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Neutron Stars Kostas Kokkotas December 17, 2012 Kostas Kokkotas Neutron Stars Bibliography I Black Holes, White Dwarfs, and Neutron Stars S.L.Shapiro and S.A. Teukolsky, see Chapters 2, 3, 8 and 9. I GRAVITY J.B. Hartle see Chapter 24 I Duncan R. Lorimer, ”Binary and Millisecond Pulsars at the New Millennium”, Living Rev. Relativity 4, (2001), http://www.livingreviews.org/lrr-2001-5 I D. R. Lorimer & M. Kramer; Handbook of Pulsar Astronomy; Cambridge Observing Handbooks for Research Astronomers, 2004 Kostas Kokkotas Neutron Stars Neutron Stars: A Laboratory for Theoretical Physics I I A neutron star is the collapsed core of a massive star left behind after a supernova explosion. The original massive star contained between 8 and 25 M . (More massive stars collapse into black holes.) The remnant neutron star compresses at least 1.4M into a sphere only about 10-16 km across. This material is crushed together so tightly that gravity overcomes the repulsive force between negatively charged electrons and positively charged protons. The resulting structure of the star is complex, with a solid crystalline crust about one kilometer thick encasing a core of superfluid neutrons and superconducting protons. Above the crust exists both an ocean and atmosphere of much less dense material. Kostas Kokkotas Neutron Stars Pauli Principle An isolated WD or NS cools down to zero temperature and it is the pressure associated with matter at T = 0 that supports these stars against gravitational collapse. A simple but very important example of nonthermal source of pressure is the Fermi pressure arising from the Pauli exclusion principle. The Pauli exclusion principle: I restricts the quantum states allowed to half-integral spin particles (e, p, n) known as fermions I prohibits any two fermions from occupying the same quantum state. I is crucial for the structure of atoms and their chemical properties I has the consequence that in the lowest-energy state of an atom, the electrons are not in the lowest-energy level near the nucleus; instead they are arranged in higher-energy-level shells. I it supports the outer electrons in an atom against the attractive electric forces of nucleus. Kostas Kokkotas Neutron Stars Electron degeneracy pressure I Electron degeneracy pressure is a force caused by the Pauli exclusion principle, which states that two electrons cannot occupy the same quantum state at the same time. This force often sets a limit to how much matter can be squeezed together. I A material subjected to ever increasing pressure will become ever more compressed, and for electrons within it, the uncertainty in position measurements, ∆x, becomes ever smaller. Then, as dictated by the Heisenberg’s uncertainty principle, ∆x ∆p ≥ ~2 , the uncertainty in the momenta of the electrons, ∆p, becomes larger. Thus, no matter how low the temperature drops, the electrons must be traveling at this ”Heisenberg speed”, contributing to the pressure. I When the pressure due to the “Heisenberg speed” exceeds that of the pressure from the thermal motions of the electrons, the electrons are referred to as degenerate, and the material is termed degenerate matter. I Electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the Chandrasekhar Limit (1.44 M ). This is the pressure that prevents a white dwarf star from collapsing. Kostas Kokkotas Neutron Stars Maximum Mass of White Dwarfs I White Dwarfs support themselves against gravity by the pressure of electrons arising from the Pauli exclusion principle. I This pressure is called Fermi pressure and the corresponding compressional energy is called Fermi energy. I A rough estimate of the maximum mass that can be supported against gravity by Fermi pressure can be made by studying the Fermi energy of a spherical configuration of radius R consisting of N electrons and N protons (electrically neutral). I The heavier protons supply most of the mass and the lighter electrons supply most of the pressure. I Since electrons exclude each other, we can think of each of them as occupying a volume of characterisc size λ ∼ R/N 1/3 I The Fermi momentum of electrons is pF ∼ ~/λ ∼ N 1/3 ~/R I (1) If the sphere is compressed, R shrinks, pF rises, the Fermi energy of the electrons rises and work has to be done to make the compression. Kostas Kokkotas Neutron Stars Maximum Mass of White Dwarfs II If the compression has been carried out to the point that electrons become relativistic with energies E = [pF2 + me2 c 2 ]1/2 c ≈ pF c. Then the total energy in this approximation is: EF ∼ N · (pF c) ∼ N 4/3 ~c/R . (2) The protons supply most of the gravitational energy EG , which is roughly EG ∼ −G (mp N)2 /R (3) Both EF and EG vary as 1/R. For large N the total energy will be negative and it will be energetically favorable for the configuration to collapse. The critical N at which gravitational collapse is favored is 3/2 Ncrit ∼ ~c/Gmp2 ∼ 2.2 × 1057 (4) and the critical mass is1 Mcrit ∼ mp Ncrit ∼ 1.85M ! (5) The exact solution of the maximum mass is called the Chandrasekhar mass and it is about 1.4M ! 1m p = 1.673 × 10−24 gr and M = 1.989 × 1033 gr Kostas Kokkotas Neutron Stars Radius of White Dwarfs The equilibrium masses associated with masses M approaching Mcrit is determined by the onset of the relativistic degeneracy: EF = pF c ≥ mc 2 (6) where m refers to either electrons or neutrons. Then by using equations (1) and (4) this condition gives R≤ ~ mc ~c Gmp2 1/2 ≈{ 5.03 × 108 cm 2.74 × 105 cm for for m = me m = mn (7) Thus there are two distinct regimes of collapse: I one for densities above white dwarf values I and another for densities above nuclear densities but in both cases M ∼ M . 2 Note: 2 mp = 1.242 × 10−52 cm, h = 2.612 × 10−66 cm2 & me = 6.764 × 10−56 cm. Kostas Kokkotas Neutron Stars Maximum Mass of White Dwarfs III An ideal fermion gas is described by Fermi-Dirac statistics (if bosons then Bose-Einstein statistics) with distribution function (µ: chemical potential) 1 e (E −µ)/kT + 1 For low particle densities and high temperatures it reduces to Maxwell-Boltzmann distribution f (E ) = e (µ−E )/kT with f (E ) 1. f (E ) = I I (8) For completely degenerate fermions (T → 0 i.e. µ/kT → ∞) µ becomes the Fermi energy, EF = [pF2 c 2 + me2 c 4 ]1/2 and f (E ) = { 1 0 E ≤ EF E > EF (9) That means that there is only one electron per allowed quantum state with momentum p up to a maximum value the pF (Fermi momentum). The number density of electrons will be: Z pF 8π 1 2 4πp 2 dp = 3 pF3 = x3 ne = 3 ~ 0 3~ 3π 2 λe (10) where x = pF /me c is the dimensionless Fermi momentum and λe = ~/me c is the electron Compton wavelength. Kostas Kokkotas Neutron Stars Maximum Mass of White Dwarfs IV The pressure will be Pe ≈ 13 ne < pv > where v = pc 2 /E is the velocity and the energy density e ≈ ne E or analytically (why?): Z pF me c 2 1 2 p2 c 2 4πp 2 dp = Pe = φ(x) 2 4 1/2 3 2 2 3 ~ 0 (p c + me c ) λ3e = e = 1.42 × 1025 φ(x) dyne/cm2 Z pF 1/2 me c 2 2 p 2 c 2 + me2 c 4 4πp 2 dp = χ(x) 3 h 0 λ3e (11) (12) where φ(x) = io h 1 n x(1 + x 2 )1/2 (2x 2 /3 − 1) + ln x + (1 + x 2 )1/2 →{ 2 8π χ(x) = h io 1 n x(1 + x 2 )1/2 (1 + 2x 2 ) − ln x + (1 + x 2 )1/2 →{ 2 8π Kostas Kokkotas Neutron Stars x5 15π 2 x4 12π 2 x3 3π 2 x4 4π 2 x 1 x 1 x 1 x 1 Maximum Mass of White Dwarfs V Even if degenerate electrons contribute most of the pressure, the density is usually dominated by the rest-mass of the ions. The density is X ne mB ρ0 = ni mi ≡ nmB = = µe mu ne = 0.974 × 106 µe x 3 g/cm3 (13) Ye i where mi is the mass of ion species i, mB is the mean baryon density, Ye is the mean number of electrons per baryon, mu = 1.66 × 10−24 g is the atomic mass unit and µe = mB /mu Ye is the mean molecular weight per electron. Equations (11) and (13) provide in a parametric form the ideal degenerate equation of state (EoS) P = P(ρ0 ) 3 . If we use a polytropic form for the (EoS) P = K ρΓ0 where K and Γ are constants we get two limiting cases: I Non-relativistic electrons ρ 106 g/cm3 ,x 1 Γ= I 5 , 3 32/3 π 4/3 ~2 = 1.004 × 1013 µ−5/3 cgs e 5/3 5/3 5 me mu µe (14) Extremely relativistic electrons ρ 106 g/cm3 , x 1 Γ= 3ρ K = 4 , 3 K = 31/3 π 2/3 ~c = 1.24 × 1015 µ−4/3 cgs e 4/3 4/3 4 m u µe (15) = ρ0 + e /c 2 is the total energy density but the last term e /c 2 is negligible. Kostas Kokkotas Neutron Stars The Tolman-Oppenheimer-Volkov (TOV) Solution The energy momentum tensor for a perfect fluid is: T µν = (ρ + p)u µ u ν − g µν p where uµ = dx µ ds Then the law for conservation of energy and momentum leads to: T µν ;ν = g µν ∂ν p + [(p + ρ)u µ u ν ];ν + (ρ + p)Γµ νλ u µ u λ = 0 (16) For a spherically symmetric and static solution ds 2 = e ν(r ) dt 2 − e λ(r ) dr 2 − r 2 dθ2 + sin2 θdφ2 the fluid velocity is: u µ = (e −ν/2 , 0, 0, 0) and thus (16) becomes T µν ;ν = ∂ν pg µν + (ρ + p)Γµ 00 u 0 u 0 = 0 where Γµ 00 = − 21 g µν g00,ν = − 12 e ν ν 0 Kostas Kokkotas Neutron Stars (17) TOV - II ...multiplying with gµλ we get 1 ∂λ p = − (ρ + p) ∂λ ν 2 ⇒ dp 1 dν = − (ρ + p) dr 2 dr (18) which is the relativistic version of the equations for hydrodynamical equilibrium, since in the Newtonian limit g00 = e ν ≈ 1 + 2U and ρ p which leads to: ∂p ∂U = −ρ ∂r ∂r Still we need to find a way, via Einstein’s equations 1 Rµν = −8π Tµν − gµν Tλλ 2 (19) (20) to estimate ν(r ) in the same way that we need to solve Poisson equation to estimate the gravitaional potential U. Kostas Kokkotas Neutron Stars TOV - III {θθ} : {rr } : {tt} : h i r 1 − e −λ 1 + (ν 0 − λ0 ) = −4πr 2 (ρ − p) 2 (ν 0 )2 ν 00 ν 0 λ0 λ0 + − − = 4πe λ (ρ − p) 2 4 4 r (ν 0 )2 ν 00 ν 0 λ0 ν0 + − + = −4πe λ (3ρ + p) 2 4 4 r which leads to the following ODE 0 re −λ = 1 − 8πρr 2 leading to e λ(r ) = (1 − 2M/r )−1 (21) (22) (23) r λ0 + e λ − 1 = 8πe λ r 2 ρ or Z where M(r ) = 4π r ρ(r 0 )r 02 dr 0 (24) 0 which together with (18) give M + 4πr 3 p dp = −(ρ + p) dr r (r − 2M) Kostas Kokkotas or dν M + 4πr 3 p =2 . dr r (r − 2M) Neutron Stars (25) TOV - IV : A uniform density star For the special case ρ = const there is an analytic solution. For example the mass function will become: M(r ) = 4 3 πr ρ 3 for r ≤R and M(r ) = 4 πR 3 ρ 3 for r ≥R The we get: (1 − 2Mr 2 /R 2 )1/2 − (1 − 2M/R)1/2 p = . ρ 3(1 − 2M/R)1/2 − (1 − 2Mr 2 /R 3 )1/2 (26) But substituting the above relations in (18) we get an analytic solution for g00 i.e. 1/2 1/2 3 2M 1 2Mr 2 e ν/2 = 1− − 1− . (27) 2 R 2 R3 Maximum allowed mass when p(r = 0) → ∞ : M 4 = R 9 Kostas Kokkotas Neutron Stars Spherical Relativistic Stars The equations of structure for a spherical relativistic star are: dm dr dP dr dν dr = 4πr 2 ρ (28) 3 = = m + 4πr P r (r − 2m) 1 dP − ρ + p dr −(ρ + p) (29) (30) together with and EoS P = P(ρ) and an equation for the metric function λ(r ) i.e. e −λ(r ) = 1 − 2m/r . If we move from geometrical units back to the cgs units G G G 2 m, ρ → 2 ρ, P → 4 P and ν(r ) → 2 Φ c2 c c c we get the following equations describing a spherical star in hydrostatic equilibrium in the non-relativitic limit m→ dm dr dP dr dΦ dr = 4πr 2 ρ = −ρ = Kostas Kokkotas Gm r2 1 dP Gm − = 2 ρ dr r Neutron Stars (31) (32) (33) (34) Polytropes I The ideal Fermi gas EoS reduces to simple polytropic form P = K ρΓ0 in the limiting cases of extreme non-relativistic (Γ = 5/3) and ultrarelativistic electrons (Γ = 4/3). Equilibrium configurations with such an EoS are called polytropes and herey are quite simple to analyze. The hydrostatic equilibrium equations (32) and (33) can be combined to give 1 d r 2 dP = −4πG ρ (35) r 2 dr ρ dr if we substitute P = K ρΓ0 , and we write Γ = 1 + 1/n (n is called polytropic index) we can get a dimensionless form of this equation the so called Lane-Emden equation 1 d 2 dθ ξ = −θn (36) ξ 2 dξ dξ where we have made the following substitutions # " 1/n−1 1/2 (n + 1)K ρc (37) ρ = ρc θn , r = aξ, a = 4πG where ρc = ρ(r = 0) is the central density. While the boundary conditions are θ(0) = 1 , Kostas Kokkotas and θ0 (0) = 0 . Neutron Stars (38) Polytropes II Lane-Emden equation can be easily integrated numerically starting at ξ = 0 with the BC (38). For n < 5 the solution decreases monotonically and has a zero at finite value ξ = ξ1 i.e. θ(ξ1 ) = 0 which corresponds to the surface of the star, where p = ρ = 0. Thus we get: R = aξ1 = Z M = = (n + 1)K 4πG 1/2 ρc(1−n)/2n ξ1 R (39) 3/2 (n + 1)K ρ(3−n)/2n ξ12 |θ0 (ξ1 )| c 4πG 0 n/(n−1) (n + 1)K (3−n)/(1−n) 2 0 4πR (3−n)/(1−n) ξ1 ξ1 |θ (ξ1 )|. 4πG 4πr 2 ρdr = ... = 4π (40) (41) For the special cases that we are interested: 5 , 3 4 Γ= , 3 Γ= 3 , 2 ξ1 = 3.65375, ξ12 |θ0 (ξ1 )| = 2.71406 (42) n = 3, ξ1 = 6.89685, ξ12 |θ0 (ξ1 )| = 2.01824 (43) n= Kostas Kokkotas Neutron Stars Polytropes II Figure: θ(ξ) for n = 0, 1, 2, 3, 4 and 5 Kostas Kokkotas Neutron Stars Maximum Mass of White Dwarfs : Chandrasekhar limit For low-density white dwarfs Γ = 5/3 we get: −1/6 µe −5/6 ρc km (44) R = 1.122 × 104 3 2 106 g/cm −3 1/2 µe −5 ρc µe −5/2 R M = 0.5 M = 0.7 M (45) 3 4 km 6 2 10 2 10 g/cm For the high-denity case Γ = 4/3 we get: −1/3 µe −2/3 ρc R = 3.35 × 104 km 3 2 106 g/cm 2 2 M M = 1.457 µe (46) (47) Note that M is independent of ρc and R in the ultra-relativistic limit. Thus as ρc → ∞, the electrons become more and more relativistic throughout the star, then R → 0 and the mass asymptotically will be given by equation (47). The mass limit (47) is called Chandrasekhar limit (MCh = 1.457M ) and is the maximum possible gas of a white dwarf. For cold perfect gas the dependence of MCh on composition is contained entirely in µe = mB /mu Ye = A/Z which is the mean molecular weight per electron, for He : A = 4 &Z = 2. Kostas Kokkotas Neutron Stars Neutron Stars I The above results can be easily scaled to determine the equation of state of an arbritrary species i of ideal fermions. For pure neutrons we get: n = mn c 2 χ(xn ) = 1.625 × 1038 χ(xn ) λ3n (48) where xn = pF /mn c. ρ0 = mn nn = mn 1 3 xn = 6.107 × 1015 xn3 g/cm3 λ3n 3π 2 (49) If we use a polytropic form for the equation of state (EoS) P = K ρΓ0 we get for the two limiting cases: I Non-relativistic neutrons ρ 6 × 1015 g/cm3 , x 1 Γ= I 5 , 3 K = 32/3 π 4/3 ~2 = 5.38 × 109 cgs 8/3 5 mn (50) Extremely relativistic neutrons ρ 6 × 1015 g/cm3 , x 1 Γ= 4 , 3 K = 31/3 π 2/3 ~c = 1.23 × 1015 cgs 4/3 4 mn (51) Notice that in this case the mass density ρ = n /c 2 is due entirely to neutrons and greatly exceeds ρ0 whenever the neutrons are extremely relativistic. Kostas Kokkotas Neutron Stars Neutron Stars II Low density neutron stas with the ideal neutron gas equation of state can be approximated by n = 3/2 Newtonian polytropes. From eqns (39) & (41) we get: −1/6 ρc km 1015 g/cm3 1/2 ρc M M = 1.102 1015 g/cm3 R = 14.64 (52) (53) Thus there is no minimum neutron star mass i.e. M → 0, ρc → 0 and R → ∞. In nature, of course, neutrons become unstable to β-decay at low densities. Kostas Kokkotas Neutron Stars Neutron Stars III : Nuclear Density In neutron stars the density is higher than the nuclear density! The maximum value for the momentum of a neutron will be ∆p ∼ mp c. Then the uncertainty principle ∆p ∆x ∼ ~ suggests that ∆x ∼ ~ ∼ 10−13 cm mp c (54) actually the exact value is ∆x ∼ 1.8 × 10−13 cm i.e. the neutrons “touch” each other. The number density will be n∼ 1 ≈ 1.7 × 1038 cm−3 ∆x 3 (55) and the density will be ρn = mp n ≈ 2.8 × 1014 g /cm3 In neutron stars the average density is higher than this value. Kostas Kokkotas Neutron Stars (56) Neutron Stars IV : Composition I NS’s surface is believed to consist of a very thin fluid/gas atmosphere. I Beneath that, the outer crust will typically consist of a lattice of 56 26 Fe. The pressure in this region is mainly provided by the iron lattice. I At higher densities more and more electrons will become free and the pressure will increasingly be due to the degenerate electron gas. In the region were ρ 106 g/cm3 the free electrons are non-relativistic and the compressibility index γ = (ρ + p)/pdp/dρ I As we go further towards the center of the star the electrons become increasingly relativistic and γ → 4/3. I At a density of about 106 g/cm3 the electrons and protons start recombining in inverse β-decay e − + p → n + νe thus forming increasingly neutron rich nuclei. I At ρ ∼ 3 × 1011 g/cm3 the nuclei gets unstable and the free neutrons are emitted. This is called the ”neutron drip” region and here the EoS will become very soft (low compressibility index). takes the value γ = 5/3. Kostas Kokkotas Neutron Stars Neutron Stars IV : Composition I As we move towards the center, the matter consists mostly of free neutrons and small abundances of free protons, electrons and muons. The pressure will be provided mainly by the degenerate neutrons and the compressibility index will decrease from the non-relativistic value 5/3 to the relativistic 4/3. I The neutrons become superfluid at some point in the neutron star. I If no phase transitions occur the EoS will approach p → ρ/3 I The actual EoS in these ultra high densities is highly unknown e.g. there might exists a quark-gluon plasma, kaon condensates and other exotic states of matter etc Kostas Kokkotas Neutron Stars Neutron Stars V Figure: Schematic cross section of a neutron star, showing the outer crust consisting of a lattice of nuclei with free electrons, the inner crust which also contains a gas of neutrons, the nuclear “pasta” phases, the liquid outer core, and the possibilities of higher-mass baryons, Bose-Einstein condensates of mesons, and possible quark matter in the inner core. Kostas Kokkotas Neutron Stars Neutron Stars VI Figure: Neutron star interior as predicted by theory Kostas Kokkotas Neutron Stars Neutron Stars VII Figure: Mass-radius curve for HWW EoS configurations. The curve is parametrized by central density measured in gr/cm3 . At the extrema (max or min) the configuration becomes unstable/stable for increasing central density. Kostas Kokkotas Neutron Stars Neutron Stars VIII Figure: Mass-radius relationship of neutron stars and strange stars. The strange stars may be enveloped in a crust of ordinary nuclear matter whose density is below neutron drip density. Kostas Kokkotas Neutron Stars Neutron Stars: History I Walter Baade and Fritz Zwicky from CalTech first predicted the existence of neutron stars in 1934, but they were not discovered until over 30 years later. I Pulsars are a special category of spinning neutron stars, discovered in 1967 by Jocelyn Bell, an astronomy graduate student working with Prof. Antony Hewish at Cambridge. Pulsars derive their name from ”pulsating radio sources” because they were first observed at radio wave frequencies. Hewish won the 1974 Nobel Prize in Physics along with Sir Martin Ryle for their ”pioneering discoveries in radio astrophysics.” Hewish was cited for his ”decisive role in the discovery of pulsars.” Kostas Kokkotas Neutron Stars Neutron Stars: Supernovae A supernova explosion is usually associated with the formation of black holes and neutron stars. I Young stars are hydrogen, and the nuclear reaction converts hydrogen to helium with energy left over. The left over energy is the star’s radiation–heat and light. I When most of the hydrogen has been converted to helium, a new nuclear reaction begins that converts the helium to carbon, with the left over energy released as radiation. I This process continues converting the carbon to oxygen to silicon to iron. Nuclear fusion stops at iron. I A very old star has layers of different elements. The outer layers of hydrogen, helium, carbon, and silicon are still burning around the iron core, building it up. I Eventually, the massive iron core succumbs to gravity and it collapses to form a NS. The outer layers of the star fall in and bounce off the neutron core which creates a shock wave that blows the outer layer outward. I This is the supernova explosion. Kostas Kokkotas Neutron Stars Neutron Stars: Supernovae (sequence of events) I I I I Within about 0.1 sec, the core collapses. After about 0.5 sec, the collapsing envelope interacts with the outward shock. Neutrinos are emitted. Within 2 hours, the envelope of the star is explosively ejected. When the photons reach the surface of the star, it brightens by 8 orders of mag. Over a period of months, the expanding remnant emits X-rays, visible light and radio waves in a decreasing fashion. Kostas Kokkotas Neutron Stars Neutron Stars: Modelling I Stars more massive than ∼ 8M end in core collapse ( 90% are stars with masses ∼ 8 − 20M ). I Most of the material is ejected I If M > 20M more than 10 % falls back and pushes the PNS above the maximum NS mass leading to the formation of BHs (type II collapsars). I If M > 40M no supernova is launched and the star collapses to form a BH (type I collapsars) I Formation rate: 1-2 per century / galaxy I 5-40% of them produce BHs through the fall back material I Limited knowledge of the rotation rate! Initial periods probably < 20ms. Maybe about 10% of pulsars are born spinning with millisecond periods. Kostas Kokkotas Neutron Stars Neutron Stars: Maximum Mass I The mass of a stable neutron star can be estimated by using the following assumptions (Rhoades & Rufinni 1974): 1. GR is the correct theory of gravity and Tolman-Oppenheimer-Volkov equations describe the equilibrium structure. 2. The EoS satisfies the “microscopic stability” condition dP/dρ ≤ 0. If this condition is violated small elements of matter will spontaneously collapse. 3. The EoS should satisfy the causality condition dP/dρ ≤ c 2 i.e. the speed of sound is less than the speed of light. 4. The EoS below some critical density ρ0 is known The most fundamental from the above statements is the 3rd one which leads into P = P0 + (ρ − ρ0 )c 2 for ρ ≤ ρ0 . Rhoades & Rufinni adopted ρ0 = 4.6 × 1014 g/cm3 and one of the very first EoS (the Harisson-Wakano) and they got Mmax ≈ 3.2 4.6 × 1014 g/cm3 ρ0 Kostas Kokkotas Neutron Stars 1/2 M (57) Neutron Stars: Maximum Mass II Actually for uniform density stars one can proves that the maximum mass can be derived by studying the stability. That is by extremising the energy (for a fixed number of baryons): 2 ∂M = 0 and ∂ M > 0 (58) ∂χ ∂χ2 A A where M is the stellar mass, A is the total baryon number and sin2 χ = 2M/R. These condition imply that there is a critical adiabatic index (3ζ + 1 ζ + 1 ∂ ln P ρ dP Γ≡ = 1+ > Γc (χ) = (ζ+1) 1 + tan2 χ − 1 ∂ ln n P dρ 2 6ζ (59) where 6 cos χ ζ(χ) ≡ −1 9 cos χ − 2 sin3 χ/(χ − sin χ cos χ) in the Newtonian limit 4 89 M + 3 105 R ∼ 3.6M . Γc = and the maximum mass is : Mmax Kostas Kokkotas Neutron Stars (60) Neutron Stars: Maximum Mass III Figure: The stability domain in the uniform-density approximation. The curve separating the stable and unstable regions is the function Γc given by eqn (59). The dashed line shows the adiabatic index of a free neutron gas. Kostas Kokkotas Neutron Stars Neutron Stars: Maximum Rotation The shortest period for a rotating star occurs when it is rotating at break-up velocity called also Kepler limit: Ω2 R ∼ GM R2 (61) where Ω is the angular velocity of rotation. This can be written also in terms of the mean density ρ as: Ω2 ∼ G ρ (62) the universal relation between dynamical time-scale and density for a self-gravitating system. I For mean densities of 108 g/cm3 (white dwarfs) this relation gives a period: P= I 2π ≥ 1 sec Ω For mean densities of 1014 g/cm3 (neutron star) this relation gives a period: 2π P= ≥ 1 msec Ω Kostas Kokkotas Neutron Stars Neutron Stars: Magnetic Field During the collapse if we assume that the initial angular momentum and magnetic flux are conserved we will end up in a neutron star with msec period and magnetic field of the order of 1012 G. I Conservation of angular momentum 2 MR Ω = 2 MRNS ΩNS ⇒ PNS = P RNS R 2 ∼ 0.5ms (63) for P ∼ 1 month and R/RNS ∼ 7 × 105 . I Conservation of magnetic flux 2 R 2 B = RNS BNS ⇒ BNS = B R RNS 2 for initial B ∼ 100 Gauss. Kostas Kokkotas Neutron Stars ∼ 5 × 1012 Gauss (64) Neutron Stars: Pulsars I I From our earthly vantage point, pulsars appear to pulse with light with each rotation. I Their light, like a lighthouse beam, sweeps across the Earth. I Some pulsars emit visible light, X-rays, and even gamma-rays. I All pulsars are NS, but (so far as we know) not all NS are pulsars. Kostas Kokkotas Neutron Stars Neutron Stars: Pulsar Observation Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars - The magnetic dipole model The pulsar emission is derived from the kinetic energy of a rotating NS. In the oblique rotator model, it is assumed that the NS rotates uniformly in ~ oriented at vacuo at a frequency Ω and possess a magnetic dipole moment m an angle α to the rotation axis. ~ by A dipole magnetic field at the pole of the star BP is related to m |~ m| = 1 BP R 3 2 R is the radius of the star (65) The radiation at infinity due to the time-varying dipole moment will be: Ė = − B 2 R 6 sin2 α 4 2 ~ 2 |m̈| = ... = − P Ω 3 3c 6c 3 (66) The energy carried away by the radiation originates from the rotational kinetic energy of the NS E = I Ω2 /2, where I is the moment of inertia. Thus Ė = I ΩΩ̇ i.e. the pulsar slows down. The characteristic age T of pulsar is: Ω 6Ic 3 T =− = 2 6 2 BP R sin αΩ20 Ω̇ 0 Kostas Kokkotas Neutron Stars (67) (68) Neutron Stars: Pulsars - The magnetic dipole mode By integration we get the initial angular velocity 1/2 Ω2i t Ωi = Ω 0 1 + 2 2 Ω0 T where t is the present age of the pulsar given by: T Ω2 T t= 1 − 02 ≈ 2 2 Ωi (69) (70) For the Crab pulsar this method predicts 1260 yr while the true is 947 yr. The magnetic field strength can be estimated √ p 6c 3 I p 19 P Ṗ ≈ 5.2 × 10 P Ṗ Gauss (71) BP = 2πR 3 sin2 α For a typical NS (M = 1.4M , R = 12km and I = 1045 g/cm2 ) we get E = 1.8 × 1049 erg , Ė = 4.6 × 1038 erg/s (72) from the above estimation we get an approximate value for the magnetic field strength of the Crab pulsar. BP = 4.4 × 1012 G which is the same as the predicted value earlier. Kostas Kokkotas Neutron Stars (73) Neutron Stars: Pulsars - Ṗ vs P Diagram Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars - Ṗ vs P Diagram Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars - Braking index If the pulsar slowdown follows a power-law then we can write Ω̇ ≈ Ωn (74) the parameter n is called the braking index and for the magnetic dipole model n = 3. In general we can define ΩΩ̈ n≡ (75) Ω̇2 which suggests that the braking index can be measured directly from the pulsar frequency and its derivative. Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars and Gravitational Radiation Slight deformations of a pulsar (eg deformed ellipsoid) can be sources of GWs and account for their slow down. The energy loss in GW is: 32 G 2 2 6 I Ω (76) ĖGW = − 5 c5 Small ellipticities can generate GW emission. Magnetic fields of the order of 1015 -1016 Gauss can generate such ellipticities. The deceleration law for constant is ĖGW = I ΩΩ̇ ∼ Ω6 By integration we get the initial angular velocity 1/4 Ω4i t Ωi = Ω0 1 + 4 4 Ω0 TGW where t is the present age of the pulsar given by: Ω4 TGW TGW t= 1 − 04 ≈ 4 4 Ωi (77) (78) (79) For the Crab pulsar this method predicts 620 yr while the true is 947 yr i.e. gravitational radiation cannot alone be responsible for the Crab slowdown. A combination of gravitational and magnetic dipole radiation can be found which may give both the correct age and the observed pulsar deceleration rate. Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars - The Aligned Rotator Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars - The Aligned Rotator I I I I I I I Goldreich-Julian(1969) suggested that strong E-fields parallel to the surface will rip off charged particles creating a dense magnetosphere The charged particles in the magnetosphere within the light cylinder, must corotate with the pulsar. The light cylinder extends out to the distance at which the corotation velocity attains the speed of light. The corotation radius is thus given by c Rc ≡ = 5 × 104 P km (P : is the pulsar period in sec) (80) Ω As the particles approach Rc in the equatorial plane, they become highly relativistic. Whether or not the dipole field is aligned with the rotation axis plasma near the pulsar can in principle carry away sufficient angular momentum and energy to supply the necessary and energy to supply the necessary braking torque. B2 R6 The energy loss will be dictated again by Ė = − Pc 3 Ω4 which leads again to a breaking index n = 3. Particles accelerate along the B-field lines. Relativistic e − emit high energy EM radiation (γ-rays) which may disintegrate γ → e − + e + and this goes on! Kostas Kokkotas Neutron Stars Neutron Stars: Pulsars II Pulsars are divided into two main categories, isolated pulsars and binary pulsars. I Isolated pulsars (which include most radio pulsars) produce radiation primarily through their rotation, as they gradually slow down and cool off. Their light is generated by electrons caught in the pulsar’s strong magnetic field, concentrated and emitted near the magnetic poles. Thus, much of an isolated pulsar’s visible energy is funneled from the rotating magnetic pole region. The precise location of the beam is debated. I Pulsars in Binarys are those in orbit with a companion star, most often hydrogen-burning stars like our Sun. Such a pulsar can pull over, or accrete, matter from its stellar companion as their orbits bring these objects in close contact with each other. The violent accretion process can heat the gas being transferred and produce X-rays. The X-rays from this matter, under the influence of magnetic fields, can also appear to pulsate at the pulsar’s rotation rate. Kostas Kokkotas Neutron Stars Neutron Stars: Significant Pulsars I I I I I I I I I I I The first radio pulsar, CP 1919 (now known as PSR 1919+21), with a pulse period of 1.337 seconds and a pulse width of 0.04 second, was discovered in 1967 The first binary pulsar, PSR 1913+16, whose orbit is decaying at the exact rate predicted due to the emission of gravitational radiation by general relativity The first millisecond pulsar, PSR B1937+21 The brightest millisecond pulsar, PSR J0437-4715 The first X-ray pulsar, Cen X-3 The first accreting millisecond X-ray pulsar, SAX J1808.4-3658 The first extrasolar planets to be discovered orbit the pulsar PSR B1257+12 The first double pulsar binary system, PSR J0737?3039 The magnetar SGR 1806-20 produced the largest burst of energy in the Galaxy ever experimentally recorded on 27 December 2004 PSR J1748-2446ad, at 716 Hz, the pulsar with the highest rotation speed. PSR J0108-1431, the closest pulsar to the Earth. It lies in the direction of the constellation Cetus, at a distance of about 86 parsecs. Kostas Kokkotas Neutron Stars Neutron Stars: X-ray binaries I Binary pulsars are often called X-ray binaries. The flow of matter from the stellar companion, the accretion disk, also glows in X-rays owing to its high temperature. While some X-ray binaries are steady X-ray sources, others are bright only for a few weeks or months at a time, lying dormant for years between outbursts. The X-ray sky is thus highly variable, unlike the visible night sky. I X-ray binaries were first discovered by R. Giacconi and collaborators in the early ’60s, with their binary nature established in the early ’70s by Giacconi and others. Giacconi was awarded the 2002 Nobel Prize in Physics for ”pioneering discoveries in astrophysics, which have led to the discovery of cosmic X-ray sources.” Kostas Kokkotas Neutron Stars Neutron Stars: Cannibals I The process of accretion can speed up the spin of a binary pulsar, since the high-velocity accreting material hits the pulsar at a grazing angle, constantly spinning it faster. I Rotation rates for such accretion-powered pulsars can reach hundreds of Hz, nearly kHz (millisecond pulsars) I The transfer of material onto the neutron star also slowly cannibalizes the stellar companion so that, over millions of years, enough matter is accreted away to completely whittle away a once healthy star. I There are pulsars orbiting stars with as little as 10 Jupiter masses. One well-known example is the pulsar B1957+20, known as the ”Black Widow” pulsar, which is emitting a stream of high-energy radiation that will soon blow away its now feeble companion so that no trace of this star remains. Kostas Kokkotas Neutron Stars Neutron Stars: Millisecond Pulsars I I I There are many known isolated millisecond pulsars detected in the radio-wave regime. The fastest known is the pulsar PSR J1748-2446ad, spinning at an astonishing 716 Hz (2nd 649Hz and 3rd at 588 Hz). It is believed that these isolated millisecond radio pulsars were once X-ray binary millisecond pulsars that accreted material, spun up, and cannibalized their companions. This link was first confirmed in 1998 when scientists using the Rossi X-Ray Timing Explorer found that the X-ray binary millisecond pulsar SAX J1808.4-3658 was accreting matter from a companion with only 0.05M . Kostas Kokkotas Neutron Stars Neutron Stars: Population I The Galaxy is thought to contain about 100,000 pulsars, yet about two thousand are known. I Only a small fraction of these are millisecond pulsars. I Isolated pulsars are hard to find because they are dim. I X-ray binary millisecond pulsars are discovered when they flare up in a rare, sporadic accretion event that lasts only a few weeks. In many X-ray binaries, however, the accreted material on the neutron star is occasionally consumed in a violent thermonuclear explosion that sweeps across the surface, generating a bright burst of X-rays lasting a few seconds. I Using the Rossi Explorer in 1996 discovered rapid X-ray flickering during some of these explosions, called ”X-ray burst oscillations,” and suggested that this flickering might offer another way to measure the spin rates of neutron stars. An advantage of this approach is that burst oscillations are bright and fairly common. Kostas Kokkotas Neutron Stars