Chapter 1 Uniformly rotating neutron stars
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.
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. ), the EMTF method requires the less rigorous condition of global charge neutrality (GCN) as follows
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]:
The inclusion of the strong interactions between the nucleons is made through the -- nuclear model following Ref. . 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.
Thus, the system of the EMTF equations [30, 2, 4, 5] is derived from the equations of motion of the above Lagrangian. The solution of the EMTF coupled differential equations gives a novel structure of the neutron star, as shown in Fig 1.: a positively charged core at supranuclear densities, g cm, surrounded by an electron distribution of thickness , which is negatively charged and a neutral ordinary crust at lower densities .
where the subscript stands for each kind of particle, is the particle chemical potential, is the particle electric charge, is the time component of the fluid four-velocity which satisfies and is the t–t component of the spherically symmetric metric. For the static case we have only the time components of the vector fields, , , .
The constancy of the Klein potentials (3) leads to a discontinuity in the density at the core-crust transition and, respectively, this generates an overcritical electric field , where Volt cm, in the core-crust boundary interface. The Klein condition (3) is necessary to satisfy the requirement of thermodynamical equilibrium, together with the Tolman condition (constancy of the gravitationally red-shifted temperature) [37, 25], if finite temperatures are included . Particularly, the continuity of the electron Klein potential leads to a decrease of the electron chemical potential and density at the core-crust boundary interface. They attain values and at the basis of the crust, where global charge neutrality is achieved.
As it has been shown in Refs. [2, 4], that the solution of the EMTF equations along with the constancy of the Klein potentials yield a more compact neutron star with a less massive and thiner crust. Correspondingly, this results in a new mass-radius relation which prominently differs from the one given by the solution of the TOV equations with local charge neutrality; see Fig. 1..
In this chapter the extension of the previous results obtained in Refs. [2, 4] are considered. To this end the Hartle formalism  is utilized to solve the Einstein equations accurately up to second order terms in the angular velocity of the star, (see section 2.).
For the rotating case, the Klein thermodynamic equilibrium condition has the same form as Eq. (3), but the fluid inside the star now moves with a four-velocity of a uniformly rotating body, , with (see , for details)
where is the azimuthal angular coordinate and the metric is axially symmetric independent of . The components of the metric tensor are now given by Eq. (6) below. It is then evident that in a frame comoving with the rotating star, , and the Klein thermodynamic equilibrium condition remains the same as Eq. (3), as expected.
This chapter is organized as follows: in section 2. we review the Hartle formalism and consider both interior and exterior solutions. In section 3. the stability of uniformly rotating neutron stars are explored taking into account the Keplerian mass-shedding limit and the secular axisymmetric instability. In section 4. the structure of uniformly rotating neutron stars is investigated. We compute there the mass , polar and equatorial radii, and angular momentum , as a function of the central density and the angular velocity of stable neutron stars both in the globally and locally neutral cases. Based on the criteria of equilibrium we calculate the maximum stable neutron star mass. In section 5. we construct the new neutron star mass-radius relation. In section 6. we calculate the moment of inertia as a function of the central density and total mass of the neutron star. The eccentricity , the rotational to gravitational energy ratio , and quadrupole moment are shown in section 7.. The observational constraints on the mass-radius relation are discussed in section 8.. We finally summarize our results in section 9..
2. Hartle slow rotation approximation
In his original article, Hartle (1967)  derived the equilibrium equations of slowly rotating relativistic stars. The solutions of the Einstein equations have been obtained through a perturbation method, expanding the metric functions up to the second order terms in the angular velocity . Under this assumption the structure of compact objects can be approximately described by the total mass , angular momentum and quadrupole moment . The slow rotation regime implies that the perturbations owing to the rotation are relatively small with respect to the known non-rotating geometry. The interior solution is derived by solving numerically a system of ordinary differential equations for the perturbation functions. The exterior solution for the vacuum surrounding the star, can be written analytically in terms of , , and [19, 22]. The numerical values for all the physical quantities are derived by matching the interior and the exterior solution on the surface of the star.
2.1. The interior Hartle solution
The spacetime metric for the rotating configuration up to the second order of is given by 
where , , and are the metric functions and mass profiles of the corresponding seed static star with the same central density as the rotating one; see Eq. (4). The functions , , and the fluid angular velocity in the local inertial frame, , have to be calculated from the Einstein equations. Expanding up to the second order the metric in spherical harmonics we have
where is the Legendre polynomial of second order. Because the metric does not change under transformations of the type , we can assume .
The functions , , have analytic form in the exterior (vacuum) spacetime and they are shown in the following section. The mass, angular momentum, and quadrupole moment are computed from the matching condition between the interior and exterior metrics.
For rotating configurations the angular momentum is the easiest quantity to compute. To this end we consider only component of the Einstein equations. By introducing the angular velocity of the fluid relative to the local inertial frame, one can show from the Einstein equations at first order in that satisfies the differential equation
where with and the metric functions of the seed non-rotating solution (4).
From the matching conditions, the angular momentum of the star is given by
so the angular velocity is related to the angular momentum as
The total mass of the rotating star, , is given by
where is the contribution to the mass owing to rotation. The second order functions (the mass perturbation function) and (the pressure perturbation function) are computed from the solution of the differential equation
where and are the total energy-density and pressure.
Turning to the quadrupole moment of the neutron star, it is given by
where is a constant of integration. This constant is fixed from the matching of the second order function obtained in the interior from
with its exterior counterpart (see ).
It is worth emphasizing that the influence of the induced magnetic field owing to the rotation of the charged core of the neutron star in the globally neutral case is negligible . In fact, for a rotating neutron star of period ms and radius km, the radial component of the magnetic field in the core interior reaches its maximum at the poles with a value , where G is the critical magnetic field for vacuum polarization. The angular component of the magnetic field , instead, has its maximum value at the equator and, as for the radial component, it is very low in the interior of the neutron star core, i.e. . In the case of a sharp core-crust transition as the one studied by  and shown in Fig. 1., this component will grow in the transition layer to values of the order of . However, since we are here interested in the macroscopic properties of the neutron star, we can ignore at first approximation the presence of electromagnetic fields in the macroscopic regions where they are indeed very small, and safely apply the original Hartle formulation without any generalization.
2.2. The exterior Hartle solution
In this subsection we consider the exterior Hartle solution though in the literature it is widely known as the Hartle-Thorne solution. One can write the line element given by eq. (6) in an analytic closed-form outside the source as function of the total mass , angular momentum , and quadrupole moment of the rotating star. The angular momentum along with the angular velocity of local inertial frames , proportional to , and the functions , , , , , proportional to , are derived from the Einstein equations (for more details see [19, 22]). Following this prescriptions the Eq. 6 becomes:
are the associated Legendre functions of the second kind, being the Legendre polynomial, and . This form of the metric is known in the literature as the Hartle-Thorne metric. To obtain the exact numerical values of , and , the exterior and interior line elements have to be matched at the surface of the star. It is worth noticing that in the terms involving and , the total mass can be directly substituted by since is already a second order term in the angular velocity.
3. Stability of uniformly rotating neutron stars
3.1. Secular axisymmetric instability
In a sequence of increasing central density in the - curve, , the maximum mass of a static neutron star is determined as the first maximum of such a curve, namely the point where /. This derivative establishes the axisymmetric secular instability point, and if the perturbation obeys the same equation of state (EOS) as the equilibrium configuration, it coincides also with the dynamical instability point (see e.g. Ref. ). In the rotating case, the situation becomes more complicated and in order to find the axisymmetric dynamical instability points, the perturbed solutions with zero frequency modes (the so-called neutral frequency line) have to be calculated. Friedman et al. (1988)  however, following the works of Sorkin (1981, 1982) [32, 33], described a turning-point method to obtain the points at which secular instability is reached by uniformly rotating stars. In a constant angular momentum sequence, the turning point is located in the maximum of the mass-central density relation, namely the onset of secular axisymmetric instability is given by
and once the secular instability sets in, the star evolves quasi-stationarily until it reaches a point of dynamical instability where gravitational collapse sets in (see e.g. ).
The above equation determines an upper limit for the mass at a given angular momentum for a uniformly rotating star, however this criterion is a sufficient but not necessary condition for the instability. This means that all the configurations with the given angular momentum on the right side of the turning point defined by Eq. (20) are secularly unstable, but it does not imply that the configurations on the left side of it are stable. An example of dynamically unstable configurations on the left side of the turning-point limiting boundary in neutron stars was recently shown in Ref. , for a specific EOS.
In order to investigate the secular instability of uniformly rotating stars one should select fixed values for the angular momentum. Then construct mass-central density or mass-radius relations. From here one has to calculate the maximum mass and that will be the turning point for a given angular momentum. For different angular momentum their will be different maximum masses. By joining all the turning points together one obtains axisymmetric secular instability line (boundary). This boundary is essential for the construction of the stability region for uniformly rotating neutron stars (see next sections and figures).
3.2. Keplerian mass-shedding instability and orbital angular velocity of test particles
The maximum velocity for a test particle to remain in equilibrium on the equator of a star, kept bound by the balance between gravitational and centrifugal force, is the Keplerian velocity of a free particle computed at the same location. As shown, for instance in , a star rotating at Keplerian rate becomes unstable due to the loss of mass from its surface. The mass shedding limiting angular velocity of a rotating star is the Keplerian angular velocity evaluated at the equator, , i.e. . Friedman (1986)  introduced a method to obtain the maximum possible angular velocity of the star before reaching the mass-shedding limit; however  and , showed a simpler way to compute the Keplerian angular velocity of a rotating star. They showed that the mass-shedding angular velocity, , can be computed as the orbital angular velocity of a test particle in the external field of the star and corotating with it on its equatorial plane at the distance .
It is possible to obtain the analytical expression for the angular velocity given by Eq. (24) with respect to an observer at infinity, taking into account the parameterization of the four-velocity of a test particle on a circular orbit in equatorial plane of axisymmetric stationary spacetime, regarding as parameter the angular velocity itself:
where is a normalization factor such that . Normalizing and applying the geodesics conditions we get the following expressions for and
Thus, the solution of the system of Eq. (22) can be written as
where stands for co-rotating/counter-rotating orbits, and are the angular and time components of the four-velocity respectively, and a colon stands for partial derivative with respect to the corresponding coordinate. To determine the mass shedding angular velocity (the Keplerian angular velocity) of the neutron stars, we need to consider only the co-rotating orbit, so from here and thereafter we take into account only the plus sign in Eq. (22) and we write .
For the Hartle external solution given by Eq. (2.2.) we obtain
where and are the dimensionless angular momentum and quadrupole moment. The analytical expressions of the functions are given by
The maximum angular velocity for a rotating star at the mass-shedding limit is the Keplerian angular velocity evaluated at the equator (), i.e.
In the static case i.e. when hence and we have the well-known Schwarzschild solution and the orbital angular velocity for a test particle on the surface () of the neutron star is given by
3.3. Gravitational binding energy
Besides the above stability requirements, one should check if the neutron star is gravitationally bound. In the non-rotating case, the binding energy of the star can be computed as
where is the rest-mass of the star, is the rest-mass per baryon, and is the total number of baryons inside the star. So the non-rotating star is considered bound if .
In the slow rotation approximation the total binding energy is given by Ref. 
where is the internal energy of the star, with the baryon number density.
4. Structure of uniformly rotating neutron stars
We show now the results of the integration of the Hartle equations for the globally and locally charge neutrality neutron stars; see e.g. Fig. 1.. Following Refs. [2, 4], we adopt, as an example, globally neutral neutron stars with a density at the edge of the crust equal to the neutron drip density, g cm.
4.1. Secular instability boundary
In Fig. 4.1. we show the mass-central density curve for globally neutral neutron stars in the region close to the axisymmetric stability boundaries. Particularly, we construct some =constant sequences to show that indeed along each of these curves there exist a maximum mass (turning point). The line joining all the turning points determines the secular instability limit. In Fig. 4.1. the axisymmetric stable zone is on the left side of the instability line.
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