Magnetar giant flare oscillations and the nuclear symmetry energy

If the observed quasi-periodic oscillations in magnetar flares are partially confined to the crust, then the oscillation frequencies are unique probes of the nuclear physics of the neutron star crust. We study crustal oscillations in magnetars including corrections for a finite Alfvén velocity. Our crust model uses a new nuclear mass formula that predicts nuclear masses with an accuracy very close to that of the finite range droplet model. This mass model for equilibrium nuclei also includes shell corrections and an updated neutron-drip line. We perturb our crust model to predict axial crust modes and assign them to observed giant flare quasi-periodic oscillation frequencies from the soft $\gamma$-ray repeater SGR 1806-20. We find magnetar crusts that match observations for various magnetic field strengths, entrainment of the free neutron gas in the inner crust, and crust-core transition densities. We find that observations can be reconciled with smaller values of the symmetry energy slope parameter, $L$, if there is a significant amount of entrainment of the neutrons by the superfluid or if the crust-core transition density is large. We also find neutron star masses and radii which are in agreement with expectations from what is known about low-density matter from nuclear experiment. Matching observations with a field-free model we obtain the approximate values of $M=1.35M_{\odot }$ and $R=11.9\mathrm{km}$. Matching observations using a model with the surface dipole field of SGR 1806-20 $\left(B=2.4\times {10}^{15}\mathrm{G}\right)$ we obtain the approximate values of $M=1.25M_{\odot }$ and $R=12.4\mathrm{km}$.

Figure 1

Equilibrium composition of the crust without a magnetic field for (a) a model that neglects shell effects and (b) a model that includes them. The blue lower curve corresponds to the proton number $Z$ and the black upper curve corresponds to the neutron number $N$.

Figure 2

Alfvén velocity (blue curve) and shear velocity (black curve) in the crust as a function of mass density. The composition is that of a $1.4M_{\odot }$ neutron star using the SLy4 crust EOS.

Figure 3

Frequency of the fundamental $l=2$ mode as a function of the magnetar mass for the core EOS probability distribution (centroid and $\pm 2\sigma )$ from Steiner et al. [17] and an SLy4 crust EOS. The dashed black line indicates the observed $29\mathrm{Hz}$ QPO of SGR 1806-20. The frequencies are evaluated for a crust-core transition density of $0.12{\mathrm{fm}}^{-3}$ with $B=0\mathrm{G}$.

Figure 4

Magnetar mass as a function of radius for the core EOS probability distribution from Steiner et al. [17]. Frequencies are evaluated using (a) the SLy4 crust EOS $\left(L=46\mathrm{MeV}\right)$ and (b) the Rs crust EOS $\left(L=86\mathrm{MeV}\right)$, both for $n_{\mathrm{t}}=0.12{\mathrm{fm}}^{-3}$. The thick red solid line indicates masses and radii for which the fundamental mode has a frequency of $29\mathrm{Hz}$ in the case of SLy4 and $18\mathrm{Hz}$ in the case of Rs. The black short-dashed line indicates masses and radii for a $626\mathrm{Hz}$ harmonic mode and $B=0\mathrm{G}$. Masses and radii from $626\mathrm{Hz}$ harmonic modes with magnetized crusts are labeled accordingly. Arrows indicate masses and radii that match both the fundamental and the harmonic modes for the field-free case and the case with the magnetic field of SGR 1806-20 $\left(B=2.4\times {10}^{15}\mathrm{G}\right)$.

Figure 5

Same as Fig. 4, but for the SLy4 crust EOS with $n_{\mathrm{t}}=0.08{\mathrm{fm}}^{-3}$. The thick red solid line indicates masses and radii determined from a fundamental mode of $29\mathrm{Hz}$. Masses and radii from harmonic modes with magnetized crusts are labeled accordingly. Arrows indicate masses and radii that match both the fundamental and the harmonic modes for the field-free case and the case with the magnetic field of SGR 1806-20 $\left(B=2.4\times {10}^{15}\mathrm{G}\right)$.

Figure 6

Magnetar mass as a function of radius for the core EOS probability distribution from Steiner et al. [17]. Frequencies are evaluated using the SLy4 crust EOS with $B=0\mathrm{G}$ and $n_{\mathrm{t}}=0.12{\mathrm{fm}}^{-3}$. The red dot-dashed, blue dotted, and black dashed lines indicate masses and radii from fundamental modes of frequency $29\mathrm{Hz}$ for different free neutron entrainment fractions $f_{\mathrm{ent}}$. The shaded band indicates masses and radii from $626\mathrm{Hz}$ harmonic modes as $f_{\mathrm{ent}}$ is varied from $0.50$ to $1.0$. Arrows indicate the masses and radii that match both the fundamental and the harmonic modes for $f_{\mathrm{ent}}=1.0,0.75$, and $0.50$.

Figure 7

Same as Fig. 6, but for the Rs crust EOS with $B=0\mathrm{G}$. Here the free neutron entrainment fraction $f_{\mathrm{ent}}$ is varied from 0.20 to 0.30, with $f_{\mathrm{ent}}$ labeled next to the corresponding curves. The red dot-dashed, blue dotted, and black dashed lines indicate masses and radii from the $29\mathrm{Hz}$ fundamental mode. The shaded band indicates masses and radii from the $626\mathrm{Hz}$ harmonic mode. Arrows indicate the masses and radii for $f_{\mathrm{ent}}=0.30,0.25$, and $0.20$ that match both the fundamental and harmonic modes.