Radial Oscillations of Viscous Stars

Abstract

Oscillation modes of neutron stars, a key target for third-generation gravitational wave detectors, encode key information about their constituent nuclear matter. In this work, we study the effect of viscosity on oscillations of cold, polytropic, spherically symmetric neutron stars. We focus on purely radial oscillations and work perturbatively to linear order within two hydrodynamic frameworks: the acausal covariant generalization of the Navier-Stokes equations proposed by Eckart, and the causal generalization formulated by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). We find that viscosity damps the radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star, reaching up to the percent level for the fundamental mode with bulk viscosities $ζ\sim10^{30}\mathrm{g}/\mathrm{cm}/\mathrm{s}$. For more viscous stars, the oscillation frequency decreases, becoming zero (i.e., an overdamped mode) for $ζ\gtrsim10^{31}\mathrm{g}/\mathrm{cm}/\mathrm{s}$. We also study the linear threshold of gravitational collapse. Consistent with recent analytic results in the small viscosity regime, we find that viscosity in Eckart theory cannot stabilize an unstable inviscid star. We provide numerical evidence that viscosity in BDNK theory is similarly unable to prevent gravitational collapse, but it slightly modifies the threshold of collapse. Overall, our results advance our understanding of the impact of viscosity on the oscillation modes of neutron stars, a key component of viscous asteroseismology with next-generation gravitational wave detectors.

Figures (9)

• Figure 1: Mode content of Gaussian initial data evolved by the perfect fluid (red), Eckart (blue) and BDNK (green) equations with a "small" central bulk viscosity, $\zeta_{c} = 1.9\times10^{29}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$. The vertical dashed lines represent the frequencies of the first five radial modes computed using the Eckart frequency-domain shooting method. We see excellent agreement between the time-domain and frequency-domain results. Time domain convergence is demonstrated in Fig. \ref{['fig:convergence']}.
• Figure 2: Lagrangian displacement (upper left) and velocity perturbation (lower left) in Eckart and BDNK simulations of fundamental mode initial data for central bulk viscosities $\zeta_{c} = 1.9\,\hat{\zeta}\times10^{31}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$ (see legend). Fourier transforms of the time series are shown in the right column, wherein the vertical line is the fundamental mode computed using the Eckart frequency-domain shooting method with $\hat{\zeta}=10^{-3}$. Significant damping occurs for the two largest viscosities, while the viscosity-induced shifts in $f$ are on the sub-Hz level.
• Figure 3: The effect of viscosity on the fundamental mode of neutron stars with central bulk viscosities $\zeta_{c}\lesssim10^{30}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$. Left: fractional frequency shifts $\Delta{f}/f_{\rm{PF}}$, where $\Delta{f}=f_{\mathrm{PF}}-f_{\mathrm{visc}}$, as a function of the central bulk viscosity. Right: corresponding damping times in milliseconds. The red curves are extracted from damped sinusoid fits to time-domain Eckart simulations. The blue, green and black curves are extracted from fits to time-domain BDNK simulations in the three causal frames A, B, and C, respectively, listed in \ref{['eq:CausalFrames']}. For $\zeta_{c}\sim10^{30}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$, viscosity shifts the fundamental mode by $\sim1\%$. The effect of changing between causal frames is much smaller than the effect of including viscosity, indicating that these stars remain in the regime of validity of BDNK theory. There is also excellent agreement between Eckart and BDNK except in the limit $\hat{\zeta}\to0$, where numerical viscosity dominates the physical viscosity in our BDNK simulations, causing the (faint) plateau of the BDNK curves.
• Figure 4: Transition to an overdamped fundamental mode in the large-viscosity regime of Eckart stars with central densities $\epsilon_{c}=\hat{\epsilon}_{c}\times10^{15}\mathop{}\!\mathrm{g}/\mathrm{cm}^{3}$. Left: fundamental mode eigenvalue $\omega=2\pi{f}-i/\tau$ in the complex plane as a function of the central bulk viscosity $\zeta_{c}$ with $\hat{\epsilon}_{c}=5.5$. As $\hat{\zeta}\to\infty$, the mode becomes aperiodically damped on infinitely long time scales. Similar behavior occurs for the higher overtones (see Tables \ref{['tbl:FDFreqsA']}--\ref{['tbl:FDFreqsB']}). Middle: fractional shifts $\Delta{f}/f_{\rm{PF}}$, where $\Delta{f}=f_{\rm{PF}}-f_{\rm{eck}}$, as a function of $\zeta_{c}$ for several central densities (see the legend in the right panel). The dashed horizontal line corresponds to unity, i.e., where the modes lie on the imaginary axis in the complex plane. Right: corresponding damping times for the modes shown in the middle panel. The transition to overdamped modes happens for all considered stellar models at central bulk viscosities within roughly an order of magnitude of $\zeta_{c}\sim5\times10^{31}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$, shaded in purple in the figure. For EoS B \ref{['eq:EoS']}, a transition to overdamped modes also occurs around this same range of viscosities, possibly hinting at some universal behavior. This transition $f\to0$ with increasing $\zeta_{c}$ suggests that sufficiently viscous stars can have arbitrarily low frequency oscillation modes.
• Figure 5: The effect of compactness on the fundamental radial mode of neutron stars. Left: fractional frequency shifts $\Delta{f}/f_{\rm{PF}}$, where $\Delta{f}=f_{\mathrm{PF}}-f_{\mathrm{visc}}$, as a function of the compactness of the star. Right: corresponding damping timescales for the modes shown in the left panel. The dimensionless bulk viscosity parameter is fixed to $\hat{\zeta}=0.05$. The least compact stars ($\epsilon_{c}=10^{15}\mathop{}\!\mathrm{g}/\mathrm{cm}^{3}$) have $\zeta_{c}=2.4\times10^{28}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$ and the most compact stars ($\epsilon_{c}=5.5\times 10^{15}\mathop{}\!\mathrm{g}/\mathrm{cm}^{3}$) have $\zeta_{c}=9.5\times10^{29}\mathop{}\!\mathrm{g}/\mathrm{cm}/\mathrm{s}$. The red and blue curves are obtained from damped sinusoid fits to Eckart and BDNK time-domain simulations of fundamental mode initial data. More compact stars exhibit a faster damping rate and larger fractional shifts in the oscillation frequency.