Rapid Cooling of Magnetized Neutron Stars

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Rapid Cooling of Magnetized Neutron Stars View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Rapid co oling of magnetized neutron stars 1 2;3 2 1 Debades Bandyopadhyay, Somenath Chakrabarty, Prantick Dey, and Subrata Pal 1 Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India 2 Department of Physics, University of Kalyani, Kalyani 741235, India 3 IUCAA, P.B. 4, Ganeshkhind, Pune 411007, India Abstract The neutrino emissivities resulting from direct URCA pro cesses in neutron stars are calculated in a relativistic Dirac-Hartree approach in presence of a magnetic eld. In a quark or a hyp eron matter environment, the emissivity due to nucleon direct URCA pro cesses is suppressed relative to that from pure nuclear matter. In all the cases studied, the magnetic eld enhances emissivity compared to the eld-free cases. PACS numb ers: 26.60.+c, 21.65.+f, 12.39.Ba, 97.60.Jd Typ eset using REVT X E 1 Neutron stars are b orn in the aftermath of sup ernova explosions with interior temp era- > 11 tures T 10 K, but co ol rapidly in a few seconds by predominant neutrino emission [1] 10 5 6 to T < 10 K. Neutrino co oling then dominates and lasts for t 10 10 yr and sub- < 8 sequently photon emission takes over when T 10 K. Since the long term co oling of the 8 10 young neutron stars (T 10 10 K) pro ceeds via emission of neutrinos primarily from matter at supranuclear densities within the core, the study of the co oling of neutron stars by examination of neutrino emissivities may provide considerable insight into their interior structure and comp osition. For a long time the dominant neutrino co oling mechanism has b een the so-called standard mo del based on the mo di ed URCA pro cesses [2,3] (n; p)+n ! (n; p)+ p + e + ; e (n; p)+p+e ! (n; p)+ n + : (1) e The ROSAT detection [4] of thermal emission from neutron stars indicates the necessity of faster co oling mechanism in some young neutron stars, in particular the Vela pulsar. Faster neutrino emission than the standard mo del was prop osed by invoking pion [3,5] or kaon [6] condensates whichhave neutrino emissivities comparable to that from the -decay of quarks [7] in quark matter (consisting of u, d and s quarks) d ! u + e + ; u + e ! d + : (2) e e A similar relation for s-quark -decaymay o ccur and is obtained by replacing d by s quark in Eq. (2). The most p owerful energy losses, exp ected to date, are pro duced by the so-called direct URCA mechanism involving nucleons [8] n ! p + e + ; p + e ! n + : (3) e e The threshold density of this pro cess is however considerably larger than that of the mo di ed URCA pro cess. 14 Observations of pulsars predict large surface magnetic eld of B 10 G [9]. In the m core the eld may b e considerably ampli ed due to ux conservation from the original weak eld of the progenitor during its core collapse. In fact, the scalar virial theorem [10] predicts 18 large interior eld B 10 G or more [11], and these elds are frozen in the highly m conducting core. It has b een demonstrated [11] that when the eld B is comparable to m (c) or ab ove a critical eld B , the energy of a charged particle changes signi cantly in the m quantum limit; the quantum e ects are most pronounced when the particle moves in the lowest Landau level. The phase space mo di cations stemming from the strong magnetic eld in the core are exp ected to in uence the neutrino emission rate from young neutron stars. In this communication weevaluate the neutrino emissivity for the nucleon direct URCA pro cess of Eq. (3) in presence of a magnetic eld B , and demonstrate that it would lead to m more rapid co oling in the core. (A straightforward extension of the emissivity for nucleons into the quark sector may also be obtained in a magnetic eld.) For this purp ose, we consider a npe matter in -equilibrium within a relativistic Dirac-Hartree approach in the linear -! - mo del [12]. 2 At the neutron star core at temp eratures well b elow the typical Fermi temp erature of 12 T 10 K, the nucleons and electrons participating in neutrino pro ducing pro cesses are all F degenerate (the and are free) and have their momenta close to the Fermi momenta p , e e F i where i = n; p; e. Since neutrino and antineutrino momenta are kT =c p , the nucleon F i direct URCA pro cess is allowed by the momentum conservation when p + p p . Since F F F p e n matter is very close to -equilibrium, the chemical p otentials of the constituents satisfy the condition = + . (Henceforth we set h = c = k = 1.) n p e Employing the Weinb erg-Salam theory for weak interactions, the interaction Lagrangian p cc density for the charged current reaction (3) may b e expressed as L =(G = 2) cos l j , F c W int 49 3 where G ' 1:435 10 erg cm is the Fermi weak coupling constant and the Cabibb o F c angle. The lepton and nucleon charged weak currents are resp ectively, l = (1 ) 5 2 4 and j = (g g ) . Here, and in other formulae to follow, the indices i =14 V A 5 1 3 W refer to the n, , p and e, resp ectively. The vector and axial-vector coupling constants are e g = 1 and g =1:226. V A The emissivity due to the antineutrino emission pro cess in presence of a uniform magnetic eld B along z-axis when b oth the electrons and protons are Landau quantized is given by m Z Z Z Z Z Z 3 3 qB L =2 p qB L =2 p m x m x F F p e Vd p Vd p L dp L dp L dp L dp 1 2 y 3y y 4y z 3z z 4z " (B )=2 m 3 3 (2 ) (2 ) 2 2 2 2 qB L =2 qB L =2 p p m x m x F F p e 0 max max X X E W f (p )[1 f (p )][1 f (p )] ; (4) 2 fi 1 3 4 0 =0 =0 0 where and are resp ectively the maximum number of Landau levels p opulated for max max protons and electrons. The prefactor 2 takes into account the neutron spin degeneracy. The p (E ; p ) are the 4-momenta and E the antineutrino energy. The functions f (E ) denote i i i 2 i the Fermi-Dirac functions for the ith particle. The transition rate per unit volume due to the antineutrino emission pro cess may be derived from Fermi's golden rule and is given by 2 W = hjM j i=(tV ). Here t represents time and V = L L L the normalization volume. fi fi x y z 2 jM j is the squared matrix element and the symbol hi denotes an averaging over initial fi spins and a sum over nal spins. The matrix element for the V A interaction is given by Z G F 4 p M = (X ) (g g ) (X ) (X ) (1 ) (X ) : (5) d X fi V A 5 3 5 4 1 2 2 In presence of a uniform magnetic eld B , the normalized proton wave function is m q (X ) = 1= L L exp (iE t + ip y + ip z ) f (x), where f (x) is the 4- 3 y z 3 3y 3z p ;p p ;p 3y 3z 3y 3z comp onent spinor solution [11]. The form of the spinor in a magnetic eld (see Ref. [11]) restricts the analytical evaluation of the neutrino emissivity to elds strong enough so as to 0 p opulate only the ground state for electrons and protons, i.e. = =0. The only p ositive energy spinor for protons in the chiral representation is then [11,13] 0 1 E + p 3z 3 B C 0 B C =0 f (x)=N B I C (x); (6) =0 =0 ;p 3y p ;p 3y 3z @ A m 0 3 q 2 1=2 H 2 ) is the e ective where N = 1= 2E (E + p ), and E = E U = (p + m =0 3z 3 3 3 3 0;p 3z relativistic Hartree energy. The function I (x) is similar in form as in Ref. [11]. The =0;p 3y nucleon e ective and rest masses are resp ectively, m and m = m = m = 939 MeV. In n p presence of the magnetic eld, the wave functions for free electrons (X ) have the same 4 form as those for protons, but with m and E for protons replaced by the bare mass m and 3 e kinetic energy for electrons, resp ectively. The neutrons and neutrinos/antineutrinos b eing una ected by B , have plane wave functions. m Using these wave functions, it is straightforward to calculate the transition rate p er unit volume and is given by " # 2 2 2 G 1 (p p ) +(p +p ) 1x 2x 3y 4y F W = exp fi 3 E E E E V L L 2qB 2 4 y z m 1 3 h i 2 2 2 2 2 (g + g ) (p p )(p p )+(g g ) (p p )(p p ) (g g )m (p p ) V A 1 2 3 4 V A 1 4 3 2 4 2 V A (E E E E ) (p p p p ) (p p p p ): (7) 1 2 3 4 1y 2y 3y 4y 1z 2z 3z 4z Substituting Eq.
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