Gravitational wave transient signal emission via Ekman pumping in neutron stars during post-glitch relaxation phase

Glitches in the rotational frequency of a spinning neutron star could be promising sources of gravitational wave signals lasting between a few microseconds to a few weeks. The emitted signals and their properties depend upon the internal properties of the neutron star. In neutron stars, the most important physical properties of the fluid core are the viscosity of the fluid, the stratification of flow in the equilibrium state, and the adiabatic sound speed. Such models were previously studied [C. A. van Eysden and A. Melatos, Classical Quantum Gravity 25, 225020 (2008); M. F. Bennett, C. A. van Eysden, and A. Melatos, Mon. Not. R. Astron. Soc. 409, 1705 (2010)] following simple assumptions on all contributing factors, in which the post-glitch relaxation phase could be driven by the well-known process of Ekman pumping [G. Walin, J. Fluid Mech. 36, 289 (1969); M. Abney and R. I. Epstein, J. Fluid Mech. 312, 327 (1996)]. We explore the hydrodynamic properties of the flow of fluid during this phase following more relaxed assumptions on the stratification of flow and the pressure-density gradients within the neutron star than previously studied. We calculate the time scales of duration as well as the amplitudes of the resulting gravitational wave signals, and we detail their dependence on the physical properties of the fluid core. We find that it is possible for the neutron star to emit gravitational wave signals in a wide range of decay time scales and within the detection sensitivity of aLIGO for selected domains of physical parameters.

Figure 1

The idealized system.

Figure 2

Frequency characteristics of emitted signals for a polar observer: three different sets of data are plotted for the rotational frequency $\mathrm{\Omega }$ of 10 Hz and ${\partial }_z\eta \sim {10}^{-11}$. The respective color-coded time scales are 9.8, $1.1\times {10}^{-4}$, and $8.5\times {10}^{-7}$ days. The corresponding resonant frequencies are $\pm 125.66371$, $\pm 125.66375$, and $\pm 126.39611\mathrm{Hz}$. Note that values of the time scales are calculated for specifically chosen physical parameters of the system—$v_{\mathrm{c}}$, ${\partial }_z\eta$, $v_{\mathrm{eq}}$, K, and F—in order to cover a large range of time scales. A similar result for an equatorial observer is shown in Fig. 3.

Figure 3

Frequency characteristics of emitted signals for an equatorial observer: three different sets of data are plotted for the rotational frequency $\mathrm{\Omega }$ of 10 Hz and ${\partial }_z\eta \sim {10}^{-11}$. The respective color-coded time scales are 9.8, $1.1\times {10}^{-4}$, and $8.5\times {10}^{-7}$ days. Moreover, the corresponding resonant frequencies for $\times$ polarization are $\pm 62.83186$, $\pm 62.83188$, $\pm 63.19805\mathrm{Hz}$, whereas the resonant frequencies for the $+$ polarization remain exactly the same as they were for the case of a polar observer. The other physical parameters are chosen to be the same as in Fig. 2.

Figure 4

Emitted gravitational wave strain (turn page sideways). (Top panel) Time scales for the $t_{21}$ mode and the corresponding gravitational wave strain amplitudes for three sets of values for ${\partial }_z{v}_{\mathrm{c}}$: 0, ${10}^{-3}c{\mathrm{L}}^{-1}$, $-{10}^{-4}c{\mathrm{L}}^{-1}$ (left to right), respectively. (Bottom panel) The $t_{11}$ modes and corresponding gravitational wave strain amplitudes. The parameters are set to $f=100\mathrm{Hz}$, $\mathrm{E}={10}^{-7}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1.0\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g=10^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}=10^{17}\mathrm{kg}/{\mathrm{m}}^3$. All positive time scales, as well as the corresponding emitted amplitudes, are marked with a •, while the negative time scales and corresponding amplitudes are marked with a $\mathbf{+}$. Negative time scales correspond to the scenario of growing modes.

Figure 5

Outlier enlarged. The characteristics of the parameter space in the vicinity of the apparent outlier in Fig. 4 for the case of ${\partial }_z{v}_{\mathrm{c}}=-{10}^{-4}c{\mathrm{L}}^{-1}$ (the rightmost panels in Fig. 4) are shown in higher resolution. All positive time scales as well as the corresponding emitted amplitudes are marked with a •, while the negative time scales and the corresponding strain amplitudes are marked with a $\mathbf{+}$.

Figure 6

Sensitivity to E. The characteristics of emitted signals for $\mathrm{E}={10}^{-14}$ and ${\partial }_z{v}_{\mathrm{c}}=0$ are shown, setting $f=100\mathrm{Hz}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g={10}^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}={10}^{17}\mathrm{kg}/{\mathrm{m}}^3$. Note the emission of loud amplitudes within the range ${\mathrm{N}}^2\in ({10}^{-4},{10}^{-2})$; the equivalent classical Brunt-Väisälä frequency ${\mathrm{N}}_{\mathrm{c}}^2$ for this range lies within permitted physical expectations.

Figure 7

Sensitivity vs $f$. The characteristics of emitted ${\mathrm{signals}}^{24}$ and $h_{\mathrm{o}}^{\min }$ are shown as a function of the neutron star’s rotational frequency $f$ as well as the emitted resonant frequency $f_{\mathrm{R}}$, given as $f_{\mathrm{R}}={\omega }_{\mathrm{R}}/2\pi$. We have set $v_{\mathrm{c}}^{\mathrm{o}}=0.1c$, ${\mathrm{N}}^2={10}^{-4}$, $\mathrm{E}={10}^{-10}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g=10^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}=10^{17}\mathrm{kg}/{\mathrm{m}}^3$, ${\partial }_z{v}_{\mathrm{c}}=0$. Note that the emitted amplitudes are largely insensitive to ${\partial }_z{v}_{\mathrm{c}}$ for the chosen points in $\{v_{\mathrm{c}}^{\mathrm{o}},{\mathrm{N}}^2\}$ parameter space. The multidetector amplitude spectral density $\sqrt{{\mathrm{S}}_h(\omega )}$ is calculated by taking the harmonic mean of the individual amplitude spectral densities of the H1 (aLIGO Hanford) and L1 detectors (aLIGO Livingston) measured during initial days of the O1 run, i.e., September 12 through October 20, 2015.

Figure 8

Energetics. We have set $\mathrm{E}={10}^{-7}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g=10^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}=10^{17}\mathrm{kg}/{\mathrm{m}}^3$, and ${\partial }_z{v}_{\mathrm{c}}=0$. We find that similar results, i.e., ${\mathrm{E}}_{\mathrm{C}}=O({10}^{-7})$, are achieved when we set $0<|{\partial }_z{v}_{\mathrm{c}}|\ll 1$; this is due to the fact that ${\kappa }_{\alpha \gamma }$ and ${\mathrm{V}}_{\alpha \gamma }$ show very weak dependence on ${\partial }_z{v}_{\mathrm{c}}$ when $|{\partial }_z{v}_{\mathrm{c}}|\ll 1$.

Figure 9

Error characteristics. Three different sets of data are plotted for the $t_{21}$ mode at a rotational frequency of 100 Hz, and for a set of values of ${\partial }_z{v}_{\mathrm{c}}$: 0, ${10}^{-3}c{\mathrm{L}}^{-1}$, and $-{10}^{-4}c{\mathrm{L}}^{-1}$ (from left to right). Other relevant physical parameters are chosen from astrophysical priors, as in all of the previous figures. The top panel represents $R_{\kappa }$, while the bottom panel represents ${\mathrm{R}}_{\mathrm{V}}$. Note that larger values of $R_{\kappa }$ and ${\mathrm{R}}_{\mathrm{V}}$ signify a large mismatch between the approximated analytic and numerical results.

Figure 10

$t_{\nu \gamma }$ characteristics. $t_{21}$ and $t_{11}$ are plotted as a function of a neutron star’s rotational frequency $f$. We have set $v_{\mathrm{c}}^{\mathrm{o}}=0.1c$, ${\mathrm{N}}^2={10}^{-4}$, $\mathrm{E}={10}^{-10}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g=10^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}=10^{17}\mathrm{kg}/{\mathrm{m}}^3$, ${\partial }_z{v}_{\mathrm{c}}=0$.

Figure 11

Emission characteristics. We have set $\mathrm{E}={10}^{-10}$, $\epsilon ={10}^{-4}$, $d_{\mathrm{s}}=1\mathrm{kpc}$, $\mathrm{L}=10^4\mathrm{m}$, $g=10^{12}\mathrm{m}/{\sec }^2$, ${\rho }_{\mathrm{o}}=10^{17}\mathrm{kg}/{\mathrm{m}}^3$, ${\partial }_z{v}_{\mathrm{c}}=0$, and three different sets of values of $v_{\mathrm{c}}^{\mathrm{o}}$ and ${\mathrm{N}}^2$ are explored. Note the appearance of dips in one or both of the contributions in frequency space, i.e., mass-quadrupole and current-quadrupole emissions, due to a variation in $v_{\mathrm{c}}^{\mathrm{o}}$ and ${\mathrm{N}}^2$.