Radial Oscillations of Viscous Neutron Stars: Zero Diffusion Case

TL;DR

This work analyzes how viscosity influences the radial oscillation spectrum of neutron stars within two relativistic dissipative frameworks: relativistic Navier–Stokes and Israel–Stewart. It develops linear perturbation theory with zero energy diffusion, derives a master equation for the Lagrangian displacement, and examines both frequency-domain and time-domain behavior, including singular points and Newtonian limits. The key finding is that bulk viscosity can strongly modify the fundamental mode and, in Israel–Stewart theory, the nonhydrodynamic mode can drive instability near the turning point, while shear viscosity largely leaves the n=0 mode unaffected. The study highlights the importance of causal dissipative formulations and redshifted relaxation times for interpreting neutron-star quasinormal modes and has implications for gravitational-wave signals from mergers and post-merger remnants.

Abstract

The spectrum of radial oscillations of neutron stars is systematically studied within two frameworks of viscous relativistic hydrodynamics: the relativistic Navier-Stokes and Israel-Stewart theories. A correspondence is established between the discrete stellar eigenmodes and the continuous dispersion relation of perturbations around a homogeneous fluid, providing a basis for interpreting our numerical results. We analyze the Newtonian limit and assess the impact of relativistic corrections, such as the gravitational redshifting of microscopic relaxation timescales. We show that bulk viscosity can significantly affect the behavior of both hydrodynamic and nonhydrodynamic fundamental modes, and that, depending on the magnitude of the viscous effects, it is the nonhydrodynamic mode that becomes unstable beyond the turning point in a sequence of equilibrium configurations. These results provide a useful step toward systematic studies of neutron star quasinormal modes in the presence of viscosity.

Figures (7)

• Figure 1: Real (upper panels) and imaginary (lower panels) parts of the eigenfrequencies $\bar{\omega}_{(n)} = \omega_{(n)}/\omega_0$ as a function of the overtone number $n$ for a Newtonian $\Gamma = 2$ polytrope subject to shear (blue circles) or bulk (red triangles) viscosity in the Navier-Stokes (left) and Israel-Stewart (right) theories. (a) We consider $\tau_\eta \omega_0 = 0.5$ in the shear viscous case (blue circles) and $\tau_\zeta \omega_0 = 2/3$ in the bulk viscous case (red triangles), which both correspond to the critical frequency $\omega_{\textrm{cr}} = -3 i \omega_0$ (dashed line). The shaded region corresponds to Eq. (\ref{['eq:omega_homogeneous_ansatz']}) with $n_\textrm{cr} \in [6,7]$. (b) We set $\omega_\pi = -i\omega_0$ in the shear viscous case and $\omega_\Pi = -i\omega_0$ in the bulk viscous case, along with the same values of $\tau_{\eta,\zeta}$ as in (a). The critical frequency is now $\omega_{\textrm{cr}} = -0.75 i \omega_0$ (dashed line). The shaded region corresponds to the roots of Eq. (\ref{['eq:omega_homogeneous_ansatz_IS_S']}), with $n_\textrm{cr}$ in the same range as before. In all panels, mode frequencies in the case of zero viscosity (gray squares) are also shown for comparison.
• Figure 2: Imaginary part of the eigenfrequencies $\bar{\omega}_n = \omega_{(n)}/\omega_0$ for a Newtonian $\Gamma = 2$ polytrope subject to shear (left panel) or bulk (right panel) viscosity in Israel-Stewart theory, shown as a function of $\bar{\tau}_\eta = \omega_0 \tau_\eta$ or $\bar{\tau}_\zeta = \omega_0 \tau_\zeta$. For the shear-viscous case (left), we set $\omega_\pi = -i\omega_0$, and recall that $\omega_\textrm{cr}$ is defined in Eq. \ref{['eq:wcr_Newt_eta']}. In the bulk viscous case (right panel), we similarly set $\omega_\Pi = -i\omega_0$, with $\omega_\textrm{cr}$ defined in Eq. \ref{['eq:wcr_Newt_zeta']}. Branches of hydrodynamic and non-hydrodynamic modes are shown with solid and dashed lines, respectively. While shear viscosity has a minimal impact on the fundamental ($n=0$) radial modes, bulk viscosity strongly modifies both the hydrodynamic and non-hydrodynamic $n=0$ branches.
• Figure 3: Real (upper panels) and imaginary (lower panels) parts of the eigenfrequencies $\bar{\omega}_{(n)} = \omega_{(n)}/\omega_0$, with $\omega_0 = \sqrt{2\pi\rho_c}$, for $n \in\{0,1,2,3,4\}$, as a function of the central rest-mass density $\rho_c$ (in units of $\rho_\textrm{sat} = 2.7 \times 10^{14} \textrm{g}/\textrm{cm}^3$), for a relativistic $\Gamma = 2$ polytrope within Navier-Stokes theory. Results for shear and bulk viscosity are presented in the left and right panels, respectively, for (a) $\bar{t}_\eta = t_\eta \omega_0 = 1$ and (b) $\bar{t}_\zeta = t_\zeta \omega_0 = 4/3$. A black dot-dashed line in the lower panels indicates the imaginary part of $\bar{\omega}_\textrm{cr} = \omega_\textrm{cr}/\omega_0 = - 1.5 i$, with $\omega_\textrm{cr}$ defined in Eq. \ref{['eq:wcr_rel']}.
• Figure 4: Real (upper panels) and imaginary (lower panels) parts of the eigenfrequencies $\bar{\omega}_{(n)} = \omega_{(n)}/\omega_0$, with $\omega_0 = \sqrt{2\pi\rho_c}$, for $n \in\{0,1,2,3\}$, as a function of the central rest-mass density $\rho_c$ (in units of $\rho_\textrm{sat} = 2.7 \times 10^{14} \textrm{g}/\textrm{cm}^3$), for a relativistic $\Gamma = 2$ polytrope within Israel-Stewart theory. Results for shear and bulk viscosity are presented in the left and right panels, respectively, for (a) $t_\pi \omega_0 = 1$ and $t_\eta \omega_0 = 1$, and (b) $t_\Pi \omega_0 = 1$ and $t_\zeta \omega_0 = 4/3$. These values are such that condition \ref{['eq:lin-causality']} is satisfied for the entire range of central densities shown in the plot. Branches of hydrodynamic and non-hydrodynamic modes are shown with solid and dashed lines, respectively. A black dot-dashed line in the lower panels indicates the imaginary part of $\bar{\omega}_\textrm{cr} = \omega_\textrm{cr}/\omega_0 = - 0.6 i$ [cf. Eq. \ref{['eq:wcr_rel']}]. The fundamental ($n=0$) mode frequencies of a perfect fluid are represented as dashed gray lines for comparison.
• Figure 5: Lagrangian displacement evaluated at the stellar surface for perturbations including shear (upper panel) or bulk (lower panel) viscosity, as a function of time. Different colors correspond to different background central density configurations. We adopt a $\Gamma = 2$ polytropic equation of state, and set $t_{\eta} = t_{\pi} = 0.2$ ms in the upper panel and $t_{\zeta} = t_{\Pi} = 0.2$ ms in the bottom panel.