Mass, Radius and Moment of Inertia of Neutron Stars

Transcription

1 Sapienza Università di Roma ICRA & ICRANet X-Ray Astrophysics up to 511 KeV Ferrara, Italy, September 2011

2 Standard approach to neutron stars (a) In the standard picture the structure of neutron stars is given by: Core: mainly neutrons and a small presence of protons and electrons in β-equilibrium local charge neutrality n e = n p in order to close the system of equations a priori neglecting of the electromagnetic interactions ρ ρ 0, where ρ g cm 3 is the nuclear saturation density for ordinary nuclei Crust: inner crust: nuclei lattice (Coulomb lattice) in a background of electrons and neutrons; ρ 0 ρ ρ drip, where ρ drip g cm 3 is the neutron-drip density outer crust: nuclei lattice in a white-dwarf-like material (free electrons); ρ ρ drip (Note that the core-crust transition happens when the uniform phase becomes energetically favorable with respect to the non-uniform Coulomb lattice)

3 Standard approach to neutron stars (b) The system of equation for such an object is given by local charge neutrality and β-equilibrium in addition to the Einstein equations: dp dr dm dr = G(E +P)(M +4πr3 P) r 2 (1 2GM, ) r = 4πr 2 E, where M = M(r) is the mass enclosed at the radius r and the energy-density E and the pressure P are given by E = E i, P = P i, i=n,p,e being E i and P i the single fermion fluid contributions E i = 2 P F i (2π) 3 0 P i = (2π) 3 P F i 0 i=n,p,e ǫ i (p)4πp 2 dp, p 2 ǫ i (p) 4πp2 dp.

4 New approach to neutron stars We follow a new description for neutron stars: Core: degenerate gas of neutrons, protons and electrons in β-equilibrium global charge neutrality N e = N p the electromagnetic interactions are taken into account gravitational-electrodynamical-strong-weak equilibrium ensures the constancy of the general relativistic Fermi energy E F i of all particles species ρ ρ 0 as in the standard picture Crust: inner crust: disappears in favor of a tiny transition layer (details in the next slides) outer crust: as in the standard description, nuclei lattice in a white-dwarf-like material (free electrons); ρ ρ drip

5 Core: Quantum Hadrodynamical Model (a) How describe electromagnetic interaction between electrons and protons plus nuclear interaction??? We follow the so-called Walecka model, in which the nucleons interact through a Yukawa coupling, so that the strong interaction is modeled by a meson-exchange of the σ, ω and ρ virtual meson-fields. σ=isoscalar meson fields attractive long-range nuclear force ω=massive vector field repulsive short-range nuclear force ρ=massive isovector field surface effects modeling a repulsive nuclear force

6 Core: Quantum Hadrodynamical Model (b) The total Lagrangian density of the system has to take into account gravitational, strong, weak and electromagnetic interactions and is given by L = L g +L f +L σ +L ω +L ρ +L γ +L int, L g = R 16πG, L γ = 1 4 FµνFµν, L σ = 1 2 µσ µ σ U(σ), L ω = 1 4 ΩµνΩµν m2 ωω µω µ, L ρ = 1 4 RµνRµν m2 ρρ µρ µ, L f = ψ i (iγ µ D µ m i )ψ i, i=e,n L int = g σσ ψ N ψ N g ωω µj µ ω g ρρ µj µ ρ + ea µj µ γ,e ea µj µ γ,n,

7 Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (a) We consider non-rotating spherically symmetric neutron stars, so we introduce the spacetime metric ds 2 = e ν(r) dt 2 e λ(r) dr 2 r 2 dθ 2 r 2 sin 2 θdϕ 2. Since the very large number of fermions ( ), we can adopt the mean-field approximation, so that the full system of general relativistic constitutive equations become ) dλ 1 e λ(r) ( 1 r 2 1 r e λ(r) ( 1 r r dr dν dr V + 2 [ r V 1 r(ν + λ ) 4 d 2 σ dr + dσ [ 2 2 dr [ 2 d 2 ω dr + dω 2 d 2 ρ dr + dρ 2 dr r 2 = 8πGT0 0, ) 1 r = 2 8πGT1 1, r 1 ( dν 2 dr + dλ dr dr r 1 ( dν 2 dr + dλ dr [ 2 r 1 ( dν 2 dr + dλ dr ] = 4πe e ν/2 e λ (n p n e), )] = e λ [ σu(σ) + g sn s], )] = e λ ( g ωj 0ω m2ω ω ), )] = e λ ( g ρj 0ρ m2ρ ρ ), E F e = e ν/2 µ e ev = constant, (1) E F p = eν/2 µ p + V p = constant, E F n = eν/2 µ n + V n = constant.

8 Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (b) T 0 0 = E +(E2 /8π)+U(σ)+(1/2)C ωn 2 b +(1/2)Cρn2 3 T 1 1 = P +(E2 /8π)+U(σ) (1/2)C ωn 2 b +(1/2)Cρn2 3 J ω 0 = (nn +np)eν/2 = n b u 0 ;J ρ 0 = (np nn)eν/2 = n 3 u 0 ;J ch 0 = (n p n e)e ν/2 = n ch u 0 E F n = E F p +E F e m N = m N +g sσ; m e = m e V p = g ωω +g ρρ+ev V n = g ωω g ρρ n e = e 3ν/2 3π 2 [ˆV 2 +2m e ˆV m 2 e (e ν 1)] 3/2,whereˆV ev +E e n s = ψ N ψ N = 2 (2π) i=n,p d 3 3 k m N ǫ i (p) e λ(r) = 1 2GM(r) r +Gr 2 E 2 (r) = 1 2GM(r) r + GQ2 (r) r 2

9 Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (c) In order to integrate the equilibrium equations we need to fix the parameters of the nuclear model, namely, fixing the coupling constants g s, g ω and g ρ, and the meson masses m σ, m ω and m ρ. Conventionally, such constants are fixed by fitting experimental properties of nuclei. Usual experimental properties of ordinary nuclei include saturation density, binding energy per nucleon (or experimental masses), symmetry energy, surface energy, and nuclear incompressibility. In the following table we present selected fits of the nuclear parameters. In particular, we show the following parameter sets: NL3, NL-SH, TM1, and TM2. NL3 NL-SH TM1 TM2 m σ (MeV) m ω (MeV) m ρ (MeV) g s g ω g ρ g 2 (fm 1 ) g c Table: Selected parameter sets of the σ-ω-ρ model.

10 Core-Crust transition layer (a) the Coulomb potential energy inside the core of the neutron star is found to be m πc 2 and related to this there is an internal electric field of order E c where E c is the critical field for vacuum polarization. at the core-crust boundary the continuity of all general relativistic particle Fermi energies guarantee a self-consistent matching of the core and the crust: a core-crust transition surface of thickness λ e = /(m ec) 100fm (electron screening scale) is developed in which an overcritical electric field appears. the continuity of the electron Fermi energy force the variation of electron chemical potential at the core-crust boundary to be of order µ e(core) µ e(crust) ev(core) m πc 2. therefore we obtain a suppression of the so-called inner crust of the neutron star if µ e(crust) µ e(core)-ev(core) 25 MeV, which is approximately the value of the electron chemical potential at the neutron drip point.

11 Core-Crust transition layer (b) At the transition-point from the core to the crust of the neutron star, the mean-field-approximation for the meson-fields is not valid any longer and therefore full numerical integration of the meson-field equations of motion, taken into account all gradient terms, must be performed. Since,as can be shown numerically, the metric functions hold almost constant on the core-crust transition layer and then have their values at the core-radius e νcore e ν(rcore) and e λcore e λ(rcore), the system of equations can be reduced to V + 2 r V = e λcore ejch 0, σ + 2 r σ = e λcore [ σu(σ)+g sn s], ω + 2 r ω = e [ λcore g ωjω 0 mωω 2 ], ρ + 2 r ρ = e [ λcore g ρjρ 0 mρρ 2 ], e νcore/2 µ e ev = constant, e νcore/2 µ p +ev +g ωω +g ρρ = constant, µ n = µ p +µ e +2g ρρe νcore/2.

12 Core-Crust transition layer (c) E 1000 Ec E 150 Ec r Rcore ΛΣ r Rcore ΛΣ Figure: Left:g ρ 0.Right : g ρ = 0 Left region: on the proton-profile we can see a bump due to Coulomb repulsion while the electron-profile decreases. Such a Coulomb effect is indirectly felt also by the neutrons due to the coupled nature of the system of equations. However, the neutron-bump is much smaller than the one of protons and it is not appreciable due to the plot-scale. Central region: the surface tension due to nuclear interaction produces a sharp decrease of the neutron and proton profiles in a characteristic scale λ π. Moreover, it can be seen a neutron skin effect, analogous to the one observed in heavy nuclei, which makes the scale of the neutron density falloff slightly larger with respect to the proton one. Right region: smooth decreasing of the electron density similary to the behavior of the electrons surrounding a nucleus in the Thomas-Fermi model

13 Crust, two different EoS (a) Due to the neutrality of the crust, the structure equations to be integrated in the crust are the Tolman-Oppenheimer-Volkoff equations. In order to see the effects of the microscopic screening on the structure of the configuration we will consider two equations of state for the crust: the locally neutral case (uniform approximation) used for instance by Chandrasekhar in the study of white dwarfs the equation of state due to Baym, Pethick and Sutherland (BPS), which is by far the most used equation of state in literature for the description of the neutron star crust

14 Crust, two different EoS (b) In the uniform approximation, both the degenerate electrons and the nucleons distribution are considered constant inside each cell of volume V ws. This kind of configuration can be obtained only imposing microscopically the condition of local charge neutrality n e = Z V ws. (2) The total pressure of the system is assumed to be entirely due to the electrons, i.e. P = P e = 2 P F e 3(2π ) 3 0 c 2 p 2 4πp 2 (3) c 2 p 2 +mec 2 4dp, while the total energy-density of the system is due to the nuclei, i.e. E=(A/Z)m N n e, where m N is the nucleon mass.

15 Crust, two different EoS (c) In the BPS model, we have a point-like nucleus surrounded by uniformly distibuted electrons. The sequence of the equilibrium nuclides present at each density in the BPS equation of state is obtained by looking for the nuclear composition that minimizes the energy per nucleon for each fixed nuclear composition (Z, A). P = P e W Ln N, where the electron energy-density is given by E = W N +W L + Ee(n bz/a), n b A n b E e = 2 P F e (2π) 3 p 2 +me4πp 2 2 dp, 0 and W N (A,Z) is the total energy of an isolated nucleus given by the semi-empirical formula W N = m nc 2 (A Z)+m pc 2 Z ba, with b being the Myers and Swiatecki binding energy per nucleon. The lattice energy per nucleus W L is given by W L = Z2 e 2. a

16 Results (a) M/M NL3 NL SH TM1 TM ρ(0)/ρ nuc M/M NL3 NL SH TM1 TM R(km) GM/(c 2 R) GM/(c 2 R) NL3 NL SH TM1 TM NL3 NL SH TM1 TM M/M R (km) Figure: Top:Mass-central mass-density relation (left panel) and mass-radius relation (right panel),in the BPS-EoS case. The mass has been plotted in solar masses, the mass density in nuclear mass density ρ nuc = g cm 3, and the star radius is given in km. Bottom: compactness of the star GM core/(c 2 R core) as a function of the star mass M core (left panel) and as a function of the star radius R core (right panel), for the BPS-EoS

17 Results (b) M NO C crust (10 5 M ) NL3 NL SH TM1 TM2 R NO C crust (km) NL3 NL SH TM1 TM M/M M/M 5 4 NL3 NL SH TM1 TM NL3 NL SH TM1 TM2 M BPS crust (10 5 M ) 3 2 R BPS crust (km) M/M M/M Figure: Top: mass versus compactness and crust-thickness versus compactness for crust without Coulumb-interactions. Bottom: mass versus compactness and crust-thickness versus compactness for crust with Coulumb-interactions.

18 Crust comparison (a) Equilibrium Nuclei Below Neutron Drip Nucleus Z ρ max(g cm 3 ) R 1 (km) R.A.1(%) R 2 (km) R.A.2(%) 56 Fe Ni Ni Se Ge Zn Ni Fe Mo Zr Sr Kr Table: ρ max is the maximum density at which the nuclide is present; R 1, R 2 and R.A.1(%), R.A.2(%) are rispectively the thickness of the layer where a given nuclide is present and their relative abundances in the outer crust for two different case, one with M core = 2.6M and R core = 12.8Km, and one with M core = 1.4M and R core = 11.8Km.

19 Crust comparison (b) Fe 56 Ni 63 Ni 64 Se 84 Ge 82 Zn 80 Ni 78 Fe 76 Mo 124 Zr 122 Sr 120 Kr Fe 56 Ni 63 Ni 64 Se 84 Ge 82 Zn 80 Ni 78 Fe 76 Mo 124 Zr 122 Sr 120 Kr 118 Figure: Relative abundances of chemical elements in the outer crust of neutron stars for the two different cores: Mcore=2.6M, Rcore=12.8 Km and Mcore=1.4M, Rcore= 11.8 Km. In both cases we obtain as average nuclear composition Br Using these value for < A > and < Z > in the Chandrasekhar approximation we obtain: M chandra crust 1.18M BPS crust and Rchandra crust 0.95 R BPS crust

20 Observational constraints M/M NL3 NL SH TM1 TM R (km) Figure: Constraints on the mass-radius relation given by J. E. Trümper and the theoretical mass-radius relation presented in this slides. The solid line is the upper limit of the surface gravity of XTE J , the dotted-dashed curve corresponds to the lower limit to the radius of RX J , the dashed line is the constraint imposed by the fastest spinning pulsar PSR J ad, and the dotted curves are the 90% confidence level contours of constant R of the neutron star in the low-mass X-ray binary X7. Any mass-radius relation should pass through the area delimited by the solid, the dashed and the dotted lines and, in addition, it must have a maximum mass larger than the mass of PSR J , M = 1.97 ± 0.04M.

21 Magnetic field and momentum of inertia (a) The upper limit on the magnetic field of a pulsar, obtained by requesting that the rotational energy loss due to the dipole field be smaller than the electromagnetic emission of the dipole, is given by ( 3c 3 ) 1/2 I B = 8π 2 R 6PṖ, where P and Ṗ are the rotational period and the spin-down rate of the pulsar which are observational properties, and the moment of inertia I and the radius R of the object are model dependent properties.

22 Magnetic field and momentum of inertia (b) The general relativistic moment of inertia of the star can be obtained from the integral I = 8π R r 4 (E + P)e λ ν 2 dr, 3 0 where the effects due to rotation and deformation are neglected. Due to the smallness of the rotation rates of these pulsars as compared to the maximum rotation rate of a neutron star, the effects of rotation can be obtained by using the Hartle-Thorne formalism which approximately describes the gravitational field of rotating stars up to second order in the angular velocity and up to first order in the quadrupole deformations. Within this formalism, the moment of inertia is given by I = 8π R r 4 (E + P)e λ ν ω Ω dr, where ω is the angular velocity of the fluid relative to the local inertial frame and Ω is the uniform angular velocity of the star. The function ω is obtained from the differential equation ( 1 d r 4 r 4 j d ω ) + 4 dj ω = 0, dr dr r dr where j(r) = exp (λ + ν)/2, being λ(r) and ν(r) the metric functions of the corresponding non-rotating configuration I (10 45 g cm 2 ) non rotating Hartle Thorne M/M Figure: Total momentum of inertia I as a function of the total mass M, for the NL3 parametrization.

23 Implications on the Magnetic-dipole model of Pulsar High-Magnetic Field Pulsar B/B c J J J J J Table: Magnetic fields of the overcritical high-magnetic field pulsars assuming a canonical neutron star of M = 1.4M, R = 10 km and I = g cm B/B c 0.8 B/B c F.D J J J J J J J J J J M/M M/M Figure: Magnetic field B, in units of the critical field B c = m 2 e c3 /(e ) G, versus the total mass of the star, for the NL3 parametrization without and with rotation effects (left and right panel respectively). In particular, we plot the magnetic fields of the high-magnetic field pulsars of previous table.

24 End of the talk Thank you