Bulk viscosity in neutron stars with hyperon cores

It is well known that r-mode oscillations of rotating neutron stars may be unstable with respect to the gravitational wave emission. It is highly unlikely to observe a neutron star with the parameters within the instability window, a domain where this instability is not suppressed. But if one adopts the “minimal” (nucleonic) composition of the stellar interior, a lot of observed stars appear to be within the r-mode instability window. One of the possible solutions to this problem is to account for hyperons in the neutron-star core. The presence of hyperons allows for a set of powerful (lepton-free) nonequilibrium weak processes, which increase the bulk viscosity and thus suppress the r-mode instability. Existing calculations of the instability windows for hyperon neutron stars generally use reaction rates calculated for the ${\mathrm{\Sigma }}^{-}\mathrm{\Lambda }$ hyperonic composition via the contact $W$-boson-exchange interaction. In contrast, here we employ hyperonic equations of state where the $\mathrm{\Lambda }$ and ${\mathrm{\Xi }}^{-}$ are the first hyperons to appear (the ${\mathrm{\Sigma }}^{-}$’s, if they are present, appear at much larger densities) and consider the meson-exchange channel, which is more effective for the lepton-free weak processes. We calculate the bulk viscosity for the nonpaired $npe\mu \mathrm{\Lambda }{\mathrm{\Xi }}^{-}$ matter using the meson-exchange weak interaction. A number of viscosity-generating nonequilibrium processes is considered (some of them for the first time in the neutron-star context). The calculated reaction rates and bulk viscosity are approximated by simple analytic formulas, easy to use in applications. Applying our results to calculation of the instability window, we argue that accounting for hyperons may be a viable solution to the r-mode problem.

Figure 1

Pressure vs density for the chosen EoS models.

Figure 2

Mass-radius relations for the chosen EoS models.

Figure 3

Particle fractions $y_j=n_j/n_b$ for species $j=n,p,e,\mu ,\mathrm{\Lambda },{\mathrm{\Xi }}^{-},{\mathrm{\Sigma }}^{-}$, which emerge in NSs of EoS models we consider.

Figure 4

Baryon Fermi velocities for baryon species $j=n,p,\mathrm{\Lambda },{\mathrm{\Xi }}^{-},{\mathrm{\Sigma }}^{-}$, for the EoS models used.

Figure 5

The maximum bulk viscosity ${\zeta }_{\max 30}={\zeta }_{\max }/({10}^{30}\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-1})$ and the optimum total reaction rate ${\lambda }_{\max 45}={\lambda }_{\max }/({10}^{45}{\mathrm{erg}}^{-1}{\mathrm{cm}}^{-3}{\mathrm{s}}^{-1})$ at $\omega ={10}^4{\mathrm{s}}^{-1}$ as functions of density ${\rho }_{14}=\rho /({10}^{14}\mathrm{g}{\mathrm{cm}}^{-3})$ for different EoS models. Squares, diamonds, and circles mark the points of ${\mathrm{\Xi }}^{-}$, ${\mathrm{\Sigma }}^{-}$, and ${\mathrm{\Xi }}^0$ onsets. Crosses show the state in the center of the maximum mass NS. The thicker gray lines are for the fit by Eq. (15). The thinner gray lines show 60% (${\zeta }_{\max }$) and 20% (${\lambda }_{\max }$) deviations from the fit [i.e., ${\zeta }_{\max }^{\mathrm{appr}}\times (1\pm 0.6)$ and ${\lambda }_{\max }^{\mathrm{appr}}\times (1\pm 0.2)$].

Figure 6

The lightest-meson-exchange Feynman diagrams for the inelastic scatterings in Eqs. (1). Open and filled circles mark weak and strong vertices, respectively.

Figure 7

Thick lines show the upper estimate of the effective pion mass for the EoS models employed. Thin lines show what happens if we do not account for the polarization operator in Eq. (33).

Figure 8

The $\mathcal{W}$ functions for the nonleptonic weak processes from Eq. (1), for the EoS models used. The thicker gray lines show ${\mathcal{W}}_{\mathrm{appr}}$ from (41) with parameters from Table 3, and the thiner ones show deviations from ${\mathcal{W}}_{\mathrm{appr}}$ that cover most of the curves.

Figure 9

The reaction rates ${\lambda }_{45}=\lambda /({10}^{45}{\mathrm{erg}}^{-1}{\mathrm{cm}}^{-3}{\mathrm{s}}^{-1})$ for different EoS models at $T={10}^8\mathrm{K}$. The thicker gray lines show ${\lambda }_{\mathrm{appr}}$ from Eq. (42) with best-fit parameters from Tables 3 and 4. The thinner ones show the deviations within errors from these Tables taken together, which allow to cover the whole domains occupied by $\lambda (\rho )$ curves for EoS models.

Figure 10

Equilibrium reaction rates ${\mathrm{\Gamma }}_0$ for the $np\leftrightarrow \mathrm{\Lambda }p$ process. Thick lines are for the OME channel, and thin lines are for the contact $W$-exchange channel multiplied by 10.

Figure 11

Optimum temperatures for the bulk viscosity at $\omega =2\pi \times (400\mathrm{Hz})$ assuming nonsuperfluid and nonsuperconducting matter [top; see Eq. (46)], strong superfluidity of charged baryons and nonsuperfluid neutral ones [middle; see Eq. (47)], and optimum temperature for the case when only the reaction $n\mathrm{\Lambda }\leftrightarrow \mathrm{\Lambda }\mathrm{\Lambda }$ operates [bottom; see Eq. (48)]. In the top panel, diamonds and circles mark the ${\mathrm{\Sigma }}^{-}$ and ${\mathrm{\Xi }}^0$ onsets, correspondingly, where the set of reactions included in the total $\lambda$ becomes incomplete. In each case, the curves are plotted at $\rho \ge 1.01{\rho }_{\text{start}}$ to avoid discontinuities.

Figure 12

Example of r-mode critical frequency curves for FSU2H (left) and TM1C (right) EoS for different neutron-star masses (shown near the curves). For each mass, the unstable region of the rotation frequency $\nu$ and the redshifted internal temperature $\tilde{T}$ (the instability window) is above the curve. On each plot, the lowest curve effectively represents the critical curve in the absence of hyperons. The upper plots show the instability windows when all the processes in the set (1) operate (no account for baryon pairing). The middle plots illustrate what happens if one switches off all reactions involving charged baryons (conservative treatment of $p$ and ${\mathrm{\Xi }}^{-}$ pairing). Finally, the bottom plots are for models that partially account for $n$ pairing ($nn\leftrightarrow \mathrm{\Lambda }n$ is switched off, while $n\mathrm{\Lambda }\leftrightarrow \mathrm{\Lambda }\mathrm{\Lambda }$ is not affected). The blue data points show the observed LMXBs with measured $\nu$ and estimated $\tilde{T}$; see Ref. [60] and footnote  for details.

Figure 13

Comparison of the critical frequency curves calculated using the exact bulk viscosity (solid lines) and fitting Eqs. (15), (41), and (42) (dashed lines). The hyperon onset density ${\rho }_{\mathrm{\Lambda }}$ is adjusted for each EoS. Using Eq. (15), ${\zeta }_{\max }$ is multiplied by 1.4 for FSU2H and by 0.8 for TM1C. All processes in the set (1) are switched on.