Magnetic field distribution in magnetars

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, which we fit by a simple eighth-order 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 Tolman–Oppenheimer–Volkoff system of equations. This new formalism produces slightly better results in terms of mass-radius diagrams than previous attempts to add magnetic terms to these equations. However, we show that such approaches are less accurate than the usual, nonmagnetized TOV models, and that consistent models must depart from spherical symmetry. Thus, our “universal” magnetic-field profile is intended to serve as a tool for nuclear physicists to obtain estimates of the magnetic field inside neutron stars as a function of radial depth to deduce its influence on composition and related properties. It possesses the advantage of being based on magnetic-field distributions from realistic self-consistent computations which are solutions of Maxwell's equations.

Figure 1

Magnetic-field lines in the $(x,z)$ plane for a $M_G=2M_{\odot }$ neutron-star model endowed with a magnetic field whose central value is $b_c=5\times {10}^{17}\mathrm{G}$ and using the SLy203a EoS of Table 1. Thick line denotes the surface of the star and the magnetic moment is along the $z$ axis.

Figure 2

Magnetic-field norm $b$ (8) as a function of the baryon density for two angular directions for the same stellar model as in Fig. 1.

Figure 3

Radial profiles of the first four even multipoles, $b_{\ell },\ell =0,2,4$ and $\ell =6$, see definition (9), of the magnetic-field norm $b(r,\theta )$ computed for the stellar model described in Fig. 1. From symmetry arguments odd multipoles are all zero.

Figure 4

Monopolar part of the magnetic-field profile $b_0\left(r\right)$, normalized to its central value for different magnetized neutron-star models, as functions of the radius expressed in Schwarzschild coordinates $\overline{r}$ [see expression (4)] divided by the star's mean radius (see text). (a) all models are using SLy230a EoS (see Table 1) but have different masses ($1.6M_{\odot }$ or $2M_{\odot }$) and different central magnetic fields $b_c$ (${10}^{15}\mathrm{G}, {10}^{17}\mathrm{G}$, or ${10}^{18}\mathrm{G}$). (b) All are $2M_{\odot }$ models, with a central magnetic field $b_c=5\times {10}^{17}\mathrm{G}$, but with different EoSs; see Table 1 for details.

Figure 5

Same as Fig. 4 but for the dipole term $b_2\left(r\right)$ of the multipolar expansion (9) of the magnetic-field norm.

Figure 6

Gravitational mass vs mean radius stellar sequences for the TOV-like solution, TOV solution using a directionally averaged energy-momentum tensor (see text) and the exact (axisymmetric) calculation for a central magnetic field of $b_c={10}^{17}\mathrm{G}$ (left) and $b_c={10}^{18}\mathrm{G}$ (right) employing the SLy230a EoS. Note that, in both figures, the thin-dotted lines ($B=0$) correspond to solutions of the TOV system with no magnetic field at all.

Figure 7

(left) Comparing the values of the maximum gravitational mass as function of the central magnetic field between our TOV-like approach, the averaged TOV and the full solution. (right) Gravitational mass as function of the central magnetic field for two different values of the baryonic mass.