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rd 3 lecture of “Compact Object and Accretion”, Master Programme at Leiden Observatory st I Neutron Stars class study material: Chapter 9, ShapiroTeukolsky these slides at http://home.strw.leidenuniv.nl/~emr/COA/ Friday, October 2, 2015 Neutron stars (NSs) a few facts we already know • Final evolutionary stage of stars with mass between 9 Msun and a not well determined mass, 30-40 Msun (depending, for example the star metallicity) • After all the nuclear reaction cycles take place, the iron core collapses and form a neutron star • During the core collapse, the core neutralisation + takes place via inverse beta reaction*p → n + e + νe − as the neutron beta decay n → p + e + ν¯e is suppressed because an electrons with an energy larger than the Fermi energy needs to be produced in an electron degenerate medium − *Note, also electron capture p + e → n + νe Friday, October 2, 2015 Neutron stars (NSs) a few facts we already know • Matter remains transparent to neutrinos until a density of ~1015 kg m-3 (R~300 km) and neutrinos escape and be detected • Formation of a neutron stars is accompanied by a supernova explosion (Type II ou Type I/b) Friday, October 2, 2015 Neutron stars (NSs) a few facts we already know • Typical values for their physical characteristics Radius • • Friday, October 2, 2015 Mass Central mass density Compactness of 0.2 Gravity is counterbalanced by pressure originated by strong force between nucleons. We will see that the equation of state corresponding to this situation is rather uncertain NSs: history Friday, October 2, 2015 Discovery of Neutron • Neutrons were discovered by James Chadwick in 1932 in Cambridge (Nobel prize in 1935). Meanwhile Chandrasekhar and Landau were working on WD and showed the existence of a maximum mass, wondering what was then the fate of more massive stars. In 1931/1932 Landau proposed that this fate would be an object “where the atomic nuclei were so close to each other to form a gigantic nucleus” James Chadwick Friday, October 2, 2015 NS concept 1934 Baade & Zwicky proposed the existence of NSs. They had discovered that some “Novea” were extragalactic and therefore much more intrinsically luminous than Galactic Novea. They called them “Supernovae” and propose: Friday, October 2, 2015 Equation of state • 1939 first calculations of NS structure by Oppenheimer & Volkoff. They use general relativity and an equation of state for a perfect gas of degenerate neutrons. we will see that this predict a maximum mass of ~0.7 Msun in contraddiction with observations. This shows that neutron degenerate pressure is not counterbalancing gravity in NSs. WWII stopped momentarily further advancements... Friday, October 2, 2015 Equation of state cont. • Friday, October 2, 2015 At the end of 1950s theoretical works on NSs restarted, in particular around the question of what is the equation of state for ultra-dense matter. Notable works by Harrison, Wikano & Wheeler (1958), Cameron (1959), Ambartsumyan & Saakyan (1960) & Hamada & Salpeter (1961). These more realistic equation of state showed that NS must have very small radii ~ 10 km. Despite their large surface temperature, the small size makes direct detection of surface emission undetectable. NSs during these years remained a study subject for (a small group of) theoreticians. The beginning of X-ray astronomy In 1962, Giacconi and co-authors discovered the first X-ray source beyond the solar system (Sco X-1, Lx~ 6 104 Lsun), following the launch of the USA Arobee rocket (Nobel prize in 2002). It is interpreted as a young and hot neutron star (Te ~2 1010 K). The question is then how fast NSs cool down. At the same time, there is a debate on whether QSOs (discovered in 1963) are neutron stars. However, it was soon realised that the observed redshift in their spectra cannot be of gravitational origin, given the NS expected compactness. We now know that Sco X-I is an accreting neutron star and QSOs are accreting supermassive black holes Friday, October 2, 2015 Pulsars discovery • In 1964, Hoyle, Narlikar & Wheeler proposed that a highly magnetised NS (~1010 Gauss) is at the centre of th Crab nebula. • In 1967, Pacini proposed that the source of energy for the Crab would be a highly magnetised and rotating NS. X-ray Optical Friday, October 2, 2015 Pulsars discovery • In 1967, Hewish and his student Bell discover the first radio pulsar: a period radio source, with an extremely short (modern value P = 1.337301 s) and stable period. (Nobel prize to Hewish in 1974). The extreme stability suggested to some an extra-terrestrial life origin... • In 1968, they published their discovery after much double checking. Name: CP 1919 (Cambridge Pulsar, right ascension 19, declination 19. The Modern name is “PSR B 1919+21”) Friday, October 2, 2015 More pulsars... • In 1968, discovery of the radio Pulsar Vela (PSR B 0833-45) and Crabe (PSR B 0531+21). They are both within supernova remnants. These discovery validate Baade, Zwicky & Pacini’s predictions. • In 1968, Gold proposes a pulsar model as magnetised, rapidly rotating NS, with which he predicts with precision the Crabe period and its derivative, that were only measured in 1969 (we will study it...) Friday, October 2, 2015 Pulsars, as NS There are very little alternative to a NS to explain a Pulsar. Given mass M, radius R and angular velocity Ω the centrifugal force at the surface, at the equator is & The maximum rotation for balance is then where So Friday, October 2, 2015 Pulsars, as NSs The first radio pulsar has Ω � 4.7 s −1 ρ̄ > 7.9 × 1010 Kg m−3 a minimum density of The Crabe pulsar has Ω � 0.20 ms −1 a minimum density of ρ̄ > 1.2 × 10 14 Kg m −3 No system is known that is so dense, not even WDs recall, for WD ρ̄ ≈ 2 × 10 Kg m 9 Friday, October 2, 2015 −3 Rotation as cause for observed period Alternatives were discussed but dismissed: • Surface radial oscillation: WDs fundamental mode is too slow, NSs is too fast • Orbital period of a binary: for two NSs with 1.4 Msun, one need a separation of 4000 km, for P~1 s. In this situation gravitational wave emission would be so intense to make the NSs coalesce in a few hours It is therefore immediately clear that the observed period is linked to the NS spin and the agreement with Gold’s model indicates a strongly magnetised NS Friday, October 2, 2015 Other notable discovery • Friday, October 2, 2015 1969, first X-ray pulsars (no radio detected). They are binary systems, that allowed to measure the NS mass. Results in agreement with theory. Other notable discovery • 1975, Hulse& Taylor discover the first binary pulsar PSR B 1913+16 (pulsar period ~56 s; orbital period of 7.8 h). Determined that they are two NSs of ~1.4 Msun. Only one is a pulsar. Monitoring the pulsar, they measure a decrease in the orbital period consistent with Gravitational Wave emission. First (indirect)detection of GW! Nobel prize in 1993 from Weisberg & Taylor 2004 Currently, agreement with GR within 0.1 % Friday, October 2, 2015 Other notable discoveries • 1982, first pulsar discovered with P ~1.56 ms (millisecond pulsar) . Such a short period is difficult to justify within the standard formation scenario. We think now that they were “re-accelerated” via accretion from a companion. Recycled pulsars • In 1992, the first discovery of (3) extrasolar planets, around a millisecond pulsar by Wolszcan & Frail • ..... Friday, October 2, 2015 WDs vs NS similarities and differences Friday, October 2, 2015 Similarities • • • Friday, October 2, 2015 Equation of state under the assumption of T=0 Therefore, fluid barotropic with P=P(ρ) To calculate the structure, one therefore needs only eq. mass conservation + eq. hydrostatic equilibrium Differences 1. NS much higher compactness, therefore General relativistic frame GR (solid) and Newtonian (dashed) WDs neutralisation, calculation not valid NSs calculations are indistinguishable Friday, October 2, 2015 Idialised NS with neutron degenerate pressure. In GR, this is in fact the Oppenheimer & Volkoff (1939) calculation differ at high ρ, maximum mass just 0.7 Msun Differences 2. NS much high densities (~8 orders of magnitude denser) Equation of state uncertain Mass and Radius measurements of NSs are used to try and constrain the equation of state Friday, October 2, 2015 Differences 3. NS radius is ~100-1000 times smaller. NS cool very rapidly after their formation and they typically have an effective temperature of a few 106 K . These imply L~ 10 Lsun but in X-ray (106 k => 0.1 keV) We cannot detect thermal radiation from the surface, unless the object is very close e.g. RX J185635-3754 at 120 pc WD can be observed in the optical band Friday, October 2, 2015 Differences 4. NSs have higher rotation and magnetic field B ~ 106 G (WDs); B~1012 G (NSs) P~ a few hr (WDs); P~ seconds (NSs) white dwarfs neutron stars (upper limit as some B and Ω can be lost during collapse) Lpulsar ∝ Ω R B (power) 4 6 2 WD ‘s pulsar emission is ~10-14 smaller ! Friday, October 2, 2015 cfr. Ch 9 Shapiro Teukolsky NSs: Equation of state Friday, October 2, 2015 Oppenheimer & Volkoff (1939) • perfect gas • degenerate neutrons only • T=0 The equation of state is the same as that discussed for WDs, exchanging the electron mass with the neutron mass in the coefficients Friday, October 2, 2015 n-p-e • 1) 2) gas A step forward is to consider a perfect gas of electrons, protons and neutrons at T=0 total density charge neutrality If β and inverse-β reactions are at equilibrium 3) n↔p+e “chemical equilibrium” where the chemical potential of neutrinos ~zero (like photons) T=0 + chemical equilibrium = cold catalysed matter Friday, October 2, 2015 n-p-e gas, cont. At T=0 we have μ=EF With We have 3 equations in 3 unknown np, nn,ne We are going to find np,nn,ne as a function of n, or: Friday, October 2, 2015 n-p-e gas, cont. Pressure an internal energy are just given by the sum of the 3 species (Dalton’s law for ideal gas) Reminder: Friday, October 2, 2015 Fermi-Dirac neutron drip 5/3 n e n electron degenerate pressure dominates Friday, October 2, 2015 from the chemical e- NR regime WD 1/8 Neutron rich above 109 g/cm3 (1011 Kg/m3) 4/3 UR regime nuclear density equilibrium with x>>1 neutron drip line = neutron rich nuclei becomes unstable for neutron ejection: nucleons drip out of unstable nuclei neutron degeneracy pressure dominates Realistic equation of state Additional facts to consider: • presence of iron nuclei that can survive at law densities, but will progressively disintegrate beyond the neutron drip density (4 1011 g/cm3) • strong force interaction beyond the nuclear density (~2.8 1014 g/cm3), when the mean particle distance is ~a Fermi (10-6 nm). Perfect gas approximation breaks down (perfect gas = non interacting gas) • Strong Magnetic field Friday, October 2, 2015 Equation of state within a NS I) Surface, depth of ~100 m. EOS strongly influenced by magnetic field II.)External crust of ~500 m. density less than neutron drip. EOS well understood: crystal structure of neutron rich nuclei (that interact via Coulomb force) + a fluid of free electrons and neutrons V) Core. Most uncertain region. some possibilities: formation of a superfluid core of n +p; pion condensation; formation of a solid core of neutron; phase transition towards a strange quark matter (see Ch. 8 S-T optional) Friday, October 2, 2015 III) Internal crust of ~1 Km. density above neutron drip and below nuclear density. Nuclei progressively dissolve and neutron pressure becomes more important IV) Interior: neutrons become superfluid, nuclei completely dissolved, small fraction of e- + p Above neutron drip • Non relativistic domain. Between nuclear density and ~1015 g/cm3 there are two complication: 1) describing strong force potential; 2) finding appropriate numerical technique for solving a many body problem. The potential is somewhat constrained by experimental data (e.g. nucleonnucleon scattering data) • Beyond 1015 g/cm3, composition should include hyperions (baryons that contain at least one strange quark) and nuclear interaction must be treated relativistically. Relativistic many-body technique for many-body interaction not fully developed. see Box 8.1 in S-T Friday, October 2, 2015 NS structure: maximum mass Note: despite the uncertainty on the equation of state, it can still be written as a barotropic relation P = P (ρ) We are thus going to solve the NS structure as we did for WDs, but in a general relativistic framework Friday, October 2, 2015 Einstein equation Source term is the energy-momentum tensor: µ dx µ u = with dτ and Einstein tensor: four-velocity where the space-time metric Gravity is a property of space-time, that depends on the local content of mass & energy Friday, October 2, 2015 Schwarzschild solution At the outside of a neutron star of M, we can assume vacuum Static, spherical symmetric solution M is the gravitational mass, as measure by an observer at infinity Minkowski space Friday, October 2, 2015 Tolman-Oppenheimer-Volkhoff (T.O.V) Inside a NS. For a static spherical system, only the diagonal elements of are non-zero. We thus have 3 independent equations: with Friday, October 2, 2015 Tolman-Oppenheimer-Volkhoff (T.O.V) cont. In the limit: 2 Gm(r)/r � c and P � ρc 2 They become the non-relativistic equations of 1) mass conservation, 2) hydrostatic equilibrium and 3) the equation for the gravitational potential Boundary conditions Non relativistic regime Relativistic regime Equivalent to dP/dr =0 ensures continuity with solution inside and outside Friday, October 2, 2015 Solutions with & we can easily solve equations M/Msun M/Msun Unstable configurations for radial perturbation there is a maximum mass for any ϒ ρ/ρnuc Friday, October 2, 2015 ρ/ρnuc M/Msun For a perfect gas of degenerate neutrons or of a mix of degenerate n+p+e the maximum mass is 0.7 Msun, in contradiction with observations ρ [ g cm-3] Friday, October 2, 2015 For realistic equation of state the maximum mass is in the range Salgado et al. 1994 mass measurements can be used to rule out e.o.s Friday, October 2, 2015