Reply to “Comment on ‘Equation of state of a dense and magnetized fermion system’ ”

Abstract

A Comment [1] to our recent article [2], expressed criticism on our estimate of the maximum magnetic field that can exist inside a neutron star and on the pressure anisotropy we found for a magnetized gas of fermions. These two points are relevant for the physics of neutron stars. With our reply we attempt not only to address the Comment's criticisms, but for the sake of the reader's understanding, we also try to clarify the connection between our and some literature's results mentioned in the comment. I. MAXIMUM FIELD STRENGTHAs known, the inner magnetic fields of neutron stars are not directly accessible to observation, so their possible values can only be estimated with the help of heuristic methods. A widely accepted estimate of the maximum star's magnetic field ∼ 10 18 G was done in Ref.[3] using a macroscopic analysis based on the equipartition of the magnetic and gravitational energies (Virial Theorem), and assuming a constant density and a uniform magnetic field throughout the entire star. As stated in the Comment [1], this estimate is in agreement with numerical simulations that used various EoS of nuclear matter [4,5], although it should be mentioned that the results of Ref.[4] were later criticized [6] because in their calculations the authors omitted the compositional changes in the EoS due to Landau quantization. Same criticism is applicable to Ref.[5], since the magnetic field was not considered in the EoS they used. Ignoring the field in the EoS of nuclear matter is not consistent at strong fields because the EoS is substantially softened by the magnetic field for fields well below the proton critical field ∼ 10 20 G [6].In Ref.[2] we argued that if the star is a hybrid star, meaning its core is made of deconfined quark matter, while its outer layers are made of nuclear matter, the previous inner field estimates based on a constant mass distribution may not be reliable because in the hybrid star a much larger difference should exist between the matter densities in the outer layers and in the core. In Section IIB of our paper, we did two independent estimates of the maximum magnetic field of a hybrid star assuming a very dense core. The Comment criticism pertains to the first of these estimates. Like in [3], our first estimate was based on the equipartition of the gravitational and magnetic energies in a star of given mass and radius. The main difference with respect to [3] was that we considered an ad-hoc model with nonuniform matter and field magnitude distributions. The only purpose we had with this simple example was to illustrate that the maximum field estimate can change just by relaxing the assumption of uniform mass and field distributions that was used in [3]. In the second paragraph following Eq.(10) we wrote "we make no claim that this ad hoc model correctly describes the real way the field varies with the radius in a hybrid star". More importantly, this ad hoc model was not used for any of the main derivations of the paper, so it has no relevance for any of the paper's main results. Noti...

Magnetic field-induced gluonic (inverse) catalysis and pressure (an)isotropy in QCD

“…However, in the above studies, the currents generating the magnetic field are not taken into account, and thus, the considered system is open, with an energy-momentum tensor that is not conserved [45,46]. If the system is defined to include these currents as well, additional contributions arise [24,47], which are the topic of an ongoing discussion [24,25]. Here, we address the problem from a somewhat different aspect, and consider the pressures as the response of the thermodynamic potential of the system against compressions in the corresponding directions.…”

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