Beta-Particle Transport and Thermalization in Kilonova Ejecta with Detailed Atomic Microphysics

Technical Summary: Beta-Particle Transport and Thermalization in Kilonova Ejecta with Detailed Atomic Microphysics

Problem Statement
Kilonova (KN) emission, powered by the radioactive decay of $r$-process nuclei in neutron star merger ejecta, is a critical probe for understanding nucleosynthesis, binary evolution, and the equation of state for dense matter. Accurate modeling of KN light curves requires precise calculations of the heating rate provided by the thermalization of decay products ($\beta$-particles, $\alpha$-particles, $\gamma$-rays, and fission fragments). While $\gamma$-ray transport is often treated with full radiation transport, the thermalization of charged particles ($\beta$ and $\alpha$) has historically relied on simplified descriptions. These simplifications introduce significant uncertainties, particularly regarding the efficiency of energy deposition, the spatial distribution of heating, and the resulting ionization state of the ejecta. Furthermore, the transport of charged particles is sensitive to the magnetic field geometry, which remains largely unconstrained. This paper addresses the lack of detailed atomic microphysics and the need for a relativistic transport framework to better model $\beta$-particle thermalization in expanding kilonova ejecta.

Methodology
The authors employ a multi-faceted approach combining detailed atomic physics with a relativistic Monte Carlo transport code:

1. Microphysical Data and Cross Sections:

   • Nuclear Yields and Spectra: Nuclear yields for strong, medium, and weak $r$-process trajectories are calculated using the Prism reaction network. $\beta$-decay rates and spectra are sourced from the IAEA Livechart (ENSDF) and computed using the QRPA+HF model, supplemented by BetaShape for nearly stable nuclei.
   • Electron-Atom Interactions: The study utilizes the Evaluated Electron Data Library (EEDL) from the EPICS2023 database to determine cross sections for electron impact ionization (EII), electron impact excitation (EIE), bremsstrahlung, and Mott scattering.
   • Ionization Corrections: Neutral atom cross sections are adjusted for ionized targets using empirical Lotz scaling and Hartree-Fock binding energies.
   • Secondary Electrons: The authors calculate the "ionization cascade," tracking the production of secondary and tertiary electrons from primary $\beta$-particles to determine the total ionization rate.
   • Other Interactions: Møller scattering (electron-electron) is calculated analytically. Synchrotron and Cherenkov losses are evaluated and found to be negligible compared to collisional losses.

2. Global Transport Model:

   • Ejecta Geometry: The ejecta is modeled as a spherically symmetric, homologously expanding flow with a power-law density profile ($\rho \propto r^{-\alpha}$).
   • Transport Scenarios: Two limiting magnetic field geometries are considered:
     • Trapped Limit: Corresponds to highly tangled or toroidal fields where particles are confined to specific velocity coordinates.
     • Radial Field Limit: Corresponds to coherent radial fields where particles stream freely along field lines. The authors demonstrate that for coherent radial fields, the 1D transport equations are mathematically equivalent to the unmagnetized free-streaming case due to the cancellation of the magnetic mirror force and geometric effects.
   • Monte Carlo Simulation: A fully relativistic Monte Carlo code tracks discrete $\beta$-particle packets. It accounts for adiabatic cooling due to expansion, collisional energy loss (stopping power), and scattering (transport opacity). Particles are terminated upon thermalization ($E < 1$ eV), escape ($u > v_{ej}$), or reaching a maximum time (1 year).

3. Ionization Structure:

   • A simplified rate equation model is applied to Neodymium (Nd) to illustrate the impact of transport and the ionization cascade on the ejecta temperature and ionization state, neglecting photoionization and coupling between different elements.

Key Contributions and Results

• Atomic Microphysics: The authors provide evaluated, per-species stopping and transport cross sections, as well as EII cross sections, available online. They show that at MeV energies, EII dominates energy loss, while at lower energies (thermalization phase), Coulomb interactions with free thermal electrons (Møller scattering) become dominant.
• Ionization Cascade: The study quantifies the contribution of secondary electrons to the total ionization rate. The cascade significantly enhances the ionization cross section, contributing as much as the primary electrons at 1 MeV.
• Thermalization Efficiency:
  • Non-local Effects: Non-local energy deposition and particle escape reduce thermalization efficiency, particularly in the innermost and outermost ejecta. This makes the efficiency more uniform across the ejecta compared to local deposition models.
  • Transport Limits: In the radial field limit, particle escape is the primary driver of inefficiency. In the trapped limit, inefficiency is driven by adiabatic expansion losses as particles persist longer in the ejecta.
  • Dependence on Parameters: Thermalization efficiency is sensitive to ejecta mass, velocity, $r$-process abundance patterns, and density profile slopes. It is less sensitive to the ionization state, suggesting that energy deposition can be modeled without self-consistent ionization calculations in some contexts.
• Analytic Prescription: The authors derive an analytic formula for the spatially dependent thermalization efficiency, $f_{thm}(t) = \ln(1+\tau)/\tau$, parameterized by an inefficiency time ($t_{ineff}$) and an asymptotic slope ($n$). They provide fitting parameters for these values as a function of ejecta mass, velocity, and position. The median residuals between this prescription and Monte Carlo results are $\sim0.04$ dex (radial) and $\sim0.02$ dex (trapped).
• Ionization Structure: The illustrative Nd calculation reveals an "inverted" ionization structure in the nebular phase (higher ionization in outer, less dense regions) due to less efficient recombination. Radial magnetic fields lead to lower ionization states due to particle escape, while including the ionization cascade raises the ionization state throughout the ejecta.

Significance and Claims
The paper asserts that the next generation of kilonova models must incorporate evaluated atomic data and explicit charged-particle transport to reduce uncertainties in light curve predictions. The authors claim that:

1. Transport Physics is Critical: Ignoring non-local transport effects leads to overestimates of heating in the inner and outer ejecta and incorrect ionization structures.
2. Atomic Data Matters: The use of evaluated atomic data (EEDL) and the inclusion of the ionization cascade are necessary for accurate ionization rates, which in turn affect opacity and cooling.
3. Analytic Utility: The provided analytic prescriptions offer a computationally cheap approximation for spatially dependent thermalization efficiency, suitable for semi-analytic and radiative transfer models, though explicit transport is recommended for high-fidelity simulations.
4. Limitations: The authors modestly note that their results are based on idealized spherical symmetry and limiting magnetic field geometries. They suggest that realistic magnetic field morphologies, non-spherical ejecta, and collective plasma effects (e.g., streaming instability) are areas for future investigation. They also emphasize that their results focus on $\beta$-particles and should be integrated with future work on $\alpha$-particles and fission fragments for a complete thermalization picture.