Lecture 4
Thomas Tauris AIfA Bonn Uni. / MPIfR Bonn, Summer 2014 1: Introduction Degenerate Fermi Gases Non-relativistic and extreme relativistic electron / (n,p,e-) gases 2: White Dwarfs Structure, cooling models, observations 3: Neutron Stars Structure and Equation-of-state Radio Pulsars Characteristics, observations, spin evolution, magnetars 4: Binary Evolution and Interactions Accretion, X-ray Binaries, formation of millisecond pulsars Black Holes Observations, characteristics and spins 5: Testing Theories of Gravity Using Pulsars Gravitational Waves Sources and detection Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 2 Structure of WDs EoS below neutron drip Basic characteristics Stability Super-Chandrasekhar mass WDs Chandrasekhar mass limit Neutron-rich nuclei Neutron drip Semi-empirical mass formula Including shell effects and lattice energy Harrison-Wheeler EoS Baym-Pethick-Sutherland (BPS) EoS Observations Bonn, Summer 2014 Surface layers Photon diffusion equation L 4 r 2 Temperature gradient Pressure gradient (via hydrostatic equilibrium) Core-surface boundary conditions Luminosity as a function of (M, T) Residual ion thermal energy Cooling age Crystallization d aT 4 3 dr Elementary treatment of WD cooling c Rapid cooling Observational support of WD cooling models Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 4 Structure of WDs Surface layers: (H) He non-degenerate layers in ”radiative equilibrium” LTE with outward energy flux by diffusion of photons (only small gradient in net flux, I: Planck function) Interior: CO, ONeMg (He WDs in close binaries) electrons are completely degenerate electrons have a large mean free path, e because of the filled Fermi sea thermal conductivity, th is large temperature, T is uniform (isothermal core) Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 5 Photon diffusion equation: L 4 r 2 c d aT 4 3 dr Temperature gradient Pressure gradient (via hydrostatic equilibrium) Core-surface boundary conditions Luminosity as a function of (M, T) c d dT aT 4 .... 3 dr dr dP m( r ) g G 2 hydrostatic equil. dr r L 4 r 2 b-f photoionization of atoms f-f inverse bremstrahlung of e- dP .... dT Integrate… 0 T 3.5 P M 17/4 (1) P T L Bonn, Summer 2014 mu Kramer's opacity x/ , I ( x) I 0 e 1 mean free path kT ideal gas Boundary conditions: P 0, T 0, m(r ) M (thin envelope) Thomas Tauris - Bonn Uni. / MPIfR 6 Core boundary condition (transition from surface layers to core region) Pgas Pdeg Pgas M 17/4 T K 5/3 L (2) L M T3.5 Typically: kT =K5/3 5/3 T5/2 mu ( L C M T3.5 ) T 106 107 K L 105 102 L core temperature Note 103 gcm3 c rsurface layers Bonn, Summer 2014 RWD Thus the assumption of a fully degenerate star (cold EoS) is valid! (M,R)-relations obtained earlier are ok! Thomas Tauris - Bonn Uni. / MPIfR 7 Loss of residual thermal energy of ions radiation thermal Eion radiation ( Eethermal gas ) cannot be deliberated because of the filled Fermi-sea neutrino emission is only important very early when T 108 K 3 2 (erg/K) for a monatomic gas (e.g. C -ions) Specific heat capacity per ion: cv k Total thermal energy: U cv T Nions L dU dt Integrate… (T0 T) C M T 3.5 M L 3 M kT 2 Amu 10 48 erg , T 107 K d 3 M kT dt 2 Amu 5/7 cooling age! Problem: estimated WD cooling ages were too large by a factor 10 WD Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR cluster 8 Crystallization of ion lattice Formation of lattice: (ri )2 ri 2 1 16 Ecoulomb Z 2 e2 / ri 171 Ethermal kT Lindemann’s empirical rule cv lattice 3k T cv (lattice vibrations) dominate over cv (free thermal motion) cv = 3k (including Epot,lattice ½ k per mode) ideal gas 3/2 k Quantum-mechanical effects D Bonn, Summer 2014 Tm T D Tg cv T Thomas Tauris - Bonn Uni. / MPIfR rapid cooling observational consequences 9 diamond WD! ”Lucy” Beatles song: ”Lucy in the Sky with Diamonds” Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 10 Althaus et al. (2007), A&A 465, 249 Massive 1.06-1.28 Msun ONeMg WDs with helium/hydrogen envelopes Hansen et al. (2007), ApJ 671, 380 The WD cooling sequence of NGC 639 Gilles Fontaine (2000) Evidence for rapid cooling… Surface layers Photon diffusion equation L 4 r 2 Temperature gradient Pressure gradient (via hydrostatic equilibrium) Core-surface boundary conditions Luminosity as a function of (M, T) Residual ion thermal energy Cooling age Crystallization d aT 4 3 dr Elementary treatment of WD cooling c Rapid cooling Observational support of WD cooling models Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 13 1: Introduction Degenerate Fermi Gases Non-relativistic and extreme relativistic electron / (n,p,e-) gases 2: White Dwarfs Structure, cooling models, observations 3: Neutron Stars Structure and Equation-of-state Radio Pulsars Characteristics, observations, spin evolution, magnetars 4: Binary Evolution and Interactions Accretion, X-ray Binaries, formation of millisecond pulsars Black Holes Observations, characteristics and spins 5: Testing Theories of Gravity Using Pulsars Gravitational Waves Sources and detection Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 14 Shapiro & Teukolsky (1983), Wiley-Interscience Curriculum - Chapter 4: p.82-87, (91-92), 100-105 Exercises: #23, 24 - Wed. May 14 15:00-16:30, AIfA raum 0.006 !! Bonn, Summer 2014 Thomas Tauris - Bonn Uni. / MPIfR 15
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