. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  38. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  39. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu Our bridging formula for Ye (solid line) Globally opaque material (dashed line) Globally transparent material (dotted line) ITPA,Xiamen University

  40. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  41. Numerical Results for the Disk Structure Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  42. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu Having the neutrino cooling rate , the neutrino radiation luminosity (before annihilation) Lν is obtained as ITPA,Xiamen University

  43. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu For the calculation of the neutrino annihilation luminosity we follow the approach in Ruffert et al. (1997), Popham et al. (1999), and Rosswog et al. (2003). The disk is modeled as a grid of cells in the equatorial plane. A cell k has its mean neutrino energy , neutrino radiation luminosity , and distance to a space point above(or below) the disk dk. The angle at which neutrinos from cell k encounter antineutrinos from another cell k’ at that point is denoted as θkk’. Then the neutrino annihilation luminosity at that point is given by the summation over all pairs of cells, ITPA,Xiamen University

  44. Workshop on AGN and BH, TIARA, Hsinchu Neutrino Radiation and Annihilation Luminosities ITPA,Xiamen University

  45. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu where ITPA,Xiamen University

  46. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu The total neutrino annihilation luminosity is obtained by the integration over the whole space outside the black hole and the disk, ITPA,Xiamen University

  47. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  48. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  49. Neutrino Radiation and Annihilation Luminosities Workshop on AGN and BH, TIARA, Hsinchu ITPA,Xiamen University

  50. Summary of Results Workshop on AGN and BH, TIARA, Hsinchu 1. Electron degeneracy is moderate 2. Electron fraction drops below 0.1 in inner region 3. Neutrino opacity: Electron neutrinos dominate over other two types Absorption dominates over scattering 4. Inner region neutrino-cooling dominates 5. Neutrino annihilation luminosity is considerably reduced by electron degeneracy and neutronization, but is still adequate for GRBs, and is likely to be anisotropic

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