Les ´ etoiles ` a neutrons, du mod` ele aux observations Luc Di Gallo S´ eminaires de physique, LAPP, Vendredi 8 juin 2012
Introduction Observables Models The inner crust Excitation spectrum 1920 Theoretical prediction of the 1967 First observation of a pulsar neutron by Rutherford by Hewish and Bell 1931 Some discussions about a ”neutron core” by Landau or Langer and Rosen 1932 Discovery of the neutron by Chadwick 1934 Theoretical prediction of neutron stars by Baade and Zwicky 1939 First realistic model of neutron stars, Tolman, Figure: Chadwick, Baade and Zwicky Oppenheimer and Volkov 2/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Models The inner crust Excitation spectrum Neutron stars are formed after a gravitational supernova type II. Figure: Crab nebula with a Pulsar in the center. 3/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Models The inner crust Excitation spectrum Neutron star characteristics are: A radius: R ≃ 10 − 15 km A mass: M ≃ 1 − 3 M ⊙ Compacity: Ξ = GM Rc 2 ≃ 0 . 2 Average density: ρ ≃ 2 . 10 14 g.cm − 3 Temperature: T ≃ 10 6 − 10 10 K Period of rotation: P ≃ 0 . 001 − 10 s Magnetic field: B ≃ 10 7 − 10 15 G = ⇒ Test for fundamental physics Figure: Neutron star structure, D. Page 4/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Radio observations Models X observations The inner crust Excitation spectrum 5/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Radio observations Models X observations The inner crust Excitation spectrum 6/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Radio observations Models X observations The inner crust Excitation spectrum Figure: Thermonuclear bursts observation at the surface of a neutron star, Guver et al. 2008 Figure: Probability for Mass-Radius, 7/30 Steiner et al. 2010 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Radio observations Models X observations The inner crust Excitation spectrum 0.08 normalized counts s −1 keV −1 0.06 0.04 0.02 0 1.2 1.1 ratio 1 0.9 0.5 1 2 5 Energy (keV) Figure: Chandra observations of Cassiopeia A Neutron Star between 2000 Figure: Temperature evolution, Heinke et and 2009, Heinke et al. 2010 al. 2010 OBSERVATIONS ⇆ MODELS 8/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Nuclear matter is largely unknown No QCD for nuclear matter Phenomenological interaction 3 )( N − Z ) 2 Z 2 2 3 + ( a I + a IS A − 1 B ( A , Z ) = a V A + a s A + a c 3 − δ p + E D (1) 1 A A Many body problem? Large number of neutrons? Composition of nuclear matter at very high density? Bulk and shear viscosities for description of macroscopic dynamics? → Hundreds Equation of States for nuclear matter 9/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Mass-Radius diagrams obtained with TOV equations are signature of Equation of States. Figure: EOS and corresponding Mass-Radius diagrams, Lattimer 2001 10/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Extracting equation of state informations from gravitational waves: t=8.276 ms 1e15 ρ g/cm 3 40 1e14 20 1e13 Y (km) 0 1e12 −20 1e11 −40 1e10 −40 −20 0 20 40 X (km) Figure: Isodensity contours for two Figure: BH-NS merger waveform for to merged NS after 8.276 ms, Rezzola et al. EOS, Lackey et al. 2012 2010 11/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Figure: Vortices in neutron stars, Grill and Pizzochero 2012 Figure: Yuan et al. 2010 → Glitches as a proof of superfluidity in neutron stars 12/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Thermal emission + estimated age = ⇒ constraint on thermal evolution models T = 10 K 9 C T = 5.5x10 K T = 0 C C 8 Figure: Cooling in Cassiopeia A (Page et Figure: Cooling curve and observational al. 2011) data (Gusakov et al. 2004) → Accurate thermal evolution model to interpret these constraints. 13/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Specific heat + thermal conductivity + ν emissivity = ⇒ Thermal identity of the matter Specific heat is a sum over the T=10 9 K different contributions from the Ions 22 Electrons Protons different excitations (nuclei, log10 (C V [erg cm -3 K -1 ]) Non-superfluid neutrons 21 Weakly paired neutrons phonons, electrons,...) Strongly paired neutrons 20 The crust is important for thermal 19 evolution models (Gnedin et al. 18 2001, Brown and Cumming 2009) 17 Nucleonic contribution in the 16 inner crust is strongly suppressed 10 11 12 13 14 15 log10 ( ρ [g cm -3 ]) → Investigation of a new contribution to the specific heat from the collective Figure: Specific heat contribution as a function of the density at T = 10 9 K excitations. (Fortin et al. 2010) 14/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum MOTIVATIONS At low temperature collective modes are present Pairing energy of the order of 10 10 K = ⇒ No excitations coming from pair breaking of nucleons at T < 10 10 K Due to pairing = ⇒ matter is Figure: Collective excitations regime VS superfluid single particle excitation regime function Superfluidity = ⇒ Collective of the temperature (Page and Reddy 2012) excitation at low energy → Hydrodynamic approximation to model collective behaviour of nucleons 15/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Two basic equations to derive non-relativistic hydrodynamics of uncharged superfluids (Prix 2004): Conservation of particle number: ∂ t n a + ∇ . j a = 0 with a = n,p Euler equation: ∂ t P a = ∇ π a with a = n,p − π a = µ a − 1 2 m a v 2 a + v a · p a (2) Characteristics of superfluids come from quantum properties: No viscosity Locally irrotational No entropy transport Entrainment between the two fluids (n,p): non dissipative interaction which misalign velocities and momenta = ⇒ coupling between fluids → Microscopic input from nuclear interaction 16/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Equation of state and Mass-radius diagram Observables Gravitational waves Models Pulsar glitches and superfluidity The inner crust Neutron star cooling Excitation spectrum Few hypothesis: 0.3 Zero temperature u − u + vitesse du son u (c) β equilibrium = ⇒ proportion of u n 0.2 n,p Hydrostatic equilibrium 0.1 Non relativistic hydrodynamics Two linearised equations: 0 Conservation of particle number: 0 0.05 0.1 0.15 ∂ t δ n a + n a ∇ . δ v a = 0 with a = n,p n B (fm -3 ) Euler equation: ∂ t δ P a = −∇ δµ a with a = n,p Figure: Sound velocities as a function of the density ⇒ Two eigenvectors ( U ± ) with = associated sound velocity ( u ± ) → Hydrodynamic modes in the inner crust 17/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
Introduction Observables Inner crust structures Models Characteristics of the model The inner crust Hydrodynamic mode propagation Excitation spectrum The inner crust structures: Figure: Neutron star inner crust, Newton et al 2011 Inner crust = transition from homogeneous matter to a lattice of atomic nuclei Inner crust = lattice of nuclei immersed in a neutron fluid Pasta phase = very deformed nuclei 18/30 Luc Di Gallo Les ´ etoiles ` a neutrons, du mod` ele aux observations
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