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Magnetars As Highly Magnetized Quark Stars: an Analytical Treatment

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Magnetars As Highly Magnetized Quark Stars: an Analytical Treatment The Astrophysical Journal, 734:41 (5pp), 2011 June 10 doi:10.1088/0004-637X/734/1/41 °C 2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A. MAGNETARS AS HIGHLY MAGNETIZED QUARK STARS: AN ANALYTICAL TREATMENT M. Orsaria1,2, Ignacio F. Ranea-Sandoval1,2, and H. Vucetich1 1 Gravitation, Astrophysics and Cosmology Group, Facultad de Ciencias Astronomicas´ y Geof´ısicas, Universidad Nacional de La Plata UNLP, Paseo del Bosque S/N (1900), La Plata, Argentina 2 CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina Received 2010 October 15; accepted 2011 March 30; published 2011 May 24 ABSTRACT We present an analytical model of a magnetar as a high-density magnetized quark bag. The effect of strong magnetic fields (B>5×1016 G) in the equation of state is considered. An analytic expression for the mass–radius relationship is found from the energy variational principle in general relativity. Our results are compared with observational evidence of possible quark and/or hybrid stars. Key words: dense matter – magnetic fields – methods: analytical – stars: individual (Quark Stars) – stars: magnetars 1. INTRODUCTION on some parameters. At this particular point astrophysical stud- ies become of great importance since they could shed some light The fundamental aspects of the physics involved in the de- on understanding fundamental aspects of matter: microphysics scription of the matter inside a white dwarf are well understood could be inferred from macrophysics. Here lies the great impor- (Shapiro & Teukolsky 1983), but in the case of neutron stars the tance of the studies related to ultracompact objects. For instance, situation is rather different because the equation of state (EoS) Lattimer & Prakash (2001, 2007) contrast some M–R relation- of neutron matter at very high densities is still unknown. ships obtained theoretically for different EoS. By varying some The interior of a neutron star is an astrophysical laboratory in parameters a difference of 4%–10% and 10%–15% in determin- which matter is compressed to high densities. The compression ing the maximum radius Rmax and mass Mmax, respectively, is of matter several times the saturation nuclear matter density, shown for the same EoS. Several papers (Chakrabarty & Sahu 1996; Gonzalez´ Felipe & ρ0, may produce a phase transition from nuclear to quark matter, i.e., an unconfined quark–gluon plasma. In addition, Perez´ Mart´ınez 2009;Perez´ Mart´ınez et al. 2010) study the M–R under suitable circumstances, a conversion d → s quarks may relationship of highly magnetized quark stars (HMQS) through happen through weak interactions, leading to what has been numerical integration of the Tolman–Oppenheimer–Volkoff called strange quark matter (SQM). It has been stated that SQM equation for different EoS. Although most studies of quark may be the absolute ground state of strong interactions (Bodmer stars’ properties have used such methods, Banerjee et al. (2000) 1971; Witten 1984), although such a hypothesis has not been have obtained a maximum mass and radius for unmagnetized confirmed yet. The natural scenario where SQM could occur is quark stars analytically by using a non-relativistic gravitational the inner core of neutron stars. Hence, if the SQM hypothesis treatment. is true, some neutron stars could be either hybrid stars, which Approximate analytical solutions play an important role in have quark cores surrounded by a hadronic shell, or quark stars. the astrophysical analysis, giving a keener insight than the Already 40 years ago, the existence of quark stars in hydrostatic numerical solutions. Moreover, they may be used as a testing equilibrium was suggested by Itoh (1970) in a preliminary work. point to check if the numerical scheme is accurate and also On the other hand, it is well known that at the surface they are the first step in the comparison between theory and of neutron stars there exist magnetic fields of the order of observation. Indeed, an approximate analytical solution for the 1012–1013 G. Compact stars with ultra strong magnetic fields M–R relationship may be all that is required when comparisons (102–103 larger than those of a typical neutron star) are called with observational limits that determine the confidence contour magnetars. In such objects the magnetic field at the surface could for the mass and radius are performed. Besides, in the high- be higher than 1015 G (de la Incera 2009). density EoS the uncertainties are of the same order as or larger Knowledge of the magnetars’ composition would help ex- than the errors in the variational method. plain some astrophysical phenomena. Soft gamma-ray repeaters The appropriate treatment for quark stars should be rela- (SGRs) and anomalous X-ray pulsars (AXPs) have been inter- tivistic, since the existence of a maximum mass is associated preted as evidence of magnetars. Besides, some authors (Cheng with the behavior of a relativistic gas and general relativistic &Daib2002; Ouyed et al. 2007) claim that magnetized hybrid corrections are dominant (Weinberg 1972). In this paper, we or quark stars could be the real sources of SGRs and AXPs. shall use the general relativistic energy variational principle The mass–radius (M–R) relationship tells us how matter com- described by Naurenberg & Chapline (1973) to obtain an an- posing the star behaves under compression, providing informa- alytic approximate formula for the mass, radius, and baryonic tion about its composition. Several EoS for neutron, hybrid, number of an HMQS. Quark stars are particularly suitable for and quark stars have been proposed but none of them is con- a variational treatment since their density profile resembles a clusive (Douchin & Haensel 2001; Lattimer & Prakash 2001, constant mass density star. We shall model an HMQS assuming 2007; Ozel¨ & Psaltis 2009). Each EoS produces a different M–R quark matter within a high density regime in the framework relationship which can be contrasted with the available observa- of a modified MIT Bag model EoS. We also assume that the tional data in order to test its range of validity and/or set bounds magnetic field B is low enough to be treated like a correction in 1 The Astrophysical Journal, 734:41 (5pp), 2011 June 10 Orsaria, Ranea-Sandoval, & Vucetich the EoS (B ¿ μ2, with μ being the baryon chemical potential) The series, Equation (2), can be approximated using the although, as we will see in the following sections, this is not a Euler–MacLaurin formula strong restriction. Z Xn n 1 The paper is organized as follows. In Section 2, we calculate f (j) = f (x) dx + [f (n)+f (0)] the thermodynamical quantities of the system and analyze the 2 j= 0 stability of quark matter with respect to decomposition in 0 1 1 baryons. In Section 3, we provide the analytic relativistic M–R + [f 0(n) − f 0(0)] − [f 000(n) − f 000(0)] + R, relationship and compare our results with the observational data. 12 720 We also check the dynamic stability of the star by calculating (5) the adiabatic index and the speed of sound. In Section 4,we present a summary of our main results and conclusions. where the remainder term, R, usually is expressed in terms of periodic Bernoulli polynomials (Spivey 2006) and can be 2. HIGH-DENSITY QUARK MATTER WITHIN A estimated by using STRONG MAGNETIC FIELD Z − 2ζ (4) νmax 1 In this section we shall discuss the analytic approximations to |R| 6 |f IV(ν)| dν, (6) (2π)4 the SQM EoS in the presence of a uniform magnetic field B k zˆ. 1 Within the framework of the MIT Bag model, we assume three where the Riemann Zeta function ζ (4) ' 1.0823. To avoid massless quarks u, d, and s, neglecting mediated interactions divergences appearing in the third term of Equation (5), in the between them. We also consider that the strong magnetic field limit of high densities or negligible quark masses, we apply the is a small contribution to the total energy, a fact that will be Euler–MacLaurin formula in the form checked later. Z νmax−1 2.1. Quark Matter in a Magnetic Field Ωi ' Ωi (νmax)+Ωi (0) + Ωi (ν) dν 1 Let us compute the grand canonical thermodynamic potential 1 Ω + [Ω (ν − 1) + Ω (1)] in the high density regime. Due the Landau quantization the 2 i max i phase space volume integral in the momentum space is replaced " ¯ ¯ # Ω ¯ Ω ¯ by 1 ∂ i ¯ ∂ i ¯ ˜ + ¯ − ¯ + R, (7) Z Z 12 ∂ν ∂ν (νmax−1) (1) 1 3 1 2 d pf(p) = dpz d p⊥f (p) (2π)3 (2π)3 where ν=∞ Z ∞ " # qB X + ¯ ¯ = − 3Ω ¯ 3Ω ¯ (2 δν0) dpz f (ν, pz), ˜ 1 ∂ i ¯ ∂ i ¯ 4π 2 −∞ R =− − + R. (8) ν=0 3 ¯ 3 ¯ 720 ∂ν (ν −1) ∂ν (1) (1) max We have considered Equation (8) as the remainder term, which where (2 − δ ) means that the zeroth Landau level is singly ν0 gives |R˜| 6 3%. In the limit μ2 À 2q B, the thermodynamical degenerate, whereas all other states are doubly degenerate. The i potential can be calculated by performing first the integral in grand canonical potential for each quark in the presence of a Equation (7) and then expanding in power series of B.The strong magnetic field is given by " result is Xνmax p µ ¶ qi Bgi 4 2 2 Ω =− − 2 − μ qi B qi B 5/2 i (2 δν0) μ μ 2νqi B Ω =− + log − 3 + O(B ). (9) 2 i 2 2 2 8π = 4π 8π 2 μ ν 0 p # 2 − Ω μ + μ 2νqi B The particle density n =−∂ i is − 2νqi B ln √ , (2) i ∂μ 2νqi B 3 2 2 μ qi B 5/2 where g = 2 × 3, taking into account spin and color ni = + + O(B ).
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