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Neutron stars: the densest state of condensed matter Nicolas Chamel Institut d’Astronomie et d’Astrophysique Université Libre de Bruxelles, Belgique Toulouse, 15 April 2011 What is a neutron star? Astrophysicist’s answer: a neutron star is the compact remnant of a type II supernova explosion. Neutron stars are tiny stars Neutron stars are the smallest and densest stars in the Universe: M ∼ 1 − 2M⊙ R ∼ 10 km ⇒ ρ̄ ∼ 1014 − 1015 g.cm−3 RCW 103 (from ESA) What is a neutron star? Nuclear physicist’s answer: a neutron star is a giant nucleus containing A ∼ 1057 nucleons (90% being neutrons). Neutron stars as nuclear liquid drops Treating a neutron star as a drop of incompressible nuclear matter at saturation density ρ0 ≃ 2.8 × 1014 g.cm−3 M ∼ Am ∼ 1 − 2M⊙ R ∼ r0 A1/3 ∼ 10 km What is a neutron star? Condensed-matter physicist’s answer: a neutron star is the largest quantum system in the Universe! The interior of neutron stars is cold and strongly degenerate. Condensed-matter issues structure and composition of dense matter phase transitions (crystallization, frustration) superfluidity, superconductivity strong magnetic fields Cassiopeia A (from NASA) Internal constitution of neutron stars From the surface of a neutron star to its center the density spans more than 14 orders of magnitude! Chamel&Haensel, Living Reviews in Relativity 11 (2008), 10 http://relativity.livingreviews.org/Articles/lrr-2008-10/ Outline Historical introduction to neutron stars Neutron star magnetic fields Superfluidity and superconductivity in neutron stars Brief history of neutron stars Compact stars in the 1930’s The history of neutron stars began in the 1930s. At this time a few white dwarfs were known. In 1915, Walter Adams spectral observations of Sirius B had revealed that white dwarfs are very compact stars. Sirius B In the 1920s, Arthur Eddington developed the theory of stellar structures. Later astrophysicists started to speculate about the ultimate fate of stars using the newly developed quantum theory. Sir Arthur Eddington Degenerate matter in white dwarfs The central density in white dwarfs was found to be much higher than that in ordinary matter. At such densities atoms are fully ionized and all electrons are free. However electrons are fermions and due to the Pauli exclusion principle (1925), they cannot occupy the same quantum state. Only four months after Dirac published his paper about the statistics of fermions, R.H. Fowler realized that this is the electron degeneracy pressure which resists the gravitational collapse. MNRAS 87, 114 (1926). First ideas about neutron stars In February-March 1931, Landau, Bohr and Rosenfeld discussed the possible existence of compact stars as dense as atomic nuclei. Landau published a paper in January 1932. In February 1932, the neutron (which was predicted by Rutherford in 1920) was discovered by James Chadwick. He was awarded the Nobel prize in 1935. Talk of D. G. Yakovlev from Ioffe Institute in St Perterburg http://www.ift.uni.wroc.pl/~karp44/talks/yakovlev.pdf Baade and Zwicky prediction In December 1933, during a meeting of the American Physical Society at Stanford, Baade and Zwicky predicted the existence of neutron stars as supernova remnants William Baade and Fritz Zwicky Phys. Rev. 45 (1934), 138 Neutron core In 1937, Gamow and Landau proposed independently that a possible stellar energy source could be the accretion of matter onto a dense neutron core. picture from K.S. Thorne George Gamow and Lev Landau But very soon it was shown that stars are powered by thermonuclear reactions (as suggested in the 20s by Eddington and others). The interest in neutron stars declined. Connection between white dwarfs and neutron stars Baade and Zwicky were apparently unaware of the work about the maximum mass of white dwarfs. This is Gamow who first made the connection in 1939 (Phys. Rev.55, 718). At a conference in Paris in 1939, Chandrasekhar also pointed out “If the degenerate core attain sufficiently high densities, the protons and electrons will combine to form neutrons. This would cause a sudden diminution of pressure resulting in the collapse of the star to a neutron core.” A neutron star should thus have a mass close to the Chandrasekhar limit, i.e. M ∼ 1.4M⊙ . Compactness criterion The compactness of an object of mass M and radius R can be estimated from the dimensionless parameter Ξ≡ Earth Sun White dwarf Neutron star stellar black hole 2GM Rc 2 10−10 10−6 −4 10 − 10−3 ∼ 0.2 − 0.4 1 PSR J1846-0258 (Chandra) ⇒ Neutron stars have to be described using Einstein’s theory of General Relativity Global structure of neutron stars In 1939, Richard Tolman, Robert Oppenheimer and his student George Volkoff solved the equations describing static spherical stars in General Relativity Richard Tolman They found Mmax ≃ 0.7M⊙ by considering a non-interacting gas of degenerate neutrons. Since this is smaller than the maximum mass of supernova cores, they concluded that neutron stars could not exist. Robert Oppenheimer and George Volkoff Theoretical developments after the Second World War The first realistic equation of state of dense matter was constructed in the 50s by John Wheeler and his collaborators (in 1939 he elaborated a liquid drop model of fission with Bohr). fission of a liquid drop Wheeler, Ann. Rev.Astr.Astrophys. 4 (1966), 393. It was later found that Mmax ∼ 1 − 2M⊙ : neutron stars could thus be formed as proposed by Baade&Zwicky. Born in supernova explosions, neutron stars were expected to be "hot". X-ray observations In the 60s, theoretical efforts focused on modeling the cooling of neutron stars motivated by the hope of detecting their thermal emission. Cooling calculations predicted T ∼ 106 K for neutron stars ∼ 103 year old (x-ray emission). e.g. Chiu and Salpeter, PRL12(1964),413. X-ray observations in space started in the 60’s with pioneer experiments by Riccardo Giacconi (Nobel 2002). Several sources were discovered but their nature remain elusive. Giacconi with Uhuru satellite, 1970 Compact X-ray sources In 1967, Iosif Shklovsky correctly proposed that Scorpius X-1 (found in 1962) is a neutron star accreting matter from a normal star. But its work attracted little attention among astrophysicists. Iosif Shklovsky By 1968, about 20 compact x-ray sources were known. Early observations of the Crab nebula and first speculations Already in 1942, Baade and Minkowski found that the central region of the Crab nebula (supernova remnant) contains an unusual star. Later a strong linearly polarized radio emission was detected. In 1953, Shklovsky interpreted this as being due to synchrotron radiation by relativistic electrons spiraling along a strong magnetic field. From Carroll and Ostlie Subsequent theoretical efforts were focused on understanding the origin of the energy powering the Crab nebula. Search for a neutron star in the Crab nebula The Crab nebula was observed during a lunar occultation on 7 July 1964. The size of the x-ray source was estimated as 1 light-year∼ 1013 km (size of the nebula 11 ly). This was much larger than the typical size of a neutron star (10-20 km). In 1965 Anthony Hewish and his student found a scintillating radio source and speculated that it "might be the remains of the original star which had exploded". John Wheeler in 1966 and Franco Pacini in 1967 proposed that a rapidly rotating neutron star with a strong dipole magnetic field could power the Crab nebula and could explain Hewish observations. Fortuituous discovery of pulsars In 1965, Jocelyn Bell started a PhD under the supervision of Anthony Hewish at the Cavendish Laboratory in Cambridge. Her research was about scintillation of radio sources. They constructed a 3.7m radiotelescope with a very good temporal resolution. The telescope (which consisted of an array of 2048 dipole antenna) was completed in July 1967. Jocelyn Bell in 1966 Fortuituous discovery of pulsars In August 1967, Jocelyn Bell discovered a pulsating radio source with a period of about 1 second. The source was later found to be extremely regular. In December its period was accurately measured : 1.3373012 seconds. By Februray 1968 when the results were published, three other sources had been found. These new pulsating stars were dubbed "pulsars" by a journalist of the Daily Telegraph. Anthony Hewish was awarded the Nobel Prize in 1974. Unmasking pulsars Pulsars are magnetized rotating neutron stars emitting a highly focused beam of electromagnetic radiation oriented long the magnetic axis. The misalignment between the magnetic axis and the spin axis leads to a lighthouse effect. Crab pulsar A pulsar (PSR B0531+21) was eventually found in the Crab nebula in 1968 by astronomers of Green Bank observatory. Its period is only 33 milliseconds! Vela pulsar In the same year, astronomers from the Sydney University discovered another pulsar in a supernova remnant with a period of 89 ms : the Vela pulsar (PSR B0833-45). The discovery of the Crab and Vela pulsars definitevely established the nature of pulsars and confirmed the predictions of Baade and Zwicky 35 years earlier that neutron stars are the compact remnants of supernova explosions. Pulsar properties Since 1967, ∼ 2000 pulsars have been discovered. http://www.atnf.csiro.au/research/pulsar/psrcat/ Their period P ranges from 1.396 ms for PSR J1748−2446ad up to 8.5 s for PSR J2144−3933. Their period increases gradually with time at a rate given by 10−20 . Ṗ ≡ dP . 10−12 dt Note that for the best atomic clocks Ṗ & 10−16 (this corresponds to a delay of 1 second every 300 millions years). Each pulsar has a specific pulse profile (with different polarization angles). Pulsar glitches Sometimes P may suddenly decrease. The variations are tiny but observable. ∆Ω ∼ 10−9 − 10−5 Ω Example of a glitch in the Crab pulsar Pulsar glitches Sometimes P may suddenly decrease. The variations are tiny but observable. ∆Ω ∼ 10−9 − 10−5 Ω Example of a glitch in the Crab pulsar ⇒ Small pulsar glitches are interpreted as starquakes thus providing a direct proof for the existence of a solid crust Pulsar glitches Sometimes P may suddenly decrease. The variations are tiny but observable. ∆Ω ∼ 10−9 − 10−5 Ω Example of a glitch in the Crab pulsar ⇒ Small pulsar glitches are interpreted as starquakes thus providing a direct proof for the existence of a solid crust ⇒ Large glitches and the observed long relaxation times bring strong evidence of superfluidity inside neutron stars Schematic pulsar emission mechanism From Maxwell-Faraday’s law of induction, the rotating magnetic field induces a huge electric field at the surface of the star. Charged particles are ejected and accelerated to relativistic speeds forming a magnetosphere. Electrons emit very energetic γ rays which are converted into electron-positron pairs, producing more radiations and leading to a cascade of pair-production. Goldreich & Julian, ApJ157(1969),869. From Carroll and Ostlie When particles reach the light cylinder Rc = cP/2π, they leave the star producing a pulsar wind. This can be seen in x-rays. X-ray observations of the Crab pulsar X-ray observations can reveal new phenomena that cannot be seen in radio or in optical range Neutron star magnetic fields Basic pulsar model The standard model consists of a rotating neutron star with a strong dipole magnetic field in vacuum The loss of energy due to electromagnetic dipole radiation (in cgs units) is given by Ė ≡ From Carroll and Ostlie dE 8π 4 B 2 R 6 sin2 θ =− ≤0 dt 3c 3 P 4 B is the field strength at the magnetic pole, R is the radius of the star and P its spin period. Pulsar magnetic field and characteristic age As the neutron star spins down, it loses kinetic energy at a rate given by Ėkin = IΩΩ̇. Assuming that this is entirely due to magnetic dipole radiation, we can infer the magnetic field strength p 6c 3 IP Ṗ 8π 2 B 2 R 6 sin2 θ ⇒ B = P Ṗ = 3c 3 I 2πR 3 sin θ If the initial period at birth is infinitively short and that B and I remain constant, we can further obtain the characteristic age τ of the pulsar Z P PdP = P Ṗ 0 Z τ dt ⇒ τ = 0 P 2Ṗ Pulsar ages For the Crab pulsar, τ ≃ 1.2 × 103 years, in good agreement with the known age of the supernova (1054 AD). The validity of the rotating magnetic dipole model can be better tested by measuring higher order time derivatives of the pulsar angular frequency Ω. The dipole model predicts n = 3. Braking index Ω̈Ω n≡− Ω̇2 PSR B1509−58 Crab Vela 2.84 2.5 1.4 Other spin-down mechanisms magnetospheric processes, pulsar wind, magnetic field decay, gravitational radiation, fallback disk, neutrino emission, electromagnetic radiation from superfluid neutron vortices, interstellar medium, etc. Pulsar magnetic fields Most pulsars have a magnetic field B ∼ 1012 G = 108 T. Seiradakis and Wielebinski, Astron.Astrophys.Rev.12(2004), 239. Discovery of millisecond pulsars The first millisecond pulsar was found in 1982 at Arecibo by Backer’s team. Today ∼ 200 millisecond pulsars are known. The fastest one PSR J1748−2446ad discovered in 2005, has a period of only 1.396 ms! Millisecond pulsars are characterised by 1.4 ≤ P ≤ 30 ms Ṗ ≤ 10−19 Most of them belong to a binary system and are found in globular clusters. Globular cluster Terzan 5 (ESO) Millisecond pulsars in the P − Ṗ diagram In the standard scenario of neutron star formation from supernova explosions, the fastest pulsars are the youngest. Millisecond pulsars (open circles) are thought to be old recycled pulsars that have been spun up by accretion from a companion star. This is supported by the fact that the fastest ones are not associated with any supernova remnant. Cyclotron emission line in Hercules X-1 The discovery of cyclotron line emission in the spectrum of the X-ray pulsar (accreting neutron star) in the binary Hercules X-1 provided the first direct measurement of neutron star magnetic fields. The spectral feature can be identified to relativistic electrons making a transition from the first excited Landau level E1 to the ground state E0 p EN = me c 2 ( 1 + 2(B/Bc )N − 1) Bc ≡ me2 c 3 /e~ ≃ 4.4 × 109 T ⇒ B ≃ 5.3 × 108 G Trümper, ApJ 219(1978), L105 . Cyclotron lines in other neutron stars Cyclotron lines have been detected in ∼ 15 x-ray pulsars ⇒ B ∼ 1 − 5 × 108 T. Makishima et al. ApJ 525(1999), 978. Coburn et al. ApJ 580(2002), 394. Cyclotron harmonics have been reported in a few of them: three in 4U 0115+63 two harmonics in X0331+63 one harmonic in Hercules X-1, Vela X-1, 4U1907+09, A0535+26. Enoto et al., PASP 60 (2008), S57. Evidence for a cyclotron line in the isolated neutron star CCO 1E 1207.4−5209 ⇒ B ∼ 8 × 106 T. Bignami et al., Nature 423 (2003), 725. Suleimanov et al., ApJ 714 (2010), 630. The intriguing case of RX J1856.5-3754 RX J185635−3754 is one of the nearest isolated neutron stars (400 ly). Its age is estimated to be . 106 years. Walter and Lattimer, Astr. J.576 (2002), L145. The intriguing case of RX J1856.5-3754 X-ray observations of RX J1856.5-3754 by Chandra The intriguing case of RX J1856.5-3754 X-ray observations of RX J1856.5-3754 by Chandra Trümper (2005), astro-ph/0502457 The intriguing case of RX J1856.5-3754 X-ray observations of RX J1856.5-3754 by Chandra Trümper (2005), astro-ph/0502457 The best fit is obtained for a pure black body spectrum (no spectral lines!) The intriguing case of RX J1856.5-3754 X-ray observations of RX J1856.5-3754 by Chandra Trümper (2005), astro-ph/0502457 ⇒ condensed magnetic surface? Origin of neutron-star magnetic fields In 1964 (before the discovery of pulsars), Lodewijk Woltjer argued that neutron stars could have very strong magnetic fields. This was also independently shown by Ginzurg. If we assume that the magnetic flux Φ = B4πR 2 is conserved during the gravitational collapse of massive stars eventually giving birth to a neutron star, we have Φi = Φf ⇒ Bf = Bi Ri Rf 2 This implies neutron-star magnetic fields up to Bf ∼ 1012 T. Woltjer, Astrophys. J. 140,1309 (1964). Ginzburg, Sov. Phys.Doklady 9, 329 (1964). Fossile fields The fossile-field scenario is supported by the fact that peculiar main sequence Ap/Bp stars, magnetic white dwarfs and neutron stars have similar magnetic fluxes as pointed by Ruderman in 1972. Ap/Bp WD NS R (in R⊙ ) ∼1 ∼ 10−2 ∼ 10−5 B (in T) ∼3 ∼ 105 ∼ 1011 The initial magnetic field might be have been produced in the convective core of main sequence stars by a dynamo mechanism. Alternatively the field might have been produced in the neutron star itself. Theory of magnetars The interior of a neutron star is an almost perfect electrical conductor. Duncan and Thompson showed that strong magnetic fields ∼ 1012 T can be generated via dynamo effects in hot newly-born neutron stars with initial periods of a few milliseconds. Huge amount of magnetic energy can be occasionally released in crustquakes producing γ-ray bursts. Thompson & Duncan, ApJ 408, 194 (1993). The March 5, 1979 event The theory of magnetars was proposed in 1992 by Robert Duncan, Christopher Thompson and Bohdan Paczynski to explain Soft-Gamma Repeaters (SGR). SGRs are repeated sources of x- and γ-ray bursts. The first such object called SGR 0525−66 was discovered in 1979. A very intense gamma-ray burst was detected on March 5, 1979 by two Soviet satellites Venera 11 and Venera 12. The burst lasted about 3 minutes and showed a periodic modulation of 8 seconds. Mazets et al., Nature 282 (1979), 587. The March 5, 1979 event The source was later found to lie inside a supernova remnant in the Large Magellanic Cloud (N49) thus suggesting that it might be a young isolated neutron star. But it was difficult at that time to explain the origin of the bursts. ROSAT Other burst sources have been found. 9 SGRs (7 confirmed, 2 candidates) are currently known (April 2011). http://www.physics.mcgill.ca/~pulsar/magnetar/main.html Anomalous X-ray pulsars Anomalous X-ray pulsars (AXP) are isolated sources of pulsed x-rays. Their periods range from 2 to 12 s and their spin-down rate Ṗ ∼ 10−11 so that B ∼ 1010 T. Some of them are bursters. SGR and AXP have much in common. Their observed x-ray luminosity is much larger than their kinetic energy loss rate suggesting these objects are powered by magnetic field decay. SGR and AXP are thought to belong to the same class of neutron stars: magnetars. CXO J164710.2-455216 (Chandra) 12 AXPs (9 confirmed, 3 candidates) are currently known. http://www.physics.mcgill.ca/~pulsar/magnetar/main.html Magnetar seismology Quasi Periodic Oscillations (QPO) have been discovered in the x-ray flux of giant flares from SGR 1806−20, SGR 1900+14 and SGR 0526−66. These QPOs coincide reasonably well with seismic crustal modes thought to arise from the release of magnetic stresses. Thompson and Duncan, MNRAS 275, 255 (1995) The huge luminosity variation suggests B & 1011 T at the star surface thus lending support to the magnetar scenario. Vietri et al., ApJ 661, 1089 (2007). Cyclotron lines in SGR and AXP Evidence for proton cyclotron lines have been found in the spectra of a few SGR and AXP during bursts: SGR 1900+14 SGR 1806−20 1E 1048−59 XTE J1810−197 4U 0142+61 Bspec (in T) 2.6 × 1011 ∼ 1011 2.1 × 1011 2 × 1011 4.75 × 1010 Bspin (in T) 7 × 1010 2 × 1010 4.2 × 1010 2.1 × 1010 1.3 × 1010 Mereghetti, Astron. Astrophys. Rev. 15, 225 (2008). The magnetic fields inferred from both spin-down and spectroscopic studies (not only cyclotron lines but also continuum) are consistent with the magnetar scenario: B> me2 c 3 e~ The puzzling case of SGR 0418+5729 SGR 0418+5729, discovered on 5 June 2009, has a period ∼ 9.1 s but a spin-down rate Ṗ < 6 × 10−15 implying B < 7.5 × 108 T like ordinary pulsars! Rea et al., Science 330, 944 (2010). Much stronger internal fields could power magnetar activity whereas the existence of a fallback disk might explain the low external field. Trümper et al., Astron. Astrophys.518, 46 (2010) But recent spectral analysis of the x-ray emission from SGR 0418+5729 suggests that its surface magnetic field B ≃ 1.1 × 1010 T. Güver et al., arXiv:1103.3024. Radio emission in strong magnetic fields Radio emission in pulsars is believed to be produced by the creation of a cascade of electron-positron pairs in the pulsar magnetosphere. In strong magnetic fields B > me2 c 3 /e~, a photon can split into two lower energy photons thus preventing the creation of pairs. Radio emission is therefore expected to be suppressed. However radio emissions have been detected from several high-magnetic field rotation-powered pulsars and even in three “magnetars” (although the radio emission is quite distinct from that of ordinary pulsars) Ng and Kaspi, arXiv:1010.4592 Superconductivity and superfluidity in neutron stars Are electrons in neutron stars superconducting? The surface layers of non-accreting neutron stars are mostly composed of iron which is not superconducting. It was found in 2001 (Shimizu et al., Nature 412, 316) that iron can become superconducting at densities ρ ≃ 8.2 g.cm−3 with Tce ≃ 2 K. But Tce is much smaller than the temperature in neutron stars. Yakovlev et al, proceedings (2007), arXiv:0710.2047 Are electrons in neutron stars superconducting? In the deeper layers of neutron stars at densities ρ & 104 gcm−3 , atoms are fully ionised. rs ≡ d/a0 a0 ≡ ~2 /me e2 d ≡ (3/4πne )1/3 Ceperley and Alder, PRL45(1980) 566. In ordinary metals rs ∼ 2 − 6 In neutron star crust rs ∼ 10−5 − 10−2 Electrons in neutron stars form an almost ideal Fermi gas Are electrons in neutron stars superconducting? The critical temperature of a uniform non-relativistic electron gas is given by (Tpi is the plasma temperature) Tce = Tpi exp −8~vFe /πe2 ⇒ Tce ∝ exp(−ζ(ρ/ρord )1/3 ) with ρord = mu /(4πa03 /3). At densities above ∼ 106 g.cm−3 , electrons become relativistic vFe ∼ c so that (α = e2 /~c ≃ 1/137) Tce = Tpi exp (−8/πα) ∼ 0 Ginzburg, J. Stat. Phys. 1(1969),3. Electrons in neutron stars are not superconducting. Nuclear superfluidity and superconductivity in neutron stars The BCS theory was applied to nuclei by Bohr, Mottelson, Pines and Belyaev Phys. Rev. 110, 936 (1958). Mat.-Fys. Medd. K. Dan. Vid. Selsk. 31 , 1 (1959). N.N. Bogoliubov, who developed a microscopic theory of superfluidity and superconductivity, was the first to explore its application to nuclear matter. Dokl. Ak. nauk SSSR 119, 52 (1958). Superfluidity in neutron stars was suggested long ago (before the discovery of pulsars) by Migdal in 1959. It was first studied by Ginzburg and Kirzhnits in 1964. Ginzburg and Kirzhnits, Zh. Eksp. Teor. Fiz. 47, 2006, (1964). Superfluidity in neutron matter Most microscopic calculations have been performed in uniform neutron matter. They show that medium effects act against pairing but pairing gaps remain poorly known. Lombardo & Schulze, Lect. Notes Phys. 578 (2001) Springer Superfluidity and superconductivity in neutron stars In spite of their names, neutron stars are not only made of neutrons! As a consequence, they could contain various kinds of superfluids and superconductors. traditional neutron star quark−hybrid star N+e N+e+n s to ns H d Σ,Λ ,Ξ,∆ neutron star with pion condensate ui π− p ro 2SC CFL color−superconducting strange quark matter (u,d,s quarks) 2SC 2SC+s ,µ rfl n,p,e u,d,s quarks pe ond u c t i r c n g p e su u n n,p,e, µ hyperon star − K CFL CFL−K + 0 CFL−K 0 CFL− π crust Fe 6 3 10 g/cm 11 10 g/cm 3 3 10 14 g/cm Hydrogen/He atmosphere strange star nucleon star R ~ 10 km picture from F. Weber Superfluidity in neutron star crusts Density functional theory in a nut shell The nuclear energy density functional theory is the only method which allows a tractable and consistent treatment of both crust and core. The energy of a lump of matter is expressed as (q = n, p) Z h i E = E ρq (rr ), ∇ ρq (rr ), τq (rr ), J q (rr ), ρ̃q (rr ) d3r (q) (q) where ρq (rr ), τq (rr )... are functionals of ϕ1k (r ) and ϕ2k (r ) ! (q) hq (r ) − λq ∆q (r ) ϕ1k (r ) (q) = Ek (q) ∆q (r ) −hq (r ) + λq ϕ2k (r ) ∇· hq ≡ −∇ δE δE δE ∇+ −i ·∇ ×σ , δτq δρq δJJ q (q) ϕ1k (r ) (q) ϕ2k (r ) ∆q ≡ ! δE δ ρ̃q Effective nuclear energy density functional In principle, one can construct the nuclear functional from realistic nucleon-nucleon forces (i.e. fitted to experimental nucleon-nucleon phase shifts) using many-body methods E= ~2 (τn + τp ) + A(ρn , ρp ) + B(ρn , ρp )τn + B(ρp , ρn )τp 2M +C(ρn , ρp )(∇ρn )2 +C(ρp , ρn )(∇ρp )2 +D(ρn , ρp )(∇ρn )·(∇ρp ) +Coulomb, spin-orbit and pairing Drut et al.,Prog.Part.Nucl.Phys.64(2010)120. But difficult task so in practice, the functional is parametrized with simple analytical functions Bender et al.,Rev.Mod.Phys.75, 121 (2003). Construction of the functional: experimental data The parameters of the functional are fitted to the 2149 measured nuclear masses with Z , N ≥ 8. The rms deviation for the best fit so far is σ = 0.581 MeV. Goriely, Chamel, Pearson, PRL102,152503 (2009). This fitting procedure ensures that the neutron-rich nuclei in neutron-star crusts will be properly described. Construction of the functional: N-body calculations The functional is free from self-interactions and is constrained to reproduce various properties of nuclear matter as obtained from many-body calculations: isoscalar effective mass Ms∗ /M = 0.8 equation of state of pure neutron matter 1S 0 pairing gaps in symmetric and neutron matter Landau parameters (stability against spurious instabilities) Chamel, Goriely, Pearson, Phys.Rev.C80,065804 (2009) Chamel, Phys. Rev. C 82, 014313 (2010) Chamel and Goriely, Phys. Rev. C 82, 045804 (2010) Chamel, Phys. Rev. C 82, 061307(R) (2010) With these constraints, the functional can be reliably applied to describe superfluid neutrons in the inner crust and the neutron-proton mixture in the liquid core Ground-state composition of the inner crust Results for BSk14 A 200 460 1140 1215 1485 1590 1610 800 780 0.1 0.05 0 0 -3 Z 50 50 50 40 40 40 40 20 20 nn(r), np(r) [fm ] nb (fm−3 ) 0.0003 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.1 0.05 0 0 0.1 0.05 0 0 0.1 0.05 0 0 0.1 0.05 0 0 nb = 0.06 2 4 8 6 10 12 14 nb = 0.04 3 9 6 12 18 15 21 nb = 0.02 4 8 12 20 16 24 nb = 0.005 10 5 20 15 30 25 35 nb = 0.0003 10 20 30 r [fm] Onsi, Dutta, Chatri, Goriely, Chamel and Pearson, Phys.Rev.C77,065805 (2008). 40 50 Anisotropic multi-band neutron superfluidity In the decoupling approximation, the Hartree-Fock-Bogoliubov equations reduce to the BCS equations ∆αkk = − Eβkk ′ ∆βkk ′ 1 X X pair tanh v̄αkk α−kk βkk ′ β−kk ′ 2 Eβkk ′ 2T ′ β pair v̄αkk α−kk βkk ′ β−kk ′ = Z k d3 r v π [ρn (rr ), ρp (rr )] |ϕαkk (rr )|2 |ϕβkk ′ (rr )|2 Eαkk = q ε2αkk + ∆2αkk εαkk and ϕαkk (rr ) are obtained from band structure calculations Chamel et al., Phys.Rev.C81,045804 (2010). Analogy with terrestrial multi-band superconductors Multi-band superconductors were first studied by Suhl et al. in 1959 but clear evidence were found only in 2001 with the discovery of MgB2 Ordinary superconductors generally involve a few bands whereas the number of bands in neutron star crust can be very large ∼ up to a thousand Neutron pairing gaps vs single-particle energies Example at ρ = 0.06 fm−3 with BSk16 2,2 ∆F [MeV] 2 1,8 1,6 1,4 1,2 1 0,8 -40 -30 -20 -10 ε [MeV] 0 10 20 ⇒ The presence of clusters reduces ∆αkk but much less than predicted by previous calculations Average neutron pairing gap vs temperature ∆(T)/∆(0) Example at ρ = 0.06 fm−3 with BSk16 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 T/Tc ⇒ ∆αkk (T )/∆αkk (0) is a universal function of T ⇒ The critical temperature is approximately given by the usual BCS relation Tc ≃ 0.567∆F Pairing field and local density approximation The effects of inhomogeneities on neutron superfluidity can be directly seen in the pairing field Λ X ∆ k 1 |ϕαkk (rr )|2 αk ∆n (rr ) = − v πn [ρn (rr ), ρp (rr )]ρ̃n (rr ) , ρ̃n (rr ) = 2 Eαkk α,kk Neutron pairing field for ρ = 0.06 fm−3 at T = 0 90 (c) 80 ρ = 0.06 fm 60 ∆(r) [MeV] ξ (r) [fm] 70 -3 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 r [fm] 2.4 2.2 (c) -3 ρ = 0.06 fm 2 1.8 1.6 LDA 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 r [fm] Observational evidence of superfluidity in neutron stars Cooling of isolated neutron stars During the first tens of seconds, the newly formed proto-neutron star with a radius of ∼ 50 km stays very hot with T ∼ 1011 − 1012 K. Within ∼ 10 − 20 s the proto-neutron star becomes transparent to neutrinos and thus rapidly cools down by powerful neutrino emission shrinking into an ordinary neutron star. After about 104 − 105 years, the cooling is governed by the emission of thermal photons due to the diffusion of heat from the interior to the surface. Puppis A (RX J0822-4300) from Chandra Cooling of isolated neutron stars The surface temperature of a neutron star depends on its mass and composition but also on superfluidity. Superfluidity reduces the heat capacity and neutrino emissivities but also opens new channels of neutrino emission. TC = 1 0 9K TC = 5 .5x10 8 K TC = 0 Recent observations of the cooling neutron star in the Cassiopeia A supernova remnant put stringent contraints on the critical temperature of the neutron superfluid Tc & 5 × 108 K. Page et al., PRL 106, 081101 (2011) X-ray binaries Neutron stars in x-ray binaries may be heated as a result of the accretion of matter from the companion star. The accretion of matter onto the surface of the neutron star triggers thermonuclear fusion reactions which can become explosive, giving rise to x-ray bursts. In soft x-ray transients, accretion outbursts are followed by long period of quiescence during which the accretion rate is much lower. In some cases, the period of accretion can last long enough for the crust to be heated out of equilibrium with the core. Thermal relaxation of soft x-ray transients The thermal relaxation during the quiescent state has been recently monitored for KS 1731−260 and for MXB 1659−29 after an accretion episode of 12.5 and 2.5 years respectively. Curves 1,4 : crystalline crust with neutron superfluidity Curve 2 : crystalline crust without neutron superfluidity Curve 5 : amorphous crust with neutron superfluidity Shternin et al., Mon. Not. R. Astron. Soc.382(2007), L43. Brown and Cumming, ApJ698 (2009), 1020. Pulsar glitches and superfluidity Sometimes pulsars may suddenly spin up. These glitches are followed by a relaxation over days to years thus revealing the superfluidity in neutron stars. Superfluidity is also expected to play a key role in the mechanism of large glitches (e.g. catastrophic unpinning of superfluid vortices). Similar phenomena were observed in superfluid helium. Anderson and Itoh, Nature 256, 25 (1975) Neutron star precession Long-term cyclical variations of order months to years have been reported in a few neutron stars: Her X-1 (accreting neutron star), the Crab pulsar, PSR 1828−11, PSR B1642−03, PSR B0959−54 and RX J0720.4−3125. Stairs et al., Nature 406(2000),484. 30 0 ∆P (ns) -30 2 1 0 -1 1 <S> Example: Time of arrival residuals, period residuals, and shape parameter for PSR 1828−11 ∆t (ms) 60 0.5 0 49,500 50,000 50,500 Modified Julian Date These variations have been interpreted as the signature of neutron star precession. 51,000 Precession and superfluidity Observations of long-period precession put contraints on the superfluid properties of neutron star interiors. For a non-superfluid star with deformation ǫ = ∆I/I, Pprec = For a superfluid star with pinned vortices P ≫P ǫ Link, Astrophys. Space Sci.308,435 (2007) Pprec = Ipin P≪P I Plate tectonics The strong interaction between neutron superfluid vortices and proton flux tubes in neutron star cores leads to movement of crustal plates. The evolution of the pulsar spin and magnetic field are intimately related to superfluidity and superconductivity Ruderman, Astrophys. Space Sci.357, 353 (2009) Superfluidity and mutual entrainment In superfluid helium at T > 0, the momentum p and the velocity v of each atom are not aligned due to the interactions between atoms and thermal excitations (phonons, rotons...) momentum vs velocity p = m⋆v + (m − m⋆ )vvN vN is the velocity of excitations. The effective mass m⋆ (T ) can be experimentally measured from Andronikashvili’s experiment Likewise in superfluid mixtures, momentum and velocity are not aligned even at T = 0: this is the Andreev&Bashkin effect Andreev and Bashkin, Sov.Phys.JETP42,164(1975) Entrainment and dissipation in neutron star cores Historically the long post-glitch relaxation was the first evidence of superfluidity in neutron stars. However... κ Due to (non-dissipative) mutual entrainment effects, neutron vortices carry a fractional magnetic quantum flux B vp vn V e− Sedrakyan and Shakhabasyan, Astrofizika 8 (1972), 557; Astrofizika 16 (1980), 727. picture from Glampedakis The core superfluid is strongly coupled to the crust due to electrons scattering off the magnetic field of the neutron vortex lines. This leads to a mutual friction force acting on the superfluid which could be an important dissipative mechanism for neutron star pulsations. Alpar, Langer, Sauls, ApJ282 (1984) 533-541 Entrainment effects in neutron star crusts Even though entrainment effects have been studied for a long time in neutron-star cores (e.g. Sauls, NATO Proceedings 1989), it has been only recently realized that similar effects should also occur in the crust. Like electrons in metals, unbound neutrons in neutron star crusts move as if they had an effective mass mn⋆ > mn . In the crust rest frame pn = mn⋆vn therefore in another frame pn = mn⋆vn + (mn − mn⋆ )vvc Carter, Chamel, Haensel, Int.J.Mod.Phys.D15(2006)777. Conclusions Pines theorem Neutron stars are superstars! The theorem was formulated at the conference on “Neutron Stars: Theory and Observation”, NATO Advanced Study Institute, Greece, 1990. Proof: neutron stars are superdense stars neutron stars are superfast rotators neutron stars are superprecise clocks neutron stars have superstrong magnetic fields neutron stars are superfluid neutron stars are superconducting