Magnetic field effects on the crust structure of neutron stars

We study the effects of high magnetic fields on the structure and on the geometry of the crust in neutron stars. We find that the crust geometry is substantially modified by the magnetic field inside the star. We build stationary and axis-symmetric magnetized stellar models by using well-known equations of state to describe the neutron star crust, namely, the Skyrme model for the inner crust and the Baym-Pethick-Sutherland equation of state for the outer crust. We show that the magnetic field has a dual role, contributing to the crust deformation via the electromagnetic interaction (manifested in this case as the Lorentz force) and by contributing to curvature due to the energy stored in it. We also study a direct consequence of the crust deformation due to the magnetic field: the thermal relaxation time. This quantity, which is of great importance to the thermal evolution of neutron stars, is sensitive to the crust properties, and, as such, we show that it may be strongly affected by the magnetic field.

Figure 1

Thickness of the crust $\mathrm{\Delta }{r}_{\mathrm{eq}}$ in the equatorial plane ($\theta =\pi /2$) as a function of central magnetic fields for stars at fixed baryon masses of $M_B=1.40\hspace{0.17em}{M}_{\odot }$ and $M_B=2.00\hspace{0.17em}{M}_{\odot }$.

Figure 2

Crust thickness $\mathrm{\Delta }{r}_{\text{mag}}$ in different angular directions $\theta$ for highly magnetized stars. These objects are the most magnetized stars as depicted in Fig. 1. For a star at a fixed baryon mass of $M_B=1.40\hspace{0.17em}{M}_{\odot }$, the central magnetic field is $\sim 0.9\times {10}^{18}\mathrm{G}$, while $M_B=2.00\hspace{0.17em}{M}_{\odot }$ has $\sim 1.3\times {10}^{18}\mathrm{G}$. The horizontal lines represent the crust thickness for spherical solutions (without magnetized fields), $\mathrm{\Delta }{r}_0^{(1.40)}=0.994\mathrm{km}$ and $\mathrm{\Delta }{r}_0^{(2.00)}=0.604\mathrm{km}$ for a star with $M_B=1.40\hspace{0.17em}{M}_{\odot }$ and $M_B=2.00\hspace{0.17em}{M}_{\odot }$, respectively.

Figure 3

Magnetic potential $M$ as a function of the coordinate radius in different directions inside the stars. The corresponding stars are depicted in Fig. 2.

Figure 4

Core thickness for the same stars as shown in Fig. 2.

Figure 5

Physical quantities presented in the equation of motion Eq. (4) as a function of the circular equatorial radius $R_{\text{circ}}$ for a star with $M_B=2.00M_{\odot }$. The upper plot represents a spherical and nonmagnetized solution, and the middle figure shows the same star, but highly magnetized. The central magnetic field is $1.3\times {10}^{18}\mathrm{G}$. We highlight the direction of the Lorentz force in the star in the lower panel. The vertical lines are the crust line and the stellar surface.

Figure 6

Average crust thickness for different values of surface magnetic fields.

Figure 7

Relaxation time for the $1.40M_{\odot }$ star. The three curves indicate the three cases investigated, namely, $t_1=28.8\mathrm{yr}$ (case 1), associated with absence of superfluidity in the crust; $t_1=11.1\mathrm{yr}$, associated with weak superfluidity in the crust (case 2); and $t_1=8.2\mathrm{yr}$ for the case of strong crustal superfluidity (case 3).

Figure 8

Same as Fig. 7 but for the $2.0\hspace{0.17em}{M}_{\odot }$ star.

Figure 9

Relaxation time normalized by the relaxation time of stars with vanishing magnetic fields, as a function of surface magnetic fields for star at 1.40 and $2.0\hspace{0.17em}{M}_{\odot }$.