Precursor flares of short gamma-ray bursts from crust yielding due to tidal resonances in coalescing binaries of rotating, magnetized neutron stars

As evidenced by the coincident detections of GW170817 and GRB 170817A, short gamma-ray bursts are likely associated with neutron star–neutron star merger events. Although rare, some bursts display episodes of early emission, with precursor flares being observed up to $\sim 10\mathrm{seconds}$ prior to the main burst. As the stars inspiral due to gravitational wave emission, the exertion of mutual tidal forces leads to the excitation of stellar oscillation modes, which may come into resonance with the orbital motion. Mode amplitudes increase substantially during a period of resonance as tidal energy is deposited into the star. The neutron-star crust experiences shear stress due to the oscillations and, if the resonant amplitudes are large enough, may become overstrained. This overstraining can lead to fractures or quakes which release energy, thereby fueling precursor activity prior to the merger. Using some simple Maclaurin spheroid models, we investigate the influence of magnetic fields and rapid rotation on tidally forced $f$ and $r$ modes, and connect the associated eigenfrequencies with the orbital frequencies corresponding to precursor events seen in, for example, GRB 090510.

Figure 1

Decay of the orbital separation $a$, normalized by stellar radius $R=13\mathrm{km}$, as a function of time for a binary with $M=M_{\text{comp}}=1.4M_{\odot }$. The vertical dashed lines mark 0.5 s and 13 s prior to coalescence (occurring when $a\lesssim 3R$ [63]), which, assuming that the main burst occurs at coalescence $t=t_{\mathrm{C}}$, are the times at which precursor flares for GRB 090510 were identified [7].

Figure 2

$\ell =m=2$ $f$-mode [black curves: the solid line shows the Maclaurin value, while the dotted line shows the realistic EOS expression (21) of Ref. [91] ] and $\ell =3$, $m=2$ $r$-mode (red dashed curve) mode frequencies, normalized by the dynamical frequency $\sqrt{\pi G\rho }$, as a function of spin frequency $\nu$.

Figure 3

Field lines for the background magnetic field $\mathit{B}$ for a rotating configuration with $\nu =300\mathrm{Hz}$. Even at this spin frequency, the stellar surface is roughly spherical; the red dashed curve shows the (true) stellar surface [Eq. (17)], while the blue curve shows the spherical surface $x^2+z^2=R^2$.

Figure 4

Field lines for the total magnetic field $\mathit{B}+\delta \mathit{B}$, where $\delta \mathit{B}$ is induced by $f$ (left panel: ${\alpha }_f=0.04$) and $r$ (right panel: ${\alpha }_r=0.02$) modes at $t=0$ for a rotating configuration with $\nu =300\mathrm{Hz}$. The field lines are laterally shifted (for $f$ modes) and twisted (for $r$ modes) relative to the background dipole configuration due to the mode perturbations. The red dashed curve shows the perturbed stellar surface, which is different for different mode perturbations, while the blue curve shows the background Maclaurin surface.

Figure 5

Eigenfrequency shifts $\delta {\omega }^B$ for $f$ modes [red dashed ($\delta {\omega }_f^B>0$) and dotted ($\delta {\omega }_f^B<0$) curves] and $r$ modes (black curve) due to the magnetic field as a function of the poloidal-to-toroidal field-strength ratio $\mathrm{\Lambda }$. We have taken $\nu =300\mathrm{Hz}$, $M=1.4M_{\odot }$, and $R=13\mathrm{km}$.

Figure 6

Unperturbed (${\omega }_{0,i}$, black curve) and perturbed [Eq. (50), red dashed curve] $f$-mode frequencies in the inertial frame as a function of spin frequency $\nu$, where we set $R=13\mathrm{km}$ (left panel) and $R=15\mathrm{km}$ (right panel). Overplotted are twice the values of the orbital frequencies relevant for GRB 090510 (cf. Fig. 1).

Figure 7

Unperturbed (${\omega }_{0,i}$, black curve) and perturbed [Eq. (50), red dashed curve] $r$-mode frequencies in the inertial frame as a function of spin frequency $\nu$, where we set $\mathrm{\Lambda }=1$ (left panel) and $\mathrm{\Lambda }=0.3$ (right panel). Overplotted are twice the values of the orbital frequencies relevant for GRB 090510 (cf. Fig. 1). Note that ${\omega }_i<0$ for these modes; they are subject to the gravitational radiation (CFS) instability [88, 89, 90].

Figure 8

Crustal strain $\sigma$ for the Lagrangian displacement induced by $r$ modes [Eq. (28)] for $\nu =100\mathrm{Hz}$ (left panel) and $\nu =500\mathrm{Hz}$ (right panel), with amplitude ${\alpha }_r=2\times {10}^{-4}$. Brighter shades indicate a greater value for $\sigma$. The black curve indicates the crust-core transition, while the red curve shows the (spin-frequency-dependent) stellar surface. (The crust has been stretched by a factor of 4 for improved visibility).