Constraining the central magnetic field of magnetars
The magnetars are believed to be highly magnetized neutron stars having surface magnetic field G. It is believed that at the center, the magnetic field may be higher than that at the surface. We study the effect of the magnetic field on the neutron star matter. We model the nuclear matter with the relativistic mean field approach considering the possibility of appearance of hyperons at higher density. We find that the effect of magnetic field on the matter of neutron stars and hence on the mass-radius relation is important, when the central magnetic field is atleast of the order of G. Very importantly, the effect of strong magnetic field reveals anisotropy to the system. Moreover, if the central field approaches G, then the matter becomes unstable which limits the maximum magnetic field at the center of magnetars.
Anomalous X-ray pulsars and soft -ray repeaters are observationally identified with highly magnetized neutron stars, known as magnetars, with surface magnetic field G . The processes of supernova collapse will leave behind a strongly non-uniform frozen-in field distribution. Also any dynamo mechanism generating fields will carry the imprint of inhomogeneous density profile in the star. Thus, to maintain the local magneto-static equilibrium, more realistic treatment of the equation of state (EoS) of matter for a magnetar requires inclusion of gradually increasing magnetic field from surface to center. Massive compact stars are likely to develop exotic cores with one possibility of appearance of hyperons with the increasing density. In the present work, considering the radial profile of the magnetic field and carefully analyzing the different components of the field, we show that the pressure of the magnetar matter parallel to the magnetic field exhibits instability.
2 Model of magnetar matter
To construct the model of dense matter, we employ non-linear Walecka mean field theory [2, 3] of nuclear matter including the possibility of appearance of hyperons and muons at higher density. In the presence of magnetic field, the Lagrangian density of the system is , with and are the baryonic and leptonic Lagrangian densities respectively in the presence of magnetic field [4, 5] and the electro-magnetic field tensor. For details, see Ref. \refcitesms.
Total energy density and pressure of the system can be obtained by considering the energy-momentum tensor of the system , where and are the matter and field parts respectively. In the absence of electric field, and in the rest frame of fluid
when the magnetic field is considered to be along -direction with , is the magnitude of magnetic field, the magnetization per unit volume, and are respectively the pressure and energy density of the matter. This clearly shows anisotropic nature of the pressure in the presence of (strong) magnetic field. is calculated using the charged single particle energy , considering the quantized phase space in the presence of magnetic field, where is the component of momentum along -axis, the mass, the total charge, with being the electron’s charge, of the particle, is the number of occupied Landau level. Then matter pressure is , where and are respectively the chemical potentials and number densities for baryons () and leptons (). The density profile of the magnetic field is modeled as 
where and are two parameters, and are respectively the number densities of matter and nuclear matter, and are respectively the magnitudes of the magnetic field in the surface and center of the underlying magnetar.
Figure 1a shows EoS for hypernuclear matter in strong and fixed magnetic field profiles. For non-zero magnetic field, the pressure splits into the parallel () and transverse () components, and exhibits anisotropy. It is seen that the low-density behavior of EoS with constant magnetic field implies unrealistically large anisotropic magnetic field up to the surface of the star, which is inconsistent with the inferred surface magnetic field ( G) of magnetars. It is seen from Fig. 1b that for a given value of , the EoS becomes softer with increasing . Consequently, beyond a certain critical and in a certain density regime, ceases to increase (and eventually decreases) with the further increase in . This implies the onset of instability of matter above that value of density for that particular and magnetic field profile. We also show the results for each with (minimum value). Note that the maximum is taken in such a way that forms a plateau as a function . Furthermore, it is evident that with the decrease of , the instability occurs at larger values of and .
The instability arises due to the negative contribution from the field energy density to the pressure of magnetized baryons and leptons in the direction of the magnetic field, which is evident from Eq. (1). With the increase of , more negative contribution is added to , and consequently at a certain , ceases to increase and then decreases with the increase of , rendering instability.
We have found that for sufficiently large magnetic fields with G, the magnetar
unstable. The instability is associated with the anisotropic effects arised due to
the magnetic field. The onset of instability depends on
the magnetic field profile and , which puts a natural upper bound for the central magnetic field
of neutron stars, which is G.
This work was partially supported by the grant ISRO/RES/2/367/10-11 (B.M.) and the Alexander von Humboldt Foundation (M.S.).
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