arXiv:astro-ph/9912301v1 15 Dec 1999 SUBMITTED TO APJ Preprint typeset using L A T E X style emulateapj EQUATION OF STATE IN STRONGLY MAGNETIZED NEUTRON STARS: EFFECTS ON MUON PRODUCTION AND PION CONDENSATION I N-SAENG SUH 1 AND G. J. MATHEWS 2 Center for Astrophysics, Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA 1 isuh@cygnus.phys.nd.edu; 2 gmathews@bootes.phys.nd.edu Submitted to ApJ ABSTRACT Recently, neutron stars with very strong surface magnetic fields have been suggested as the site for the origin of observed soft gamma repeaters (SGRs). In this paper we investigate the influence of such strong magnetic fields on the properties and internal structure of these magnetized neutron stars (magnetars). We study properties of a degenerate equilibrium ideal neutron-proton-electron (npe) gas model in a magnetic field. The presence of a sufficiently strong magnetic field changes the ratio of protons to neutrons as well as the neutron drip density. We also study the appearance of muons as well as pion condensation in strong magnetic fields. We discuss the possibility that boson condensation in the interior of magnetars is a source of starquakes. Subject headings: stars: interiors — stars: magnetic fields — stars: neutron 1. INTRODUCTION Among the more than two thousand observed cosmologi- cal gamma-ray bursts (GRBs), four recurrent sources, so-called soft gamma repeaters (SGRs), have been identified and a fifth has probably been observed (Hurley 2000). They are believed to be a new class of γ -ray transients separate from the source of classical GRBs. Recently, observations of SGR 0526-66 (Mazets et al. 1979), SGR 1806-20 (Murakami et al. 1994), and SGR 1900+14 (Kouveliotou et al. 1998) with RXTE, ASCA, and BeppoSAX have confirmed the fact that these SGRs are newly born neutron stars that have very large surface mag- netic fields (up to 10 15 G) based upon measurements of the spin-down timescale. Recently SGR 1627-41 has also been discovered by BATSE (Woods et al. 1999). It is estimated that its magnetic field could be B > ∼ 5 × 10 14 G. The most re- cent source is SGR 1801-23 (Cline et al. 1999) observed by Ulysses, BATSE, and KONUS- Wind . Such stars have been named magnetars (Duncan & Thompson 1992, Thompson & Duncan 1995). They are probably a remnant of a supernova explosion. These magnetars also include anomalous X-ray pul- sars (AXP) (van Paradijs, Taam, & van den Heuvel 1995) such as 1E 1841-045 (Kes 73) (Gotthelf, Vasisht, & Dotani 1999), RX J0720.4-3125 (Haberl et al. 1997), and 1E 2259+586 (Rho & Petre 1997). As relics of stellar interiors, the study of the magnetic fields in and around degenerate stars should give important informa- tion on the role such fields play in star formation and stellar evolution. However, the origin and evolution of stellar magnetic fields remains obscure. As early as Ginzburg (1964) and Wolt- jer (1964) it was proposed that the magnetic flux (Φ B ∼ BR 2 ) of a star is conserved during its evolution and subsequent col- lapse to form a remnant white dwarf or neutron star. A main sequence star with radius on the order of R ∼ 10 11 cm and sur- face magnetic field B ∼ 10 - 10 4 G [magnetic A-type stars have typical surface fields < ∼ 10 4 G (Shapiro & Teukolsky 1983)] would thus collapse to form a white dwarf with R ∼ 10 9 cm and B ∼ 10 5 - 10 8 G, or a neutron star with R ∼ 10 6 cm and B ∼ 10 11 - 10 14 G. Indeed, shortly after their discovery (Hewish et al. 1968) pulsars were identified as rotating neutron stars (Gold 1968) with magnetic fields B ∼ 10 11 - 10 13 G consistent with magnetic field amplification by flux conservation. Recently, Thompson and Duncan (1993) have invoked a con- vective dynamo mechanism, to suggest that the magnetic dipole field of young neutron stars could realistically reach a magnetic field of the order of 10 14 - 10 15 G, i.e., 10 2 - 10 3 times stronger than in ordinary pulsars. Moreover, the internal magnetic field of a star may not nec- essarily be reflected in its surface magnetic field (Ruderman 1980, Galloway, Proctor, & Weiss 1977). Therefore, the total strength of internal magnetic fields remains unknown. Never- theless it is expected that appreciably higher magnetic fields can exist in the interiors of neutron stars (Ruderman 1980). Ultimately, the allowed internal field strength of a star is constrained by the scalar virial theorem (cf. Lai & Shapiro 1991, Shapiro & Teukolsky 1983), 2T + W + 3Π + M =0, where T is the total kinetic energy, W is the gravitational po- tential energy, Π is the internal energy, and M is the magnetic energy. For a star of size R and mass M, This gives a maxi- mum interior field strength of B ∼ 2 × 10 8 (M/M ⊙ )(R/R ⊙ ) -2 G. For neutron stars with R ≈ 10 km and M ≈ M ⊙ , the maximum interior field strength could thus reach B < ∼ 10 18 G (Lerche & Schramm 1977). Numerical studies (Bocquet et al. 1995) have confirmed that neutron stars with ultrastrong magnetic fields are stable up to the order of 10 18 G. They also have found that for such values the maximum mass of neutron stars increases by 13 - 29 % with relative to the maximum mass of non-magnetized neutron stars. This is similar to the case of magnetic white dwarfs (Suh & Mathews 1999). The strength of the internal magnetic field in a neutron star could, in principle, be constrained by any observable conse- quences of a strong magnetic field. For example, rapid motion of neutron stars may be due to anisotropic neutrino emission induced by a strong magnetic field (e.g., see Janka 1998). One could also consider the effect of magnetic fields on the thermal evolution (Heyl & Hernquist 1997, Leinson & Perez 1998) and the maximum mass (Vshivtsev & Serebryakoba 1994) of neu- tron stars. Recently, Chakrabarty et al. (1995) have investigated the gross properties of cold nuclear matter in a strong magnetic field in the context of a relativistic Hartree model and have ap- plied their equation of state to obtain the maximum masses and 1