Magnetic field distribution in magnetars
Abstract
Using an axisymmetric numerical code, we perform an extensive study of the magnetic field configurations in nonrotating neutron stars, varying the mass, magnetic field strength and the equation of state. We find that the monopolar (spherically symmetric) part of the norm of the magnetic field can be described by a single profile, that we fit by a simple eighthorder polynomial, as a function of the star’s radius. This new generic profile applies remarkably well to all magnetized neutron star configurations built on hadronic equations of state. We then apply this profile to build magnetized neutron stars in spherical symmetry, using a modified TolmanOppenheimerVolkov system of equations. This new formalism satisfactorily reproduces the correct behavior of the neutron star total mass with increasing magnetic field. Our “universal” magnetic field profile is intended to serve as a tool for nuclear physicists to obtain estimates of magnetic field inside neutron stars, as a function of radial depth, in order to deduce its influence on composition and related properties. It possesses the advantage of being based on magnetic field distributions from realistic selfconsistent computations, which are solutions of Maxwell’s equations.
pacs:
97.60.Jd, 26.60.c, 26.60.Dd, 04.25.D, 04.40.DgI Introduction
The macroscopic structure and observable astrophysical properties of neutron stars depend crucially on its internal composition and thus the properties of dense matter. The Equation of State (EoS) determines global quantities such as observed mass and radius. Transport properties such as thermal conductivity and bulk viscosity have an effect on cooling observations as well as emission of gravitational waves. As we enter an era of multimessenger astronomy, it is crucial to construct consistent microscopic and macroscopic models in order to correctly interpret astrophysical observations.
There are a large number of astrophysical observations, e.g. softgamma repeaters (SGR) or anomalous Xray pulsars (AXP), that indicate the existence of ultramagnetized neutron stars or magnetars kaspi17. While such observations only probe the surface magnetic field, there is no way to measure directly the maximum magnetic field in the interior. Using the simple virial theorem, one may estimate the maximum interior magnetic field to be as high as G. If such large fields exist in the interior, they may strongly affect the energy of the charged particles by confining their motion to quantized Landau levels and consequently modify the particle population, transport properties as well as the global structure avancini18; tolos17; franzon16b; franzon16; gomes18; gomes17b; gomes17; gomes14; gomes13; dexheimer12; dexheimer14; wei17b; wei17. However, the problem is that one requires to know the magnetic field amplitude at a given location in the star, i.e. a magnetic field distribution, in order to determine its effect on the internal composition and EoS.
The ideal way to tackle that problem would of course be to selfconsistently solve the neutron star structure equations endowed with a magnetic field, i.e. combined Einstein, Maxwell and equilibrium equations, together with a magnetic field dependent EoS, as done by Chatterjee et al. chatterjee15. This solution is complicated by the fact that in presence of a magnetic field, the neutron star structure strongly deviates from spherical symmetry and the spherically symmetric TolmanOppenheimerVolkov (TOV) equations are no longer applicable for obtaining the macroscopic structure of a the neutron star bocquet95; chatterjee15; dexheimer17c. For small magnetic fields, perturbative solutions have been developed konno99, but can no longer be applied for field strengths which might influence matter properties.
There have been several attempts to determine neutron star structure assuming an ad hoc profile of the magnetic field, without solving Maxwell equations within the TOV system (see e.g. casali14; sotani17; chu18). To that end, many authors employ the parameterization introduced twenty years ago by Bandyopadhyay et al. bandyopadhyay97, where the variation of the magnetic field norm with baryon number density from the centre to the surface of the star is given by the form
(1) 
with two parameters and , chosen to obtain the desired values of the maximum field at the centre and at the surface.
Lopes and Menezes lopes15 later introduced a variable magnetic field, which depends on the energy density rather than on the baryon number density:
(2) 
where is the energydensity of the matter alone, is the central energy density of the maximum mass nonmagnetic neutron star and a parameter , arguing that this formalism reduces the number of free parameters from two to one. The authors put forward as additional motivation the fact that it is the energy density and not the number density that is relevant in TOV equations for structure calculations. To account for anisotropy in the shear stress tensor, they applied the above field profile in the chaotic magnetic field formalism, taking the shear stress tensor as diag() menezes16.
There have also been suggestions of the magnetic field profile being a function of the baryon chemical potential dexheimer12 as:
(3) 
with , and given in MeV. In contrast to the profiles in Eqs. (1,2), such a formula avoids that a phase transition induces a discontinuity in the effective magnetic field. Dexheimer et al. dexheimer17 suggested, too, a fit to the shapes of the magnetic field profiles in the polar direction as a function of the chemical potentials (as in dexheimer12) by quadratic polynomials instead of exponential ones as
(4) 
where are coefficients determined from the numerical fit.
However, it was subsequently pointed out by Menezes and Alloy menezes16b that such ad hoc formulations for magnetic field profiles are physically incorrect since they do not satisfy Maxwell’s equations. In particular it is obvious that assuming such a magnetic field profile in a spherically symmetric star implies a purely monopolar magnetic vector field distribution, which is incorrect. The inconsistency of this type of approach can be seen, too, by inspecting the most general solution of the equations of hydrostatic equilibrium in general relativity for a spherically symmetric star. In Schwarzschild coordinates, :
(5) 
the resulting coupled system of equations for the star’s structure has been derived by Bowers and Liang bowers74 and reads
(6) 
with an energymomentum tensor of the form , where and are the radial and tangential pressure components. This is the most general energymomentum tensor one can use assuming spherical symmetry and it goes beyond the perfectfluid model, for which . One may be tempted to cast a general electromagnetic energymomentum tensor assuming a perfect conductor and isotropic matter, and for a magnetic field pointing in direction (see e.g. chatterjee15) into this form. However, in the case of the electromagnetic energymomentum tensor (look at Eqs. (23d)(23e) of chatterjee15), in clear contradiction with the assumption of Bowers and Liang (6) in spherical symmetry. Another problem arises from the fact that and thus, the last term in Eq. (6) diverges at the origin. This discussion shows that there cannot be any correct description of the magnetic field in spherical symmetry.
In Ref. mallick, a density dependent profile is applied within a perturbative axisymmetric approach à la Hartle and Thorne hartle68. It remains, however, that the star’s deformation due to the magnetic field implies that such a density (or equivalent) dependent profile depends on the direction, thus will be different looking e.g. in the polar or the equatorial direction.
In view of all these intrinsic difficulties, we will not propose here a simple scheme for solving structure equations of magnetized stars – to that end we refer to the publicly available numerical codes assuming axial symmetry chatterjee15; xns. Instead, since in many cases it might be sufficient to have an idea of the order of the value of the magnetic field strength to test its potential effect on matter properties, our aim is to provide a “universal” magnetic field strength profile from the surface to the interior obtained from the field distribution in a fully selfconsistent numerical calculation from one of these codes. Further, we probe the applicability of this profile for determining the structure of magnetized neutron stars in an approximate way in spherical symmetry compared with full numerical structure calculations. As we will show, qualitatively the correct tendency can be reproduced for some NS properties, but to reproduce quantitatively correct results, the full solution has to be applied.
The paper is organized as follows. Sec. II describes our physical models, including the EoSs we use in this manuscript, together with the numerical techniques applied to solve the models. Sec. III provides the magnetic field profiles derived numerically by varying certain physical parameters, to achieve a generic profile for the monopolar part of the norm of the magnetic field. This profile is then applied in Sec. IV to a modified TOV system, to see its effect on NS masses and radii. Finally, Sec. V gives a summary of our work, together with some concluding remarks.
Ii Formalism and models
In this section, we summarize the numerical approach for selfconsistently modelling magnetized neutron stars. More details can be found in in chatterjee15; bocquet95; chatterjee17.
Model  

(MeV)  (MeV)  (MeV)  (MeV)  ()  (km)  
HS(DD2)  0.149  16.0  243  31.7  55  2.42  13.2  810 
SFHoY  0.158  16.2  245  31.6  47  1.99  11.9  399 
STOS  0.145  16.3  281  36.9  111  2.23  14.5  1420 
BL_EOS  0.17  15.2  190  35.3  76  2.08  12.3  466 
SLy9  0.15  15.8  230  32.0  55  2.16  12.5  533 
SLy230a  0.16  16.0  230  32.0  44  2.11  11.8  401 

ii.1 Nonrotating magnetized neutron stars in general relativity
Due to the high compactness of neutron stars, we consider models within the theory of general relativity and solve coupled EinsteinMaxwell partial differential equations. We follow the scheme described in bonazzola93, who considered the general case of rotating neutron stars, with the assumptions of stationarity, axial and equatorial symmetry, and circular spacetime, where the metric is given in the quasiisotropic gauge, different from that used in TOV systems (5), by:
(7)  
where and are the gravitational potentials which are, as all other fields in Secs. IIIII, functions of the coordinates only (independent from the coordinate).
In this work, we shall restrict ourselves to the case without rotation, which in particular implies that there is no electric field in the models (perfect conductor). Nevertheless, as said in the introduction, the presence of a magnetic field induces a distortion of the stellar structure, which cannot remain spherically symmetric. Due to spacetime symmetries and circularity condition, only two magnetic field geometries can be described within this framework: a purely poloidal magnetic field (see bocquet95) or a purely toroidal one (see frieben12). In this work, we consider only purely poloidal magnetic fields, meaning that the only nontrivial components are and . This choice results in an asymptotically dipolar magnetic field distribution.
Matter is supposed to be composed of a perfect fluid coupled to the magnetic field. Since, as demonstrated in chatterjee15 the magnetic field dependence of the EoS and magnetization have only small effects on neutron star structure, we neglect them here. They could be included in a straightforward way chatterjee15. Matter is also assumed to be perfectly conducting and the magnetic field originates from free currents, moving independently from the perfect fluid. Equilibrium equations are obtained from the divergencefree condition of the energymomentum tensor, and can be written as a first integral of motion, bonazzola93; chatterjee15. It is mostly the Lorentz force term in this equilibrium equation which distorts the stellar structure and makes it deviate from spherical symmetry. To summarize, given an equation of state (EoS) for nuclear matter (see Sec. II.3 hereafter), we thus solve the system of coupled EinsteinMaxwell equations, together with magnetostatic equilibrium. These models are then characterized by their gravitational mass (, see bonazzola93 for a definition), their EoS (see Sec. II.3) and the central magnetic field, .
ii.2 Numerical methods
The equations to be solved to get axisymmetric solutions form a set of six nonlinear elliptic (Poissonlike) partial differential equations, coupled together with noncompact support (sources for gravitational field extend up to spatial infinity). These equations are solved using the same procedure as described in bocquet95, employing the numerical library lorene lorene based on spectral methods for the representation of fields and the resolution of partial differential equations (see grandclement09).
Numerical accuracy of the axisymmetric solutions is checked through an independent test, the socalled relativistic virial theorem (bonazzola94; gourgoulhon94). This gives an upper bound on the relative accuracy of the obtained numerical solution, and we checked that it always remained lower than for the axisymmetric models presented in Sec. III.
ii.3 Equations of state
The system of equations described above is closed by the EoS for nuclear matter relating, the pressure to the baryon density . Our selection of EoSs for the present work has been guided by the idea to represent a large variety of different neutron star compositions and nuclear properties, derived from completely different nuclear physics formalisms. This was done in order to achieve an unbiased universal parameterization applicable to any realistic nuclear EoS. We consider a BHF calculation with chiral interactions (“BL_EOS”) bombaci18^{1}^{1}1The model calculation exist only for homogeneous matter and a crust has been added, see the CompOSE entry for details., two nonrelativistic Skyrme mean field models (“SLy9”and “SLy230a”)chabanat95; chabanat97; gulminelli15, two relativistic mean field models (“STOS” and “HS(DD2)”) shen98; typel09; hempel09 and one model with hyperons (“SFhoY”) fortin17. Some nuclear and neutron star properties of the different EoS models are listed in Table 1. All EoS data are available from the online database CompOSE compose.
Iii Generic magnetic field profile
The numerical models of neutron stars endowed with a magnetic field described in Sec. II.1 consider two components ( and ) of the magnetic field vector, as measured by the Eulerian observer (see bocquet95 for details). In the case of nonrotating stars considered here, this magnetic field is the same as that measured in the fluid restframe, denoted as and in chatterjee15. As an example, the magnetic field distribution of a full neutron star model is displayed in Fig. 1, for a central value of the magnetic field G. The surface of the star (thick line) does not exhibit any significant deviation from spherical shape, but it is clear that the magnetic field distribution is dominated by the dipolar structure and cannot be accurately described by any sphericallysymmetric model.
When trying to parameterize the magnetic field profile, the simplest approach is to consider the norm of the magnetic field, namely
(8) 
where the metric potential has been defined in Eq. (7). Note that is the quantity that enters the EoSs which take into account magnetization, as explained e.g. in chatterjee15. The central value of this magnetic field norm is denoted as (independent of ). In the rest of this work, we will consider this field as the main object of our study.
As stated in the introduction, several authors have considered a parameterization of the magnetic field norm by the baryon density . In Fig. 2 we have plotted, for the same neutron star model of and G as in Fig. 1, the norm of the magnetic field as a function of baryon number density , along two radial directions: for (passing through the pole) and for (passing through the equator). As these two curves show noticeable differences, including close to the center of the star (), it seems that this type of parameterization can induce some inconsistency when describing magnetic field in a neutron star. We therefore try to improve it and adopt a different approach, taking a multipolar expansion of the magnetic field norm ( being the spherical harmonic functions):
(9) 
In Fig. 3 we have plotted the first four nonzero terms of this multipolar decomposition as functions of the coordinate radius . Note that, because of the symmetry with respect to the equatorial plane, odd terms in the decomposition (9) are all zero. It appears that, at least in the highdensity central regions of the star, the monopolar term , which is spherically symmetric, is dominant over the others. It is important to stress here that, contrary to the magnetic (vector) field, which has no monopolar part in terms of vector spherical harmonics, the norm of the vector field considered here is a scalar field which can possess a monopolar component.
We then look at the behavior of the radial profile of when varying the neutron star model in Fig. 4. On the left panel, we vary the gravitational mass of the star (either or ), as well as the amplitude of the magnetic field central value ( G, G and G). These profiles are no longer displayed as functions of the quasiisotropic coordinate radius , defined by the line element (7), but in view of the application to TOVsystems in Sec. IV, we consider here the Schwarzschild coordinate radius , defined by the line element (5). The gauge transformation is obtained numerically and profiles are displayed as function of this radius divided by the star’s mean radius which is such that the integrated (coordinateindependent) surface of the star reads . Indeed, when the star gets distorted because of the magnetic field, it is difficult to define uniquely a relevant radius. In that sense, is directly connected to the star’s surface and some of its emission properties.
It is remarkable that, although all possible parameters defining a magnetized stellar model (mass, central magnetic field, EoS) have been varied, all profiles are quite similar and deviate one from another only by a few percent. The only case where a noticeable difference appears is when using quark matter EoS. Therefore, we make the following conjecture: the monopolar part of the norm of the magnetic field follows a universal profile, up to minor variations, when considering different neutron star models with realistic hadronic EoSs. This “universal” profile has been fitted using a simple polynomial:
(10) 
where is the ratio between the radius in Schwarzschild coordinates (5) and the star’s mean (or areal) radius. Let us stress that the aim of the present investigation is to obtain a universal profile for realistic EoS and that we have therefore excluded polytropic EoSs. A preliminary calculation showed that the general parameterization applicable to the family of realistic EoSs is not applicable directly to the case of polytropes without specific fine tuning.
Iv Application to a TOVlike system
As discussed in the introduction, it is fundamentally inconsistent to solve spherically symmetric equations for magnetized neutron star models since it completely neglects the star’s deformation due to the electromagnetic field. It is, however, tempting, to have a simple approach at hand which allows at least to qualitatively reproduce the effects of the magnetic field on (some) neutron star properties performing calculations only slightly more complicated than solving TOV equations. To that end, we modify the TOV system by adding the contribution from the magnetic field to the energy density and a Lorentz force term to the equilibrium equation:
(11) 
denotes here the Lorentz force contribution, which is noted in bonazzola93 (see this reference for more details).
Similar to the magnetic field norm , we found from the full numerical calculations the following parametric form
(12) 
where and the central magnetic field, , is given in units of G. For the magnetic field in Eqs. (11), the profile (10) is applied.
In order to get an idea of the quality of this “TOVlike” approach, we show in Figs. (5,6) a comparison between results obtained with the TOVlike approach in spherical symmetry and the full numerical solution in axial symmetry. In Fig. (5) the gravitational mass vs. the mean radius is displayed for a central magnetic field of G. As expected, deviations become larger at smaller masses since the ratio of magnetic to matter pressure increases and thus the stars are more strongly deformed and the relation between the mean radius and the radius of a spherically symmetric configuration is no longer obvious. The same is true if the central magnetic field is increased.
Masses should be less sensitive to the deformation. As can be seen in Fig. (6), indeed the TOVlike solution for the maximum gravitational mass as well as the gravitational mass for fixed baryon mass as function of the central magnetic field show the correct qualitative behavior. Both increase with and the difference to the exact solution remains reasonably small up to central fields of G. This difference of course becomes more pronounced with increasing magnetic field since magnetic effects become more important and the limits of the TOVlike approach can be clearly seen.
Thus, although our investigations can serve as a guideline and reproduce at least for gravitational mass as function of magnetic field the correct qualitative tendency, it should be stressed that it is strongly recommended to use a correct axisymmetric approach (e.g. employing publicly available software), to determine properties of magnetized neutron stars.
V Conclusions
Many attempts can be found in the literature trying to study strongly magnetized neutron stars and to include magnetic field effects on the matter properties. As mentioned in the introduction, most of these investigations suffer from different assumptions and approximations motivated by the complexity of the full system of equations. First, in order to avoid solving Maxwell’s equations in addition to equilibrium and Einstein equations, often an ad hoc profile for the magnetic field is assumed, which has no physical motivation. Second, spherical symmetry is assumed for modelling the star.
In this work, we tackle the first point: we proposed a “universal” parameterization of the magnetic field profile (Eq. 10) as a function of dimensionless stellar radius, obtained from a full numerical calculation of the magnetic field distribution. We tested this profile against several realistic hadronic EoSs, based on completely different analytic approaches, and with different magnetic field strengths in order to confirm its universality. For the case of quark matter EoSs, preliminary investigations showed that although MIT bag models conform to the universality, other quark matter EoSs may not necessarily do so. The profile is intended to serve as a tool for nuclear physicists for practical purposes, namely to obtain an estimate of the maximum field strength as a function of radial depth (within error bars), in order to deduce the composition and related properties.
We applied the proposed magnetic field profile in a modified TOVlike system of equations, that include the contribution of magnetic field to the energy density and pressure, and account for the anisotropy by introducing a Lorentz force term. Compared with full numerical structure calculations, we find that qualitatively the correct tendency is reproduced and quantitatively the agreement is acceptable for large masses and small magnetic fields. Thus, although we encourage to employ the profile proposed here to conclude about the importance of magnetic field effects on matter properties, we can only recommend the use of a full axisymmetric numerical solution for modelling magnetized neutron stars.