Physics of Neutron Star Crusts Nicolas Chamel Institut d’Astronomie et d’Astrophysique Universit´eLibre de Bruxelles CP226 Boulevard du Triomphe B-1050 Brussels, Belgium email: [email protected] http://www-astro.ulb.ac.be/ Pawel Haensel Nicolaus Copernicus Astronomical Center Polish Academy of Sciences Bartycka 18, 00-716 Warszawa, Poland email: [email protected] http://www.camk.edu.pl Abstract The physics of neutron star crusts is vast, involving many different research fields, from nu- clear and condensed matter physics to general relativity. This review summarizes the progress, which has been achieved over the last few years, in modeling neutron star crusts, both at the microscopic and macroscopic levels. The confrontation of these theoretical models with obser- vations is also briefly discussed. arXiv:0812.3955v1 [astro-ph] 20 Dec 2008 1 Contents 1 Introduction 5 2 Plasma Parameters 9 2.1 Nomagneticfield.................................. ... 9 2.2 Effectsofmagneticfields. ...... 12 3 The Ground State Structure of Neutron Star Crusts 14 3.1 Structureoftheoutercrust . ....... 14 3.2 Structureoftheinnercrust . ....... 17 3.2.1 Liquiddropmodels.............................. .. 18 3.2.2 Semi-classicalmodels. ..... 21 3.2.3 Quantumcalculations . ... 24 3.2.4 Goingfurther: nuclearbandtheory. ....... 29 3.3 Pastas.......................................... 32 3.4 Impuritiesanddefects .. .. .. .. .. .. .. .. .. .. .. .. ...... 35 4 Accreting Neutron Star Crusts 36 4.1 Accreting neutron stars in low-mass X-ray binaries . .............. 36 4.2 Nuclear processes and formation of accreted crusts . .............. 37 4.3 Deepcrustalheating .............................. ..... 40 4.4 Thermal structure of accreted crusts and X-ray bursts . .............. 44 5 Equation of State 46 5.1 Groundstatecrust ................................ .... 46 5.2 Accretedcrust ................................... ... 50 5.3 EffectofmagneticfieldsontheEoS . ...... 51 5.4 Supernovacoreatsubnucleardensity . .......... 51 6 Crust in Global Neutron Star Structure 54 6.1 Sphericalnonrotatingneutronstars. ........... 55 6.2 Approximateformulae .. .. .. .. .. .. .. .. .. .. .. .. ..... 56 6.3 Crustinrotatingneutronstars . ........ 58 6.4 Effects of magnetic fields on the crust structure . ............ 61 7 Elastic Properties 63 7.1 Isotropicsolid(polycrystal) . .......... 64 7.2 Nuclearpasta.................................... ... 66 8 Superfluidity and Superconductivity 68 8.1 Superconductivity in neutron star crusts . ............ 68 8.2 Static properties of neutron superfluidity . ............. 69 8.2.1 Neutron pairing gap in uniform neutron matter at zero temperature . 70 8.2.2 Critical temperature for neutron superfluidity . ............ 73 8.2.3 Pairinggapinneutronstarcrusts . ...... 74 8.3 Superfluidhydrodynamics . ...... 77 8.3.1 Superflowandcriticalvelocity . ...... 77 8.3.2 Rotatingsuperfluidandvortices . ...... 80 8.3.3 Type II superconductors and magnetic flux tubes . ......... 83 8.3.4 Superfluid vortices and magnetic flux tubes in neutron stars......... 83 2 8.3.5 Dynamicsofsuperfluidvortices . ...... 84 8.3.6 Superfluid hydrodynamics and entrainment . ........ 86 8.3.7 Entrainmenteffectsinneutronstars . ....... 88 9 Conductivity and Viscosity 89 9.1 Introduction.................................... .... 89 9.2 Boltzmann equation for electrons and its solutions . .............. 89 9.3 Thermaland electricalconductivities . ............ 92 9.4 Viscosity....................................... ... 94 9.5 Transport in the presence of strong magnetic fields . ............. 97 9.5.1 Nonquantizing magnetic fields . ...... 99 9.5.2 Weakly-quantizing magnetic fields . ....... 99 9.5.3 Strongly-quantizing magnetic fields . ......... 99 9.5.4 Possible dominance of ion conduction . ....... 99 10 Macroscopic Model of Neutron Star Crusts 100 10.1 Variational formulation of multi-fluid hydrodynamics ................. 100 10.2 Two-fluidmodelofneutronstarcrust . ......... 101 10.3 Entrainmentandeffectivemasses . ......... 103 10.4 Neutronsuperfluidity. ....... 104 11 Neutrino Emission 107 11.1 Neutrino emission processes – an overview . ............ 107 11.2 Electron-positronpairannihilation . ............. 108 11.3Plasmondecay ................................... ... 109 11.4 Photoneutrinoemission . ....... 109 11.5 Neutrino Bremsstrahlung from electron-nucleus collisions .............. 109 11.6 Cooperpairingofneutrons . ....... 110 11.7 Synchrotronradiationfromelectrons . ............ 111 11.8 Other neutrino emission mechanisms . .......... 111 11.8.1 Direct Urca process in the pasta phase of the crust . ........... 111 11.9 Neutrinos from the crust – summary in the T ρ plane ............... 112 − 12 Observational Constraints on Neutron Star Crusts 117 12.1 Supernovae and the physics of hot dense inhomogeneous matter........... 117 12.2 Coolingofisolatedneutronstars . .......... 120 12.2.1 Thermalrelaxationofthecrust . ....... 121 12.2.2 Observational constraints from thermal X-ray emission............ 122 12.3 r-process in the decompression of cold neutron star crusts .............. 124 12.4Pulsarglitches ................................. ..... 126 12.4.1 Starquakemodel ............................... 128 12.4.2 Two-componentmodels . ... 129 12.4.3 Recenttheoreticaldevelopments . ........ 130 12.4.4 Pulsar glitch constraints on neutron star structure .............. 131 12.5 Gravitationalwaveasteroseismology . ............ 133 12.5.1 Mountainsonneutronstars . ..... 133 12.5.2 Oscillationsandprecession . ....... 135 12.5.3 Crust-core boundary and r-mode instability . ........... 136 12.6 GiantflaresfromSoftGammaRepeaters . ........ 136 12.7 LowmassX-raybinaries. ...... 140 12.7.1 Burstoscillations. ..... 140 3 12.7.2 SoftX-raytransientsinquiescence . ......... 141 12.7.3 Initial cooling in quasi-persistent SXTs . ........... 142 13 Conclusion 147 14 Acknowledgments 148 A List of Notations 149 B List of Abbreviations 151 4 1 Introduction Constructing models of neutron stars requires knowledge of the physics of matter with a density significantly exceeding the density of atomic nuclei. The simplest picture of the atomic nucleus is a drop of highly incompressible nuclear matter. Analysis of nuclear masses tells us that nuclear matter at saturation (i.e. at the minimum of the energy per nucleon) has the density ρ0 = 2.8 14 −3 × 10 g cm , often called normal nuclear density. It corresponds to n0 =0.16 nucleons per fermi cubed. The density in the cores of massive neutron stars is expected to be as large as 5–10ρ0 and in spite of decades of observations of neutron stars and intense theoretical studies, the∼ structure of the matter in neutron star cores and in particular its equation of state remain the well-kept secret of neutron stars (for a recent review, see the book by Haensel, Potekhin and Yakovlev [183]). The physics of matter with ρ 5–10ρ0 is a huge challenge to theorists, with observations of neutron stars being crucial for selecting∼ a correct dense-matter model. Up to now, progress has been slow and based overwhelmingly on scant observation [183]. The outer layer of neutron stars with density ρ<ρ0 – the neutron star crust – which is the subject of the present review, represents very different theoretical challenges and observational opportunities. The elementary constituents of the matter are neutrons, protons, and electrons – like in the atomic matter around us. The density is “subnuclear”, so that the methods devel- oped and successfully applied in the last decades to terrestrial nuclear physics can be applied to neutron star crusts. Of course, the physical conditions are extreme and far from terrestrial ones. The compression of matter by gravity crushes atoms and forces, through electron captures, the neutronization of the matter. This effect of huge pressure was already predicted in the 1930s (Sterne [390], Hund [201, 202]). At densities ρ & 4 1011 g cm−3 a fraction of the neutrons is unbound and forms a gas around the nuclei. For a density× approaching 1014 g cm−3, some 90% of nucleons are neutrons while nuclei are represented by proton clusters with a small neutron fraction. How far we are taken from terrestrial nuclei with a moderate neutron excess! Finally, somewhat above 1014 g cm−3 nuclei can no longer exist – they coalesce into a uniform plasma of nearly-pure neutron matter, with a few percent admixture of protons and electrons: we reach the bottom of the neutron star crust. The crust contains only a small percentage of a neutron star’s mass, but it is crucial for many astrophysical phenomena involving neutron stars. It contains matter at subnuclear density, and therefore there is no excuse for the theoretical physicists, at least in principle: the interactions are known, and many-body theory techniques are available. Neutron star crusts are wonderful cosmic laboratories in which the full power of theoretical physics can be demonstrated and hopefully confronted with neutron star observations. To construct neutron star crust models we have to employ atomic and plasma physics, as well as the theory of condensed matter, the physics of matter in strong magnetic fields, the theory of nuclear structure, nuclear reactions, the nuclear many-body problem, superfluidity, physical kinetics, hydrodynamics, the physics of liquid crystals, and the theory of elasticity. Theories have to be applied under extreme physical conditions, very far from the domains where they were originally developed and tested. Therefore, caution is a must! Most of this review is devoted to theoretical descriptions of various