Calculation of the transport coefficients in neutron star

Using a relativistic kinetic theory approach with a modified BUU equation and the relaxation time approximation within a relativistic mean field model, this study calculates the shear viscosity and thermal conductivity in neutron star cores, finding that the former is predominantly influenced by neutrons while the latter is primarily dominated by electrons.

The Gist

Imagine a neutron star as the ultimate "cosmic pressure cooker." It's a dead star so dense that a single teaspoon of its material would weigh as much as a mountain. Inside this cosmic giant, matter is crushed into a state we can't replicate on Earth. It's a soup of subatomic particles—mostly neutrons, with a splash of protons, electrons, and muons—squeezed together so tightly that they behave like a super-fluid, super-conductive fluid.

This paper is about calculating two specific "traffic rules" for this cosmic soup: Shear Viscosity and Thermal Conductivity.

Here is the breakdown of what the authors did, using simple analogies.

1. The Goal: Measuring the "Stickiness" and "Heat Flow"

Think of the neutron star's core as a giant, chaotic dance floor.

• Shear Viscosity is like the stickiness of the dance floor. If the floor is very sticky (high viscosity), dancers (particles) can't slide past each other easily. If it's slippery (low viscosity), they glide effortlessly. In a neutron star, this "stickiness" determines how the star wobbles after a collision or how it spins.
• Thermal Conductivity is like the speed of a heat wave. If you light a match at one end of the dance floor, how fast does the heat reach the other side? High conductivity means the heat zips through instantly; low conductivity means the heat gets stuck.

2. The Method: A Relativistic Traffic Simulation

The authors didn't just guess; they built a complex computer simulation based on Relativistic Kinetic Theory.

• The Players: They modeled the particles (neutrons, protons, electrons, muons) not as tiny billiard balls, but as "quasi-particles." Imagine these particles are wearing heavy winter coats (effective mass) that change size depending on how crowded the room is (density).
• The Rules: They used a set of rules called the Relativistic Mean Field (RMF) model. Think of this as the "physics engine" of their simulation. They tried three different versions of this engine (named IUFSU, FSU2, and FSUGold) to see which one best matched reality.
• The Calculation: They calculated how often these particles bump into each other (collisions) and how long it takes them to recover from a bump (relaxation time).

3. The Big Discoveries

The "Heavy Hitters" vs. The "Speedsters"

The most surprising finding is about who is doing the work:

• For Stickiness (Viscosity): The Neutrons are the main bosses. Even though they are heavy and slow, there are so many of them that they dominate the "traffic jams." They are the ones determining how sticky the star is.
• For Heat Flow (Conductivity): The Electrons are the super-highways. Because electrons are tiny and light, they zip around much faster than the heavy neutrons. They are the ones carrying the heat across the star.

The Density Effect

As you go deeper into the star, the pressure increases (density goes up).

• The Paradox: Usually, when a crowd gets denser, traffic slows down (relaxation time decreases). However, the authors found that because there are so many more particles in a denser crowd, the overall "stickiness" and "heat flow" actually increase. It's like a highway: even if cars are bumper-to-bumper, if you have 100 lanes of traffic, the total amount of stuff moving is huge.

The Temperature Factor

The authors also looked at what happens if the star gets hotter.

• The Result: As the temperature rises, the particles get more jittery and collide more often. This makes them less efficient at moving in a straight line. Consequently, both the stickiness and the heat flow decrease. It's like a crowded dance floor where everyone is jumping around wildly; it becomes harder to slide smoothly or pass a message quickly.

4. Why Does This Matter?

You might ask, "Why do we care about the stickiness of a dead star?"

• Gravitational Waves: When two neutron stars crash, they create ripples in space-time (gravitational waves). The "stickiness" of the star determines how much energy is lost as heat during the crash, which changes the shape of those ripples.
• Pulsar Glitches: Sometimes, spinning neutron stars (pulsars) suddenly speed up. Understanding how the internal fluid flows helps explain these "glitches."
• Cooling: Young neutron stars are incredibly hot. Knowing how fast they conduct heat tells us how quickly they cool down, which helps astronomers figure out how old a star is just by looking at its temperature.

Summary

In short, this paper is a detailed "traffic report" for the inside of a neutron star. The authors used advanced math to figure out that neutrons make the star "sticky," while electrons act as the "heat pipes." They found that as the star gets denser, these effects get stronger, but as it gets hotter, they get weaker. This helps us understand the invisible physics of the most extreme objects in the universe.

Technical Summary

1. Problem Statement

Neutron stars (NS) represent extreme states of matter where density and temperature conditions are unattainable in terrestrial laboratories. Understanding the dynamics of these stars, particularly during post-merger events and pulsar glitches, requires accurate knowledge of transport coefficients, specifically shear viscosity ($\eta$) and thermal conductivity ($\lambda$).

These coefficients govern the dissipation of energy and momentum, influencing:

• The damping of $r$-mode instabilities.
• The thermal evolution and cooling rates of young neutron stars.
• The interpretation of gravitational wave signals from binary mergers.

While previous studies have estimated these values, there is a need for calculations that rigorously incorporate relativistic kinetic theory with effective masses and effective chemical potentials derived from a Relativistic Mean Field (RMF) model. The authors aim to bridge the gap between microscopic particle interactions and macroscopic hydrodynamic properties using a quasi-particle formalism.

2. Methodology

The authors employ a relativistic kinetic theory approach based on the Boltzmann-Uehling-Uhlenbeck (BUU) transport equation. The calculation proceeds through the following steps:

A. Equation of State (EOS) and Hadronic Model

• Model: The hadronic matter (neutrons, protons) is described using the Relativistic Mean Field (RMF) theory.
• Lagrangian: Includes nucleon fields interacting with scalar ($\sigma$), vector ($\omega$), and isovector ($\rho$) meson fields. Leptons (electrons, muons) are included to ensure charge neutrality and $\beta$-equilibrium.
• Parameterizations: Three distinct RMF parameter sets are used to test model dependence: IUFSU, FSU2, and FSUGold.
• Key Variables: The model calculates density-dependent effective masses ($m^*_a$) and effective chemical potentials ($\mu^*_a$), which significantly alter the structure of the Boltzmann equation compared to free-particle approximations.

B. Quasi-Particle Transport Theory

• Formalism: The distribution function is expanded around equilibrium ($f_a = f^0_a + \delta \tilde{f}_a$). The authors account for the fact that in a medium with effective mass, the energy variation $\delta E_a$ includes contributions from changes in the mean fields.
• Relaxation Time Approximation (RTA): The collision integral in the BUU equation is approximated as $-\delta f_a / \tau_a$.
• Scattering Cross-Sections:
  • Nucleons: A parameterized vacuum cross-section is used for nucleon-nucleon ($NN$) scattering.
  • Leptons: Cross-sections are calculated using Quantum Electrodynamics (QED) amplitudes. To handle infrared divergences at small scattering angles, a Debye mass is introduced to screen the interactions.
  • Interactions: Nucleon-lepton interactions are neglected (assumed zero cross-section) as they are not part of the standard RMF hadronic model.

C. Derivation of Coefficients

Using the RTA and the derived distribution functions, the authors derive analytical expressions for:

1. Shear Viscosity ($\eta$): Proportional to the integral of $|p^*|^4 \tau_a / E^{*2}_a$.
2. Thermal Conductivity ($\lambda$): Proportional to the integral of $|p^*|^2 \tau_a / E^*_a$, weighted by the deviation from the baryon current.

• Approximation: Since the system is degenerate, the integrals are evaluated at the Fermi surface ($E^*_a \approx \mu^*_a$).

3. Key Contributions

1. Incorporation of Effective Mass/Chemical Potential: Unlike earlier works that treated particles as free or used Landau Fermi liquid theory directly, this study explicitly integrates the density-dependent effective mass and chemical potential into the relativistic kinetic framework. This modifies the energy-momentum tensor and the irreversible parts of the transport equations.
2. Comprehensive Model Comparison: The study systematically compares results across three modern RMF parameterizations (IUFSU, FSU2, FSUGold), highlighting how nuclear symmetry energy and slope parameters affect transport properties.
3. Analytical Fitting Formulas: The authors provide fitted parametric formulas for $\eta(\rho, T)$ and $\lambda(\rho, T)$ valid for temperatures $0.1 \le T \le 5.0$ MeV and baryon densities $0.1 \le \rho \le 1.5$ fm$^{-3}$. These are crucial for implementing transport coefficients into large-scale hydrodynamic simulations.
4. Direct Comparison with Established Formalisms: The paper compares their quasi-particle results against the Shternin et al. (SH) formalism for viscosity and the Baiko et al. (BHY) formalism for conductivity, using the same EOS (IUFSU) to isolate the effects of the kinetic theory approach.

4. Results

A. Microscopic Properties

• Relaxation Times ($\tau$):
  • $\tau$ decreases as baryon density increases.
  • Neutrons have higher relaxation times than protons.
  • Electrons have higher relaxation times than muons.
• Effective Mass: The nucleon effective mass decreases with increasing density.

B. Shear Viscosity ($\eta$)

• Trend: $\eta$ increases with density.
• Dominant Species: Neutrons provide the dominant contribution to shear viscosity, followed by electrons.
  • Reasoning: Although electrons have higher mobility, neutrons are far more numerous. The contribution scales with particle density and momentum transfer.
• Comparison: The calculated values ($\sim 10^{13} - 10^{15}$ g cm$^{-1}$ s$^{-1}$) are significantly lower than some previous estimates ($\sim 10^{17} - 10^{20}$), but the trend of increasing viscosity with density matches established formalisms when using the same EOS.

C. Thermal Conductivity ($\lambda$)

• Trend: $\lambda$ increases with density.
• Dominant Species: Electrons are the primary carriers of thermal conductivity.
  • Reasoning: Thermal conductivity depends on particle velocity ($|p^*|/E^*$). Lighter leptons (electrons) have much higher velocities than nucleons, making them the dominant heat carriers.
• Model Dependence: The FSU2 model yields significantly higher thermal conductivity at high densities compared to IUFSU and FSUGold. This is attributed to the FSU2 model predicting a much higher electron chemical potential ($\mu_e$), leading to a higher electron density.
• Comparison: Values range from $10^{23}$ to $10^{28}$ erg s$^{-1}$ cm$^{-1}$ K$^{-1}$. The results are generally higher than BHY formalism predictions at lower densities but converge at higher densities.

D. Temperature Dependence

• Both $\eta$ and $\lambda$ decrease as temperature increases.
• Mechanism: Higher temperatures increase the collision frequency among particles, reducing their mobility and relaxation times.

5. Significance

• Astrophysical Modeling: The provided fitting formulas allow for the direct inclusion of realistic, density-dependent transport coefficients in numerical simulations of neutron star mergers and cooling. This is essential for interpreting gravitational wave data (e.g., from LIGO/Virgo) regarding post-merger oscillations and damping.
• Theoretical Refinement: By demonstrating how effective masses and chemical potentials alter the Boltzmann equation structure, the paper refines the theoretical understanding of non-equilibrium processes in dense QCD matter.
• Model Constraints: The sensitivity of thermal conductivity to the choice of RMF parameterization (specifically the symmetry energy slope) suggests that observational constraints on neutron star cooling could potentially help constrain the nuclear equation of state.

In conclusion, this work provides a robust, relativistic kinetic framework for calculating transport coefficients in neutron stars, offering both detailed numerical results and practical analytical fits for the astrophysics community.