Torsional oscillations of magnetized neutron stars with mixed poloidal-toroidal fields

The quasiperiodic oscillations found in the three giant flares of soft gamma-ray repeaters observed to date have been interpreted as torsional oscillations caused by a starquake related to a magnetospheric reconnection event. Motivated by these observations, we study the influence of the magnetic field geometry in the frequencies of the torsional oscillations of magnetized neutron stars. We use realistic tabulated equations of state for the core and crust of the stars and model their magnetic field as a dipole plus a toroidal component, using the relativistic Grad-Shafranov equation. The frequencies of the torsional modes are obtained by the numerical solution of the eigenvalue problem posed by the linear perturbation equations in the Cowling approximation. Our results show how the asteroseismology of these stars becomes complicated by the degeneracy in the frequencies due to the large relevant parameter space. However, we are able to propose approximately equation-of-state–independent relations that parametrize the influence of the magnetic field in the torsional oscillations, as well as a testable scenario in which the rearrangement of the magnetic field causes an evolution of the frequencies. Finally, we show that there is a magnetic field configuration that maximizes the energy in the perturbation at linear order, which could be related to the trigger of the giant flare.

Figure 1

Mass-radius relation for the $\mathrm{crust}+\mathrm{core}$ equations of state used in our study. The core EOS determines the bulk properties of the star, but the crust EOS will be the most important ingredient for the characteristics of the torsional oscillations.

Figure 2

Polynomial fits for the shear modulus $\mu$ as a function of the density $\rho$ for our set of crust equations of state.

Figure 3

Magnetic field lines for typical configurations with a dipolar field and an increasing toroidal field component. All four stars have the $\mathrm{APR}+\mathrm{Gs}$ EOS, $M=1.4M_{\odot }$, and magnetic field amplitude $B={10}^{15}\mathrm{G}$ at the pole. The outer black circle indicates the surface of the star, while the inner circle indicates the base of the crust. The ratio $\zeta$ between the poloidal and toroidal components is 0.00 (top left), 0.26 (top right), 0.28 (bottom left), and 0.30 (bottom right). We restrict our range for the parameter $\zeta$ for each EOS in order to prevent the formation of disjoint domains for the magnetic field lines as shown in the bottom right plot.

Figure 4

Radial profiles of the magnetic field components $B_r$ (at the axis $\theta =0$), $B_{\theta }$, and $B_{\varphi }$ (at the equatorial plane $\theta =\pi /2$), represented with solid, dashed, and dotted lines, respectively, calculated inside the star with Eqs. (4a)–(4c) for the same configurations presented in Fig. 3. The poloidal components $B_r$, $B_{\theta }$ extend continuously outside the star, but the toroidal component $B_{\varphi }$ is set to zero at the surface. The configuration with higher $\zeta$ and disjoint domains for the field lines presents a node in the $B_r$ component inside the star, as can be seen in the bottom right panel.

Figure 5

(Upper) Frequency ${{}_{2}f}_0$ of the fundamental ($n=0$) $l=2$ mode of torsional oscillations for nonmagnetized stars with different masses. (Lower) Same as the upper plot, but for magnetized stars with a fixed mass $M=1.4M_{\odot }$ as a function of the magnetic field strength at the pole $B$, normalized by $B_{\mu }=4\times {10}^{15}\mathrm{G}$.

Figure 6

Influence of the magnetic field geometry on the frequencies of the first modes of torsional oscillations for a sequence of $1.4M_{\odot }$ $\mathrm{SLy}+\mathrm{SLy}$ stars. The sequence has a fixed amplitude $10^{15}\mathrm{G}$ of the magnetic field at the pole and increasing toroidal field starting with a pure dipole configuration at $\zeta =0$. The lowest modes are more sensitive to $\zeta$, but the effect of increasing the toroidal field component becomes nonmonotonic for higher modes.

Figure 7

Same as Fig. 6, but only for the $l=2$ mode and for our set of nine combinations of $\mathrm{core}+\mathrm{crust}$ equations of state. The crust EOS is the most relevant for the behavior of the frequencies of the torsional oscillations.

Figure 8

Behavior of the coefficient ${{}_2\alpha }_0$ of the magnetic field correction to the fundamental torsional frequency, defined in Eq. (13), as a function of the magnetic field geometry as parametrized by $\zeta$.

Figure 9

Coefficients ${\alpha }_1$ and ${\alpha }_2$, defined in Eq. (14), as functions of the stellar parameters. For the case of a purely dipolar field, the correction to the torsional frequency is entirely due to ${\alpha }_1$. The correlation found for ${\alpha }_1$ is close to EOS independent, but the spread is larger in the case of ${\alpha }_2$.

Figure 10

Same as Fig. 7, but for sequences of stars with fixed total magnetic energy. The frequencies still increase for configurations with larger toroidal field contributions, but the fractional variation is approximately half of that in the case with a constant $B$ field at the pole of the star presented in Fig. 7.

Figure 11

Energy in the fundamental $l=2$ mode integrated inside the stellar crust for the sequences of stars with fixed total magnetic energy presented in Fig. 10, normalized by the configuration with maximum energy for each $\mathrm{core}+\mathrm{crust}$ EOS. The local maxima could be linked to the emission of the giant flare.