[Paper Review] Bulk viscosity in superfluid neutron star cores. I. Direct Urca processes in npeμmatter

P. Haensel, К. П. Левенфиш+1 more|ArXiv.org|Apr 13, 2000

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper investigates bulk viscosity in superfluid neutron star cores dominated by direct Urca processes in npeμ matter, focusing on how nucleon superfluidity—specifically singlet (1S0) or triplet (3P2) pairing—suppresses viscosity. Using analytical and numerical methods, it derives reduction factors for bulk viscosity under various superfluid pairing scenarios, showing strong suppression when nucleon pairing gaps are large, especially in the presence of muons which contribute comparably to electron-mediated processes.

ABSTRACT

The bulk viscosity of the neutron star matter due to the direct Urca processes involving nucleons, electrons and muons is studied taking into account possible superfluidity of nucleons in the neutron star cores. The cases of singlet-state pairing or triplet-state pairing (without and with nodes of the superfluid gap at the Fermi surface) of nucleons are considered. It is shown that the superfluidity may strongly reduce the bulk viscosity. The practical expressions for the superfluid reduction factors are obtained. For illustration, the bulk viscosity is calculated for two models of dense matter composed of neutrons, protons,electrons and muons. The presence of muons affects the bulk viscosity due to the direct Urca reactions involving electrons and produces additional comparable contribution due to the direct Urca reactions involving muons. The results can be useful for studying damping of vibrations of neutron stars with superfluid cores.

Motivation & Objective

• To understand how nucleon superfluidity affects bulk viscosity in neutron star cores, particularly in the context of direct Urca processes.
• To quantify the suppression of bulk viscosity due to pairing in singlet (1S0) or triplet (3P2) states of neutrons and protons.
• To assess the role of muons in enhancing bulk viscosity contributions through muon-mediated Urca processes.
• To derive practical analytical expressions for superfluid reduction factors applicable to realistic neutron star models.

Proposed method

• Analytical derivation of bulk viscosity in npeμ matter using the Urca process framework, considering electron and muon capture/emission by nucleons.
• Incorporation of nucleon superfluidity via pairing gaps in both 1S0 and 3P2 channels, with distinct treatment for gapped and nodal superfluids.
• Use of asymptotic expansions in the strong superfluidity limit (large gap parameters) to derive reduction factors for bulk viscosity.
• Derivation of reduction factors R_AB, R_AC, etc., via angular averaging over the superfluid gap orientation, using modified Bessel functions and exponential integrals.
• Numerical evaluation of reduction factors across different (v1, v2) parameter spaces representing neutron and proton gap strengths.
• Inclusion of muons in the Urca process calculation, showing their contribution is comparable to electrons due to similar kinematic and statistical factors.

Experimental results

Research questions

• RQ1 How does nucleon superfluidity in the 1S0 or 3P2 channel suppress bulk viscosity in neutron star cores?
• RQ2 What are the analytical expressions for the bulk viscosity reduction factor in the presence of mixed superfluid pairing (e.g., A and B types)?
• RQ3 How do muons contribute to bulk viscosity in direct Urca processes, and is their contribution comparable to that of electrons?
• RQ4 What are the asymptotic behaviors of the reduction factor in the strong superfluidity limit for different pairing configurations?
• RQ5 How does the presence of nodes in the superfluid gap (e.g., in 3P2 pairing) affect the bulk viscosity suppression compared to fully gapped states?

Key findings

• Bulk viscosity is strongly suppressed by nucleon superfluidity, with the suppression factor depending on the pairing gap strength and pairing channel (1S0 vs. 3P2).
• For strong superfluidity, the reduction factor R_AB in the case of A-type (1S0) and B-type (3P2, nodal) pairing scales as R_AB ∼ v₁² exp(−v₁) in the limit v₁ = v₂, with detailed dependence on gap parameters.
• The presence of muons contributes comparably to bulk viscosity as electrons in direct Urca processes, due to similar phase space and statistical weights.
• In the strong superfluidity limit, the reduction factor for A-type (1S0) pairing alone behaves as R_A ∼ v exp(−v) for v ≫ 1, consistent with known results.
• For mixed pairing (e.g., A and C types), the suppression is dominated by the stronger pairing channel, with R_AC ∼ v₁⁵ / v₂² exp(−v₁) when v₂ > v₁ ≫ 1.
• The derived reduction factors are valid in the strong pairing regime and break down near v₁ ≈ v₂ or when |v₂ − v₁| ≲ √v₁, where higher-order corrections become important.