Journal of Physics: Conference Series PAPER OPEN ACCESS Cooling of neutron stars and emissivity of neutrinos y the direct Urca process under influence of a strong magnetic field To cite this article: E L Coelho et al 2016 J. Phys.: Conf. Ser. 706 052011 View the article online for updates and enhancements. Related content - Effects of delta mesons on aryonic direct Urca processes in neutron star matter Huang Xiu-Lin, Wang Hai-Jun, Liu Guang- Zhou et al. - Effects of Tensor Couplings on Nucleonic Direct URCA Processes in Neutron Star Matter Yan Xu, Xiu-Lin Huang, Cheng-Zhi Liu et al. - Cooling of hyrid stars with spin down compression Miao Kang, Xiao-Dong Wang and Na-Na Pan This content was downloaded from IP address 46.3.197.84 on 11/02/2018 at 08:15
IOP Pulishing Cooling of neutron stars and emissivity of neutrinos y the direct Urca process under influence of a strong magnetic field E L Coelho 1,2, M Chiapparini 1 and R P Negreiros 3 1 Instituto de Física, Universidade do Estado do Rio de Janeiro, 20559-900, Rio de Janeiro, Brazil 2 Current address: Universidade Veiga de Almeida, 20271-020, Rio de Janeiro, Brazil 3 Instituto de Física, Universidade Federal Fluminense, 24210-346, Rio de Janeiro, Brazil E-mail: eduardo.coelho@uva.r Astract. Neutron stars are orn with high temperatures and during a few seconds suffer rapid cooling y emission of neutrinos. The direct Urca process is the main mechanism to explain this loss of energy. In this work we study the influence of a strong magnetic field on the composition of nuclear matter at high densities and zero temperature. We descrie the matter through a relativistic mean-field model with eight light aryons (aryon octet), electrons, muons magnetic field. As output of the numerical calculations, we otain the relative population for a parametrized magnetic field. We calculate the cooling of neutron stars with different mass and magnetic fields due to direct Urca process 1. Introduction Strong magnetic fields of magnitudes up to 10 14 G are suppose to exists at the surface of pulsars. So, it is of interest to study the properties of nuclear matter in the presence of such strong magnetic fields. In this work we study the influence of a strong magnetic field on the composition of nuclear matter at T = 0, and investigate the cooling of neutron stars due to the direct Urca process using the resulting equation of state. The matter at high densities is descried using a relativistic mean field (MF) theory which descries correctly the nuclear ground state properties and elastic scattering of nucleons. The Lagrangian that descries this model, with a uniform magnetic field B along the z axis, is given y [1 L = ψ [iγ D m + g σ σ g ω γ ω 1 2 g ργ τ ρ ψ + 1 2 σ σ 1 2 m2 σσ 2 U(σ) 1 4 ω νω ν + 1 2 m2 ωω ω 1 4 ρ ν ρ ν + 1 2 m2 ρρ ρ + l=e, ψl [iγ ( + iq l A ) m l ψ l 1 4 F νf ν, (1) where D = + iq A, A 0 = 0, A = (0, xb, 0), q and q l are the electric charge of aryons and leptons, ψ is the Dirac spinor for aryon in the octet {n, p, Λ, Σ, Ξ} with mass m ; m σ, m ω, Content from this work may e used under the terms of the Creative Commons Attriution 3.0 licence. Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI. Pulished under licence y IOP Pulishing Ltd 1
IOP Pulishing m ρ and g σ, g ω, g ρ are the masses and coupling constants of mesons σ, ω, ρ respectively. The summation in the first line represents the free Lagrangian of aryons together with the interaction etween aryons and mesons; The mesonic and electromagnetic field strength tensors are ω ν = ω ν ν ω, (2) ρ ν = ρ ν ν ρ, (3) F ν = A ν ν A. (4) The aryon octet, showing their quantum numers, charge q and isospin projection I 3 are shown in Tale 1. Tale 1. The aryon octet, showing their quantum numers, charge q, isospin projection I 3. Species Mass q I 3 (MeV) (e) p 939 1 1/2 n 939 0 1/2 Λ 1115 0 0 Σ + 1190 1 1 Σ 0 1190 0 0 Σ 1190 1 1 Ξ 0 1315 0 1/2 Ξ 1315 1 1/2 The Lagrangian of leptons (electrons and muons) is written in the third line while the scalar self-interactions term is given y U(σ) = 1 3 m n(g σn σ) 3 + 1 4 c(g σnσ) 4, (5) where and c are constants. The dynamic equations of nucleon and mesons (Dirac and Klein-Gordon equations respectively) are otained from the Euler-Lagrange equations L φ(x) L ( φ) = 0, (6) where φ(x) is the corresponding field. The resulting equations of motion in the mean field approximation are given y ( ) 2 gωn g ωn ω 0 = χ ω ρ, (7) g ρn ρ 03 = m ω ( ) 2 gρn χ ρ I 3 ρ, (8) m ρ ( ) 2 m gσn n = m n + [ m n (m n m m n) 2 + c(m n m n) 3 σn χ σ n s, (9) where I 3 is the 3-component of the isospin of the aryon, m n = m n χ σ g σn σ, χ σ = g σ /g σn, χ ω = g ω /g ωn, χ ρ = g ρ /g ρn ; The scalar density is given y n s = n q = 0 s + n q 0 s, (10) 2
IOP Pulishing where n q = 0 s = m 2π 2 [ ( ) k m 2 ln + k m, (11) n q s 0 = m q ν max() [ B 2π 2 g ν ln + k,ν m ν=0,ν, (12) with m 2,ν = m 2 + 2ν q B, (13) k 2,ν = 2 m 2 2ν q B, (14) where q and are the electric charge and effective chemical potential of aryon, k and k,ν are the Fermi momentum of neutral and charge aryons respectively. ν is the Landau principal quantum numer, which can take all possile positive integer values including zero. The upper limit ν max() is defined y the condition k 2,ν 0, then ν max() = int [ 2 m 2 2 q B The effective chemical potential of aryons are given y. (15) = χ ω g ωn ω 0 χ ρ g ρn I 3 ρ 03. (16) They are constrained due to the β-equilirium condition, which reads = n q e, (17) e =, (18) where n and e are the chemical potentials of neutron and electron respectively. The aryon densities are ρ q=0 = k3 3π 2, (19) ρ q 0 = q B 2π 2 ν max() while the lepton (electrons and muons) densities are with ρ l = q l B 2π 2 ν=0 ν max(l) ν=0 g ν k, ν (20) g ν k l,νl, (21) k 2 l,ν l = u 2 l m 2 l 2ν l q l B (22) and ν max(l) = int [ 2 l m 2 l 2 q l B, (23) 3
IOP Pulishing where m 2, ν = m 2 + 2ν q B, ν is the Landau principal quantum numer and the Landau level degeneracy g ν is 1 for ν = 0 and 2 for ν > 0. Neutron star matter satisfies the constraints of conservation of aryon numer and neutrality of electric charge, which reads ρ = 0 = ρ, (24) q ρ + q l ρ l. l=e, (25) The energy density due to the matter is given y [2 ( mσn ( mωn ) 2 (g ωn ω 0 ) 2 where ) 2 (g σn σ) 2 + 1 2 ε m = 1 3 m n(g σn σ) 3 + 1 4 c (g σnσ) 4 + 1 2 g σn g ωn + 1 ( ) 2 mρn (g ρnρ 03) 2 + 1 2 g ρn 8π 2 [2 3 k m 2 k m 4 ln (q=0) + q B ν max() { } 4π 2 g ν [ k, ν + m 2, ν ln + k, ν + q B 4π 2 (q 0) l=e, ν =0 ν max(l) ν l =0 g ν [ l k l, νl + m 2 l, ν l ln m, ν } { l + k l, νl m l, νl, { } + k m (26) m 2 l, ν l = m 2 l + 2ν l q l B. (27) The matter pressure is given y P m = n ρ ε m. (28) Next, we add to these the contriution from electromagnetic field tensor, otaining the total energy density and pressure as The magnetic field is parametrized y ε = ε m + B2 2, (29) P = P m + B2 2. (30) B(ρ/ρ 0 ) = B surf + B 0 [1 exp{ β(ρ/ρ 0 ) γ }, (31) where B surf = 10 8 G and B 0 = 10 19 G are the magnetic fields at the surface and the center of the star respectively, with the parameters β = 10 4 and γ = 17, and ρ 0 is the saturation density. The system of coupled nonlinear equations (7-9) with constraints (24-25) is solved numerically y iteration. We use the Newton-Raphson method with gloal search of the solution. We adopt natural units. As output, we otain the relative population of each specie of particles as a function of the aryon density, and the energy density and pressure with and without magnetic field. The coupling constants are given in Tales 2, 3 e 4. 4
IOP Pulishing Tale 2. Nucleon-meson coupling constants to compression K = 300 MeV and m /m = 0.70 [1. ( g σn ) 2 m σ ( g ωn ) 2 m ω ( g ρn ) 2 m ρ c (fm 2 ) (fm 2 ) (fm 2 ) 11.79 7.148 4.410 0.002947-0.001070 Tale 3. Parametrizations used for the hyperon coupling constants [3. χ σλ χ σσ χ σξ χ ωλ = χ ωσ χ ωξ 0.6106 0.4046 0.3195 2/3 1/3 Tale 4. Parametrizations used for the hyperon coupling constants [3. χ ρλ χ ρσ χ ρξ 1 1 1 1 n 0.1 p e - Relative Population (ρ i /ρ ) 0.01 0.001 - Λ Ξ - 0.0001 1e-05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Baryon Density (fm -3 ) Figure 1. The particle fractions in cold β-equilirated neutron star without magnetic field. The Fig 1 and Fig 2 show the results without and with an parametrized magnetic field, respectively. It can e seen that the incorporation of strong magnetic field increases the proton and electron fraction. This effect is important for the neutrino emissivity due to the direct Urca process, having an impact on the cooling of neutron stars. The direct Urca process is the most powerful mechanism of neutrino emission in the core of neutron stars. Neutron star cooling y Urca process may provide important informations aout the interior composition of the star. The reaction is given y n p + e + ν e, 5
IOP Pulishing 1 n 0,1 p e 0,01 (ρ i /ρ ) 0,001 Λ Ξ 0,0001 1e 05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 ρ (fm 3 ) Figure 2. The particle fractions in cold β-equilirated neutron star for the parametrized magnetic field. p + e n + ν e. This process may occur if the proton fraction is large enough where k F n k F p + k F e, (32) k F α = (3π 2 ρ α ) 1/3, (33) in order to conserve momentum in the reaction. As we have showed aove, strong magnetic fields lead to an increase of the proton fraction and the cooling of neutron stars is more efficient. In the Weinerg-Salam theory for weak interactions, the interaction Lagrangian is given y [4 L weak = G F 2 cos θ c l j, where G F is the Fermi weak coupling constant and θ c is the Caio angle. The Lepton and nucleon charged weak currents are l = ψ 4 γ 3 (1 γ 5 )ψ 2, (34) j = ψ 3 γ (g V g A γ 5 )ψ 1, (35) g V and g A are vector and axial-vector coupling constants and the indices i = 1 4 refer to the n, ν e, p and e, respectively. The wave functions for neutron and antineutrino are plane wave functions. The wave functions for oth protons and electron in the presence of a magnetic field 6
IOP Pulishing strong enough that only the ground Landau level is occupied are given y ψ 3 (X) = ψ 4 (X) = 1 Ly L z exp( ie 3t + ik 3y y + ik 3z z)f ν 3=0 k 3y,k 3z, (36) 1 Ly L z exp( ie 4 t + ik 4y y + ik 4z z)f ν 4=0 k 4y,k 4z, (37) where f ν 3=0 k 3y,k 3z and f ν 4=0 k 4y,k 4z are the 4-component spinor solutions of the corresponding Dirac equation. The only positive energy spinor for protons in the chiral representation is [5 f ν 3=0 k 3y ;k 3z (x) = N ν3 =0 E 3 + k 3z 0 m 3 0 I ν 3 =0;k 3y, (38) where N ν3 =0 = I ν3 =0;k 3y = 1 [2E3 (E 3 + k 3z) 1/2, E 3 = (k3z 2 + m 3 2 ) 1/2, (39) ( ) eb 1/4 exp [ 1 ( π 2 eb x k ) 2 [ 3y 1 2eB ( eb ν3! H ν 3 x k ) 3y, eb and are the Hermite polynomial and e is the electron charge. The emissivity due to the antineutrino emission process in presence of a uniform magnetic field B along z-axis is [? V d 3 k 1 ɛ ν = 2 (2π) 3 V d 3 k 2 (2π) 3 qblx/2 qbl x/2 L y dk 3y 2π Lz dk 3z 2π qblx/2 qbl x/2 L y dk 4y 2π Lz dk 4z 2π E 2W fi f 1 [1 f 3 [1 f 4, (40) where the pre-factor 2 takes into account the neutron spin degeneracy and f i is the Fermi-Dirac distriution functions. By the Fermi s golden rule, the transition rate per unity volume W fi is W fi = M fi 2, (41) tv where t is the time, V = V x V y V z is the normalization volume and the matrix element for the V-A interaction is given y M fi = G F 2 d 4 X ψ 1 (X)γ (g V g A γ 5 )ψ 3 (X) ψ 2 (X)γ (1 γ 5 )ψ 4 (X), (42) where. denotes an averaging over the initial spin of n and a sum over spins of final particles (p, e). Then, the transition rate per unit volume is W fi = G 2 ( F 1 E1 E 2E3 E 4 V 3 exp (k 1x k 2x ) 2 + (k 3y + k 4y ) 2 ) L y L z 2eB [(g V + g A ) 2 (k 1.k 2 )(k 3.k 4 ) + (g V g A ) 2 (k 1.k 4 )(k 3.k 2 ) (g 2 V g 2 A)m 2 (k 4.k 2 )(2π) 3 δ(e 1 E 2 E 3 E 4 ) δ(k 1y k 2y k 3y k 4y )δ(k 1z k 2z k 3z k 4z ). 7
IOP Pulishing Then, the emissivity is ɛ ν = 457π ( 5040 G2 F cos 2 θ c (eb)[(g V + g A ) 2 (g 2 V g 2 A) m 2 3 1 1 k F 3 3 exp [ (kf3 + k F4 ) 2 k 2 F 1 2eB ) + (g V g A ) 2 ( 1 k F 1 1 3 4 k F3 k F4 T 6 Θ, n cos θ 14 ) cos θ 14 = (k 2 F 1 + k 2 F 4 k 2 F 3 )/2k F1 k F4, T is the temperature, k F i is the Fermi momentum and the threshold factor is Θ = θ(k F3 + k F4 k F1 ), with θ(x) = 1 (x > 0), θ(x) = 0 (otherwise). 2. Results Figure 3 shows the cooling due to the direct Urca process of neutron stars with 1.4 and 1.6 solar masses (continuous and dashed lines, respectively) for the cases B = 0 (lue line) and B(ρ/ρ 0 ) (red line). We can see that the cooling is more intense with the increase in the mass of the star and for the case B(ρ/ρ 0 ). Theses differences may e attriuted to the increase of proton and electron fractions with the mass and magnetic field of the star and to the phase space modifications. 10 7 B (ρ/ρ 0 ), Μ = 1.4 M sun 10 6 B (ρ/ρ 0 ), Μ = 1.6 M sun B = 0, M = 1.4 M sun B = 0, M = 1.6 M sun T (K) 10 5 10 4 10-1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Age (years) Figure 3. Cooling of a neutron star mass with 1.4 and 1.6 solar masses (continuous and dashed lines, respectively) for the cases B = 0 (lue line) and B(ρ/ρ 0 ) = B surf +B 0 [1 exp{ β(ρ/ρ 0 ) γ } (red line). [1 Glendenning N K 2000 Compact Stars (New York: Springer). [2 Chakraarty S, Bandyopadhyay D and Pal S 1997 Phys. Rev. Letter 78 2898. [3 Chiapparini M, Bracco M E, Delfino A, Malheiro M, Menezes D P and Providência C 2009 Nuclear. Phys. A 826 178. 8
IOP Pulishing [4 Bandyopadhyay D, Chakraarty S, Dey P and Pal S 1998 Phys. Rev. D 58 121301. [5 Koayashi M and Sakamoto M 1983 Progress of Theoretical Phys. 70 5. 9