Persistent gravitational radiation from glitching pulsars. II. Updated scaling with vortex number

TL;DR

This work analyzes gravitational radiation from nonaxisymmetric current flows inside glitching neutron stars by studying metastable vortex configurations between glitches. It combines ${N_{ m v}}$-scale N-body simulations with analytic bounds to determine how the current quadrupole moment $Q$ and the wave strain $h_0$ scale with vortex number, extrapolating from ${N_{ m v}} \lesssim 5\times10^3$ to the astrophysical regime ${N_{ m v}}\sim 10^{17}$. The main finding is a provisional scaling $Q \propto {N_{ m v}}^{-0.15}$ from simulations, yielding a central extrapolated strain $h_0 \approx 7.3\substack{+7.9 \-5.4} \times 10^{-42} (f/30\;\mathrm{Hz})^{1.9} (R_*/10\;\mathrm{km})^{0.7} (M/1.4 M_\odot) (D/1\;\mathrm{kpc})^{-1}$, with upper and lower analytic bounds spanning $h_0$ by many orders of magnitude. The bounds correspond to Poisson-like (no inter-vortex repulsion) and regular-array (strong repulsion) vortex configurations. The results underscore substantial uncertainties in far-from-equilibrium vortex physics and the need for higher-$N_{ m v}$ and 3D effects to assess detectability of such signals with current detectors.

Abstract

Superfluid vortices pinned to nuclear lattice sites or magnetic flux tubes in a neutron star evolve abruptly through a sequence of metastable spatial configurations, punctuated by unpinning avalanches associated with rotational glitches, as the stellar crust spins down electromagnetically. The metastable configurations are approximately but not exactly axisymmetric, causing the emission of persistent, quasimonochromatic, current quadrupole gravitational radiation. The characteristic gravitational wave strain $h_0$ as a function of the spin frequency $f$ and distance $D$ from the Earth is bounded above by $h_0 = 1.2\substack{+1.3 \\ -0.9} \times 10^{-32} (f/30\;{\rm Hz})^{2.5} (D/1\;{\rm kpc})^{-1}$, corresponding to a Poissonian spatial configuration (equal probability per unit area, i.e. zero inter-vortex repulsion), and bounded below by $h_0 = 1.8\substack{+2.0 \\ -1.5} \times 10^{-50} (f/30\;{\rm Hz})^{1.5} (D/1\;{\rm kpc})^{-1}$, corresponding to a regular array (periodic separation, i.e.\ maximum inter-vortex repulsion). N-body point vortex simulations predict an intermediate scaling, $h_0 = 7.3\substack{+7.9 \\ -5.4} \times 10^{-42} (f/30\;{\rm Hz})^{1.9} (D/1\;{\rm kpc})^{-1}$, which reflects a balance between the randomizing but spatially correlated action of superfluid vortex avalanches and the regularizing action of inter-vortex repulsion. The scaling is calibrated by conducting simulations with ${N_{\rm v}} \leq 5\times10^3$ vortices and extrapolated to the astrophysical regime ${N_{\rm v}} \sim 10^{17} (f/30\;{\rm Hz})$. The scaling is provisional, pending future computational advances to raise ${N_{\rm v}}$ and include three-dimensional effects such as vortex tension and turbulence.

Figures (3)

• Figure 1: Snapshot of the vortex distribution extracted from an N-body simulation of a pinned, decelerating superfluid with the code developed by Howittetal2020. The snapshot is displayed in the frame corotating with the pinning sites. The vortex distribution is shown after the avalanche ends. Vortices that move in the avalanche are drawn as red dots, with a blue tracer showing their discretized path throughout the avalanche. Vortices that do not move are drawn as grey dots. The simulation is initialized by drawing positions of $2\times10^3$ vortices from a spatially uniform distribution within container radius $R_\ast=10.0$. Pinning sites are arranged in a rectangular lattice (lattice size $a_{\rm pin}=0.1$, site width $\xi=0.02$) (all lengths measured in simulation units; see Appendix \ref{['sec:AppxSimParam']}). The snapshot is taken at simulation time-step $T=826$ (midway between the 34th and 35th glitches) and the track shows movement starting from $T=798$ (midway between the 33th and 34th glitches; time in simulation units). Control parameters are recorded in Appendix \ref{['sec:AppxSimParam']}; see also Sections 2 and 3 in Howittetal2020. Vortices often annihilate at the boundary during an avalanche, although none does so in this figure.
• Figure 2: Statistics of the dimensionless vortex configuration factor $Q$ evaluated in eight independent N-body simulations starting from ${N_{\rm v}} = 5 \times 10^3$ with the parameters in Appendix \ref{['sec:AppxSimParam']}. Top panel: Scatter plot of snapshots $({N_{\rm v}}, Q)$ taken after every 10-th glitch in the range $5 \times 10^2 \leq {N_{\rm v}} \leq 5 \times 10^3$ for all eight runs. $Q$ is calculated by evaluating equation (\ref{['eq:Q']}) at midpoints between avalanches. The red and blue lines each track snapshots from arbitrarily chosen independent runs. Snapshots at more than 90 per cent or less than 10 per cent of the initial container frequency are discarded both in the plot and the parameter estimation. The dark grey line and the grey band displays the mean and 90 per cent credible interval of the posterior predictive distribution at each $N_{\rm v}$, respectively. Middle three panels: Corner plot displaying the two-dimensional and marginalized one-dimensional posterior distributions of the estimated PDF parameters $a$ and $b$ in equation (\ref{['eq:Q']}). Parameter estimates in the corner plots are reported with their 5-th, 50-th, and 95-th percentiles (vertical dashed lines). Bottom panel: Normalized histogram of $Q/s$ with $s=a{N_{\rm v}}^b$ for every $({N_{\rm v}}, Q)$ point in the top panel for $10^3$ samples of $(a, b)$ drawn from the posterior. The Rayleigh distribution $p(Q/s) = (Q/s) \exp{\left[-Q^2/2s^2\right]}$ is overlaid as a solid black curve.
• Figure 3: Scaling of the dimensionless vortex configuration factor $Q$ as a function of the number of vortices $N_{\rm v}$. Top panel: Samples of $Q$ for $10^3$ random configurations generated by a uniform Poisson point process for $5\times10^2 \leq {N_{\rm v}} \leq 5\times10^3$. For each configuration, $N_{\rm v}$ is drawn from a discrete uniform distribution on the interval $[5\times10^2, 5\times10^3]$, and $Q$ is calculated by evaluating equation (\ref{['eq:Q']}) for each configuration. The dark grey curve correspond to the median of the theoretical PDF in equation (\ref{['eq:Q_rayleigh_uniform']}) at each $N_{\rm v}$. The grey band shows the central 90 percentile of the same PDF. Bottom panel: Samples of $Q$ for $10^3$ realizations of a regular periodic array with randomized centre for $5\times10^2 \leq {N_{\rm v}} \leq 5\times10^3$. For each configuration, the intervortex spacing $d$ is drawn from a continuous uniform distribution on the interval $[2\times10^{-2}, 9\times10^{-2}]$, $c_x$ and $c_y$ are drawn from a uniform distribution on $[0, d)$, and $Q$ is calculated by evaluating equation (\ref{['eq:Q']}) for each configuration.