Thermal conductivity of dense $np\Lambda$ matter in neutron star cores

Technical Summary: Thermal Conductivity of Dense npΛ Matter in Neutron Star Cores

Problem Statement
Neutron stars (NS) are born with high temperatures ($\sim 10^{11}$ K) and undergo a thermal relaxation phase where the core and crust cool via different mechanisms. The core, containing the bulk of the stellar mass and thermal energy, is the primary site for neutrino emission, which drives the cooling evolution. While the thermal conductivity of nucleonic matter (neutrons and protons) has been extensively modeled, the transport properties of dense matter containing strange baryons (hyperons) remain under-explored. Recent astrophysical observations of massive neutron stars ($>2 M_\odot$) suggest that the high densities in their cores may allow for the appearance of hyperons, specifically $\Lambda$ hyperons. The presence of these additional degrees of freedom introduces new scattering channels that could alter the equation of state (EOS) and, crucially, the thermal conductivity. A dedicated analysis of the thermal conductivity for a specific $\beta$-equilibrated $np\Lambda$ mixture has been absent from the literature, creating a gap in understanding how hyperons influence the thermal relaxation timescale and global cooling behavior of massive neutron stars.

Methodology
This work investigates the thermal conductivity of dense, $\beta$-equilibrated $np\Lambda$ matter using a variational linearized Boltzmann kinetic approach. The study employs the Density-Dependent Meson-Exchange 2 (DDME2) equation of state to model the composition and effective masses of particles across a baryon density range of $0.5n_0$ to $4.5n_0$ (where $n_0 \approx 0.152 \text{ fm}^{-3}$).

Key methodological components include:

• Kinetic Framework: The authors utilize the formalism developed by Flowers, Itoh, and Yakovlev, solving the exact linearized Boltzmann kinetic equations via a variational method. The system is treated as a multi-component Fermi-liquid.
• Species Treatment: Protons are treated as passive scatterers due to their low abundance, while neutrons and $\Lambda$ hyperons are the active heat carriers. The electron-muon subsystem is considered independent of the baryonic subsystem.
• In-Medium Effects: The calculation incorporates in-medium effects by replacing bare particle masses with density-dependent effective masses derived from the DDME2 EOS. The matter is assumed to be in a non-superfluid and non-superconducting state.
• Scattering Cross-Sections:
  • For nucleon-nucleon ($nn$, $np$) scattering, the study uses fitted forms of the squared matrix elements based on experimental data and parametrized results from previous works.
  • For $\Lambda n$ scattering, the total cross-sections from the Nijmegen group are used, assuming isotropic scattering to provide an order-of-magnitude estimate.
  • For $\Lambda \Lambda$ scattering, the $1S_0$ partial wave is considered dominant, with the isotropic assumption applied due to the scarcity of experimental data on hyperon-hyperon interactions.
• Numerical Solution: The authors solve a system of coupled linear equations to determine the thermal relaxation times ($\tau_n$ and $\tau_\Lambda$). These times are derived from collision frequencies ($\nu_{12}$ and $\nu'_{12}$), which depend on the effective masses, number densities, temperature, and scattering integrals ($S_{12}$ and $S'_{12}$). The final thermal conductivity coefficients are obtained by applying a variational correction factor ($C_{var} = 1.2$) to the variational solutions.

Key Results
The study presents the thermal conductivity and relaxation times for neutrons and $\Lambda$ hyperons as functions of normalized baryon density ($n_B/n_0$) at temperatures of $10^7$, $10^8$, and $10^9$ K.

1. Composition and Onset: $\Lambda$ hyperons appear at a threshold density of $n_B/n_0 \approx 2.10$. Beyond this point, the $\Lambda$ density rises rapidly, becoming comparable to the neutron density at higher densities, while the proton fraction remains low.
2. Relaxation Times:
   • Below the $\Lambda$ onset, neutron relaxation times in hyperonic matter are identical to those in pure nucleonic matter.
   • Above the threshold, the neutron relaxation time in $np\Lambda$ matter becomes smaller than in pure $np$ matter due to the opening of additional $n\Lambda$ scattering channels, which enhance the collision rate.
   • $\Lambda$ relaxation times appear only above the threshold and are consistently smaller than neutron relaxation times, indicating more frequent collisions for $\Lambda$ hyperons.
   • Relaxation times decrease significantly with increasing temperature (by several orders of magnitude from $10^7$ K to $10^9$ K).
3. Thermal Conductivity:
   • Neutrons remain the dominant heat carriers throughout the density range.
   • The presence of $\Lambda$ hyperons induces a reduction in neutron thermal conductivity compared to pure nucleonic matter. This suppression becomes more pronounced at higher densities as the $\Lambda$ fraction increases.
   • The thermal conductivity of $\Lambda$ hyperons emerges above the threshold but remains approximately two orders of magnitude lower than that of neutrons across the entire density range.
   • Overall thermal conductivity increases with density (due to higher particle densities and Fermi momenta) but decreases strongly with temperature.

Significance and Claims
The paper concludes that while the inclusion of $\Lambda$ hyperons introduces new scattering channels and modifies the microscopic transport processes, their net impact on the overall thermal conductivity of the neutron star core is negligible. Specifically:

• Neutrons continue to dominate thermal transport even in the presence of a significant $\Lambda$ population.
• The reduction in neutron conductivity caused by $\Lambda$ hyperons is "remarkably small."
• Consequently, the core thermal relaxation timescale remains practically unaltered by the presence of $\Lambda$ hyperons within the DDME2 model framework.

The authors assert that standard nucleon-dominated transport frameworks remain sufficiently accurate for modeling the thermal evolution of dense neutron star cores, even when massive strange baryons are present. The study highlights that the uncertainty in hyperon-hyperon interaction data limits the precision of these transport coefficients, but the qualitative conclusion regarding the dominance of neutrons holds under the current theoretical assumptions.