Chapter 1 Uniformly rotating neutron stars

# Chapter 1 Uniformly rotating neutron stars

## Abstract

In this chapter we review the recent results on the equilibrium configurations of static and uniformly rotating neutron stars within the Hartle formalism. We start from the Einstein-Maxwell-Thomas-Fermi equations formulated and extended by Belvedere et al. (2012, 2014). We demonstrate how to conduct numerical integration of these equations for different central densities and angular velocities and compute the static and rotating masses, polar and equatorial radii, eccentricity , moment of inertia , angular momentum , as well as the quadrupole moment of the rotating configurations. In order to fulfill the stability criteria of rotating neutron stars we take into considerations the Keplerian mass-shedding limit and the axisymmetric secular instability. Furthermore, we construct the novel mass-radius relations, calculate the maximum mass and minimum rotation periods (maximum frequencies) of neutron stars. Eventually, we compare and contrast our results for the globally and locally neutron star models.

PACS 97.60.Jd, 97.10.Nf, 97.10.Pg, 97.10.Kc, 26.60.Dd, 26.60.Gj, 26.60.Kp, 04.40.Dg.
Keywords: Neutron stars, equations of state, mass-radius relation.

## 1. Introduction

Conventionally, in order to construct the equilibrium configurations of static neutron stars the equations of hydrostatic equilibrium derived by Tolman-Oppenheimer-Volkoff (TOV) [38, 28] are widely used. In connection with this, it has been recently revealed in Refs. [2, 29, 30] that the TOV equations are modified once all fundamental interactions are taken into due account. It has been proposed that the Einstein-Maxwell system of equations coupled with the general relativistic Thomas-Fermi equations of equilibrium have to be used instead. This set of equations is termed as the Einstein-Maxwell-Thomas-Fermi (EMTF) system of equations. Although in the TOV method the condition of local charge neutrality (LCN), is imposed (see e.g. [18]), the EMTF method requires the less rigorous condition of global charge neutrality (GCN) as follows

 ∫ρchd3r=∫e[np(r)−ne(r)]d3r=0, (1)

where is the electric charge density, is the fundamental electric charge, and are the proton and electron number densities, respectively. The integration is performed on the entire volume of the system.

The Lagrangian density accounting for the strong, weak, electromagnetic and gravitational interactions consists of the free-fields terms such as the gravitational , the electromagnetic , and the three mesonic fields , , , the three fermion species (electrons, protons and neutrons) term and the interacting part in the minimal coupling assumption, given as in Refs. [30, 2]:

 L=Lg+Lf+Lσ+Lω+Lρ+Lγ+Lint, (2)

where1

 Lg =−R16π,Lf=∑i=e,N¯ψi(iγμDμ−mi)ψi, Lσ =∇μσ∇μσ2−U(σ),Lω=−ΩμνΩμν4+m2ωωμωμ2, Lρ =−RμνRμν4+m2ρρμρμ2,Lγ=−FμνFμν16π, Lint =−gσσ¯ψNψN−gωωμJμω−gρρμJμρ+eAμJμγ,e−eAμJμγ,N.

The inclusion of the strong interactions between the nucleons is made through the -- nuclear model following Ref. [9]. Consequently, , , are the field strength tensors for the , and fields respectively, stands for covariant derivative and is the Ricci scalar. The Lorentz gauge is adopted for the fields , , and . The self-interaction scalar field potential is , is the nucleon isospin doublet, is the electronic singlet, stands for the mass of each particle-species and , where are the Dirac spin connections. The conserved currents are given as , , , and , where is the particle isospin.

In this chapter we adopt the NL3 parameter set [26] used in Ref. [2] with MeV, MeV, MeV, , , , plus two constants that give the strength of the self-scalar interactions, fm and .

### Footnotes

1. We use the spacetime metric signature (+,-,-,-) and geometric units unless otherwise specified.

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