Deformation of a magnetized neutron star

Magnetars are compact stars, which are observationally determined to have very strong surface magnetic fields of the order of ${10}^{14}–{10}^{15}$ G. The center of the star can potentially have a magnetic field several orders of magnitude larger. We study the effect of the field on the mass and shape of such a star. In general, we assume a nonuniform magnetic field inside the star, which varies with density. The magnetic energy and pressure as well as the metric are expanded as multipoles in spherical harmonics up to the quadrupole term. Solving the Einstein equations for the gravitational potential, one obtains the correction terms as functions of the magnetic field. Using a nonlinear model for the hadronic EoS the excess mass and change in equatorial radius of the star due to the magnetic field are quite significant if the surface field is ${10}^{15}$ G and the central field is about ${10}^{18}$ G. For a value of the central magnetic field strength of $1.75\times {10}^{18}$ G, we find that both the excess mass and the equatorial radius of the star changes by about $3–4\%$ compared to the spherical solution.

Figure 1

$\delta M$ as a function of central energy density ${\varepsilon }_c$ for fixed $B_c$ (a) and as a function of central magnetic field $B_c$ for fixed central density $n_c$ (b). Curves are plotted for two different EoS (TM1 and NW). The central magnetic field $B_c$ and central density $n_c$ are specified in the figures.

Figure 2

The eccentricity $e$ as a function of ${\varepsilon }_c$ for fixed $B_c$ and as a function of $B_c$ for fixed $n_c$ is plotted. For comparison we have plotted curves for two different EoS.

Figure 3

Radii $R_e,R$ plotted as functions of central energy density (for $B_c=1.75\times {10}^{18}$ G) and as a function of central magnetic field (for $n_c=6n_0$). Curves for two different EoS, TM1 and NW model are shown.

Figure 4

Mass-radius curve for TM1 and NW models.