Effect of strong magnetic fields on the crust-core transition and inner crust of neutron stars
The Vlasov equation is used to determine the dispersion relation for the eigenmodes of magnetized nuclear and neutral stellar matter, taking into account the anomalous magnetic moment of nucleons. The formalism is applied to the determination of the dynamical spinodal section, a quantity that gives a good estimation of the crust-core transition in neutron stars. We study the effect of strong magnetic fields, of the order of G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron () matter at subsaturation densities is determined within a relativistic mean field model. It is shown that a strong magnetic field has a large effect on the instability region, defining the crust-core transition as a succession of stable and unstable regions due to the opening of new Landau levels. The effect of the anomalous magnetic moment is non-negligible for fields larger than 10 G. The complexity of the crust at the transition to the core and the increase of the crust thickness may have direct impact on the properties of neutrons stars related with the crust.
Magnetars form a class of strongly magnetized neutron stars that includes soft -ray repeaters and anomalous X-ray pulsars (1); (2); (3). These stars have strong surface magnetic fields of the order of G (4) and, if isolated, present relatively large spin periods, of the order of 2–12 s. In fact, until now, no isolated X-ray pulsar has been detected with a spin period longer than 12 s. This feature has recently been attributed in Ref. (5) to the fast decay of the magnetic field, if the existence of a resistive amorphous layer at the bottom of the inner crust is confirmed. It was discussed in Ref. (5) that this amorphous matter, characterized by a large impurity parameter, corresponds to the matter formed by the pasta phases proposed in Ref. (6), which result from the competition between the long-range Coulomb repulsion and short-range nuclear attraction. The inner crust pasta phases have been calculated within different formalisms, including the compressible liquid drop model (6), classical and quantum molecular dynamics models (7), the Thomas-Fermi approximation within relativistic nuclear models (8); (9); (10), Hartree-Fock calculations with both non-relativistic and relativistic models(13); (11); (12), see Ref. (14) for a review of more recent works, including calculations of interest for core-collapse supernova. A recent investigation of the conductivity properties of the pasta phases has shown that topological defects affect the electrical conductivity of the system, originating a larger impurity parameter (15). However, in Ref. (16), the author has analyzed the electron transport properties in nuclear pasta phases in the mantle of a magnetized neutron star, and obtained an enhancement of the electrical conductivity. Nevertheless, it was stressed that further studies are necessary, since the contribution of non-spherical pasta clusters introduce uncertainties, and possible impurities and defects in nuclear pasta should also be considered.
Nuclear matter at subsaturation densities is characterized by a liquid-gas phase transition (17). Moreover, since nuclear matter is formed by two different types of particles, protons and neutrons, mechanical and chemical instabilities may lead both to the fragmentation of a nuclear system in heavy ion collisions and an isospin distillation effect (18); (19). The region of instability in the isospin space characterized by the proton and neutron densities is limited by the spinodal surface (20). This surface is defined, from a thermodynamic perspective, as the locus where the free energy curvature goes to zero, and, from a dynamical perspective, as the surface where the eigenmodes of matter go to zero. Both surfaces coincide if perturbations of infinity wave length are considered in the dynamical description. Spinodal decomposition has been applied to study the fragmentation of finite nuclear systems within a self-consistent quantum approach in Refs. (18); (19), and it was shown that the liquid-gas phase transition of asymmetric systems would induce a fractional distillation of the system.
Stellar matter in neutron stars is composed of protons, neutrons, electrons and possibly muons at subsaturation densities. These components are in -equilibrium, and leptons are necessary to neutralize matter. As referred above, at subsaturation density, stellar matter is essentially clusterized, possibly forming pasta phases at the higher densities. The crust-core phase transition may be determined from the pasta phase calculations, but it has also been shown that a very good estimation is obtained from the thermodynamical spinodal of proton-neutron matter, or even better, the dynamical spinodal of proton-neutron-electron matter (10); (21). In fact, the same crust-core transition density has been obtained in Ref. (10), when applying a Thomas-Fermi description of the pasta phase and a dynamical spinodal calculation.
Understanding the properties of the crust of neutron stars is essential because observations of neutron stars are directly affected by them. In particular, an important quantity is the fractional moment of inertia of the crust (22); (23). This quantity, as suggested in Ref. (24), is crucial for the interpretation of the so-called glitches, sudden breaks in the regular rotation of the star. Presently, it is still not clear whether the crust is enough to describe the glitches correctly, or if the core also contributes, since entrainment effects couple the superfluid neutrons to the solid crust (26); (25).
We will study in the present work the effect of a strong magnetic field on the crust-core transition, applying a dynamical spinodal formalism, which has shown to give a good prediction of this transition for zero magnetic field. This will be carried out using the relativistic Vlasov formalism, applied to relativistic nuclear models (27); (20); (28), and based on a field theoretical formulation (29). The normal modes of stellar matter will be calculated and special attention will be given to the unstable modes. We will only consider longitudinal modes that propagate in the direction of the magnetic field. The effect of the magnetic field on the spinodal surface, the crust-core transition of -equilibrium matter, the size of the clusters in the clusterized phase, and the fractional moment of inertia of the crust will be studied.
Previously, there have already been studies that analyze the effect of the magnetic field on the thermodynamical spinodal (30), and the pasta phases in the inner crust (31), however, both studies have been performed for magnetic fields more intense than the ones expected to exist in the crust of a magnetar. The effect of the magnetic field on the outer crust was analyzed in Ref. (32), within an Hartree-Fock-Bogoliubov calculation, and it was shown that the Landau quantization of the electron motion could affect the outer crust equation of state, giving rise to more massive outer crusts than the expected in usual neutron stars. Also, the neutron drip density and pressure are affected by a strong magnetic field, showing typical quantum oscillations, which shift the transition outer-inner crust to larger or smaller densities (33), according to the field intensity. The present work aims at studying the effect of the magnetic field on the crust-core transition, and completes the one in Ref. (34), where this study was first introduced. We present the formalism that was not introduced in Ref. (34), and we discuss the importance of including the anomalous magnetic field. A relativistic mean-field (RMF) model, that satisfies several accepted laboratory and astrophysical constraints (35); (36), will be considered. This is important because, depending on the proton fraction, which is determined by the density dependence of the symmetry energy, the magnetic field will have a weaker or stronger effect. We will also choose realistic proton fractions in the range of densities of interest. The paper is organized as follows: in section II, the formalism is introduced, in section III, the results of the calculations are presented and discussed, and, finally, in the last section, the main conclusions are drawn.
In this work, we describe stellar matter within the nuclear RMF formalism under the effect of strong magnetic fields (37); (30). We also analyze the effect of the anomalous magnetic moment (AMM) in the calculation of the dynamical spinodals. In Subsection II.1, the Lagrangian density of the RMF model is presented and in subsection II.2, the Vlasov formalism is discussed in detail.
ii.1 RMF model under strong magnetic fields
We consider a system of nucleons with mass that interact with and through meson fields. This system is neutralized by electrons because we also want to describe stellar matter. The charged particles, protons and electrons, interact through the static electromagnetic field , so that = and 0. We consider that the electromagnetic field is externally generated, which means that only frozen-field configurations are considered in the calculations.
The Lagrangian density of our system, with 1, reads
where is the nucleon Lagrangian density, given by
and the electron Lagrangian density, , together with the electromagnetic term, , are given by
The electromagnetic coupling constant is given by , and is the isospin projection for protons () and neutrons (). The nucleon AMM are introduced via the coupling of the baryons to the electromagnetic field tensor, , with , and strength , with for the neutron, and for the proton. is the nuclear magneton. The contribution of the anomalous magnetic moment of the electrons is negligible (38), hence it will not be considered.
We consider three meson fields, where the isoscalar part is associated with the scalar sigma () field with mass , and the vector omega () field with mass , whereas the isospin dependence comes from the isovector-vector rho () field (where stands for the four-dimensional spacetime indices and is the three-dimensional isospin direction index) with mass . The associated Lagrangians are
where , and .
The NL3 model (39), that we are going to consider throughout the calculations, has an additional nonlinear term, , that mixes the and mesons, allowing to soften the density dependence of the symmetry energy above saturation density. This term is given by
Some of the saturation properties of NL3 are: the binding energy, MeV, the saturation density, fm, the incompressibility, MeV, the symmetry energy, MeV, and its slope, MeV. The model satisfies the constraints imposed by microscopic calculations of neutron matter (40), and it predicts stars with masses above 2, even when hyperonic degrees of freedom are considered (35).
ii.2 Dynamical spinodal under strong magnetic fields
In this subsection, we show in detail the formalism already introduced in Ref. (34), where the dynamical spinodals are calculated within the Vlasov formalism, as previously discussed in (20); (27); (41).
The distribution function for matter at position , instant , and momentum , is given by
and is the corresponding one-body hamiltonian, where
with , and
enumerates the Landau levels of the fermions with electric charge , and is the quantum number spin, with for spin up, and for spin down. The vectors () are defined along parallel () and perpendicular () directions, since the magnetic field is taken in the -direction.
The Vlasov equation is given by
and describes the time evolution of the distribution function. denotes the Poisson brackets.
The equations, describing the time evolution of the fields , , , and the third component of the -field , are derived from the Euler-Lagrange formalism:
where the scalar densities are given by
and the components of the four-current density are
As explained in Ref. (34), the summation in in the above expressions terminates at , which is the largest value of for which the square of the Fermi momenta of the particle is still positive, and which corresponds to the closest integer from below, defined by the ratio
where and are the Fermi energies of protons and electrons, respectively.
At zero temperature, the ground state of the system is characterized by the Fermi momenta and is described by the equilibrium distribution function
are the Fermi momenta of protons, neutrons and electrons, with
and , being the polar angle. The equilibrium state is also defined by the constant mesonic fields, that are given by the following equations
where , , are the equilibrium scalar density, the nuclear density, and the isospin density, respectively. The spatial components of and are zero because there are no currents in the system.
The collective modes, which are obtained considering small oscillations around the equilibrium state, are given by the solutions of the linearized equations of motion. The deviations from equilibrium are described by
The fluctuations are written as
where are the components of a generating function defined in space,
The linearized Vlasov equations for ,
are equivalent to the following time evolution equations (27):
In the present work, only the longitudinal modes are considered, with momentum in the direction of the magnetic field, and a frequency . They are described by the following ansatz
where , represent the vector fields, and is the angle between and .
with , and . From the continuity equation for the density currents, we get for the components of the vector fields
with and .
Substituting the set of equations (29)-(32) into Eqs. (26)-(28), we get a set of five independent equations of motion in terms of the amplitudes of the proton and neutron scalar density fluctuations, , , respectively, and in terms of the amplitudes of the proton, neutron and electron vector density fluctuations, , , , respectively. These equations are given by
The eigenmodes of the system correspond to the solutions of the dispersion relation obtained by the equations written above. The coefficients and the amplitudes are given in the Appendix. The density fluctuations can be written as
At low densities, the system has unstable modes, that are characterized by an imaginary frequency, . The dynamical spinodal surface in the () space, for a given wave vector , is obtained by imposing . Inside this unstable region, we also calculate the mode with the largest growth rate, , defined as . This mode is the one responsible for the formation of instabilities. By taking its half-wavelength, we can get a good estimation of the size of the clusters (liquid) that appear in the mixed (liquid-gas) phase, i.e. in the inner crust of the stars (28).
Iii Numerical results and discussions
In the present section, we discuss the effects of strong magnetic fields on the structure of the inner crust of magnetars. In particular, we analyse the dynamical spinodals for the NL3 model for three different values of the magnetic field: G, G, and G. These values correspond to , , and , where , with G, being the electron critical magnetic field. In fact, the most intense fields detected on the surface of a magnetar are not larger than G, i.e. one or two orders of magnitude smaller than the two more intense fields considered in this study. However, in Refs. (42); (43), the authors obtained toroidal fields more intense than 10G in stable configurations, meaning that in the interior of the stars, stronger fields may be expected.
In Fig. 1, we show the dynamical spinodal sections in the (,) space for the magnetic fields mentioned above, with (top) and without (bottom panels) AMM. The black lines represent the spinodal section when the magnetic field is zero. The calculations were performed with MeV, which is a value of the transferred momentum that gives a spinodal section very close to the envelope of the spinodal sections. These sections have been obtained numerically by solving the dispersion relation (36) for . This was performed looking for the solutions at a fixed proton fraction, and for each solution, a point was obtained. The solutions form a large connected region for the lower proton and neutron densities, plus extra disconnected domains that do not occur at . The point-like appearance of the sections is a numerical limitation. A higher resolution in (,) would complete the gaps.
First we compare the results obtained omitting the AMM contribution (bottom panel). The structure of the spinodal section obtained for the strongest field considered, , clearly shows the effect of the Landau quantization, as already shown in Ref. (34): there are instability regions that extend to much larger densities than the spinodal section, while there are also stable regions that at would be unstable. This is due to the fact that the energy density becomes softer, just after the opening of a new Landau level, and harder when the Landau level is most filled. The spinodal section has a large connected section at the lower densities and extra disconnected regions. If smaller fields are considered, the structure found for is still present, but at a much smaller scale due to the increase of the number of Landau levels, see detail in the inset of the middle panel of Fig. 2, for . It is clear that the spinodal section tends to the one, as the magnetic field intensity is reduced.
In the top panel of Fig. 1, we show the same three spinodal sections, but with the inclusion of the AMM for the protons and neutrons. The overall conclusions taken for the spinodals without the AMM are still valid, although the section acquires more structure when the AMM is included since, for each Landau level, the proton spin up and spin down levels have different energies. This difference originates a doubling of the bands, which are easily identified for . Besides, these bands are also affected by the neutron AMM. The spinodal sections obtained with AMM are smaller, as it is seen from Fig. 2, where, for each field intensity, (top), (middle), and (bottom), the spinodal section without (red) and with (green) AMM are plotted. Although the inclusion of the AMM does not have a very strong effect because the proton and neutron anomalous magnetic moments are small, these effects are not negligible, and, in fact, they reduce the instability sections. In the three panels of Fig. 2, we include an inset panel where we have zoomed in the spinodal with AMM in a small range of densities to show that, although in a smaller scale, the structure is similar to the one shown for .
For neutron-rich matter, as the one occurring in neutron stars, the
instability regions extend to densities almost 40% larger than the
crust-core transition density for . The effect of the magnetic
field is larger precisely when the proton fraction is smaller.
We have included in the three panels of Fig. 2 a curve that
represents the densities () at -equilibrium,
including the contribution of the AMM. The curves cross an alternation of stable and
unstable regions, indicating the existence of a complex crust-core transition, see the insets for detail. The beginning of an homogeneous matter is shifted to larger densities, 0.100 fm for
, 0.103 fm for , and 0.105 fm
for , corresponding to the pressures 0.818 MeV fm,
0.833 MeV fm, and 0.863 MeV fm, respectively.
This complex transition region with a thickness of fm, even for the weaker fields, will have strong
implications in the structure of the inner crust of magnetars.
We will discuss later in more detail the crust-core transition region in the presence of magnetic fields.
The solution of the dispersion relation inside the spinodal section gives pure imaginary frequencies, indicating that the system is unstable to the propagation of a perturbation with the corresponding wave number in the density range where this occurs. The modulus of the frequency, designated as growth rate, indicates how the system evolves into a two-phase configuration. The evolution will be dictated by the largest growth rate (19); (20).
As an example, in Fig. 3, we show the growth rates, , as a function of the transferred momentum , for fixed values of the baryonic density: fm (top), fm (middle), and fm (bottom panels). We consider a fixed proton fraction of 0.035, which is an average value found for NL3 within a Thomas-Fermi calculation of the inner crust (44), and we choose the same values for the magnetic field as in the previous figures. The growth rates with (solid) and without (dashed) AMM are plotted together with the growth rate at (black line).
We first consider fm, far from the transition to homogeneous matter. The smaller the field, the smaller the effect of the AMM, and for , the two curves superimpose, and are almost coincident with the result. The effect of the AMM for the two larger fields is non-negligible, and may go in opposite direction because its behavior is closely related with the filling of the Landau levels. The instability does not exist for the two smaller fields at close to zero. This is the behavior discussed in Ref. (20) and is directly related to the divergence of the Coulomb field. However, for , and since the electron and proton densities are small, the attractive nuclear interaction is strong enough to drag the electrons, keeping the instability until . This is not anymore the case for the two larger densities considered, because in these two cases, the nuclear interaction is not able to compensate for the larger densities of charged particles. The stronger nuclear attraction for is also observed for the large values of : the instability is still present for MeV, well above the maximum attained for , indicating that the attractiveness of the nuclear force is stronger at short ranges.
The two larger densities have been chosen because they are at the crust-core transition or above, and this is the most sensitive region to the presence of a strong magnetic field. Due to the alternation between stable and unstable regions, it is highly probable that for one of the field intensities, no instability is present for the particular density value considered. This explains the non appearance of the curve with AMM for and fm. It also explains why the behaviors with and without AMM are so different for : the value of the density considered picks up the instability region more or less close to the limit of the instability region. In this case, also the maximum growth rates occur for different wave numbers. For , the results with and without AMM differ, and are not anymore coincident with the result, as seen for fm. However, the maximum growth rate occurs at similar wave numbers in the three cases.
Finally, we consider the larger density, fm, approximately 10% above the crust-core transition density, when no field is considered. For , this density belongs to the core, and corresponds to homogeneous matter. However, for the three intensities of the magnetic field we have been considering,