Core-wing transitions and the breakdown of diffusion in Lyman-$α$ radiative transfer

TL;DR

The paper investigates Lyα radiative transfer in static, dust-free, homogeneous clouds, revealing a breakdown of diffusion descriptions near the line core and introducing a physically motivated core–wing transition $x_\text{cw}^*$. Using diffusion theory, it derives spectral distributions for the trapping time, number of scatterings, and force multiplier, and benchmarks them against no-core-skipping MCRT results to delineate the diffusion regime. A key finding is that diffusion models accurately capture wing-dominated quantities but significantly underestimate core contributions to the force, highlighting limitations of core skipping for internal feedback predictions; energy-density and pressure-based estimators offer robust alternatives. The study also uncovers anomalous spatial diffusion with fat-tailed jump distributions, proposes scaling relations for wing excursions, and discusses implications for incorporating Lyα radiation pressure in simulations, including pathways toward hybrid or fractional-diffusion approaches.

Abstract

The Lyman alpha (LyA) line of neutral hydrogen plays a central role in observations of star-forming galaxies. However, resonant scattering makes it difficult to directly interpret LyA signatures. Monte Carlo radiative transfer (MCRT) calculations have become the gold standard for modeling LyA, but it becomes extremely computationally expensive in optically thick environments. Workarounds, such as core-skipping to avoid repetitive low-transport scatterings, greatly increase the efficiency of MCRT simulations but introduce errors in the solutions. While core-skipping is designed to preserve emergent spectra, the internal radiation field, most importantly, the momentum imparted, is not properly preserved. On the other hand, to make analytical and numerical progress, it is often assumed that photons diffuse in both frequency and physical space. We find that these diffusion approximations break down for frequencies near the core and positions at finite optical depths. We propose a more physically-motivated definition for the core-wing transition frequency to isolate such effects. We derive new spectral distributions of internal radiation properties and compare the results with simulations. We analyze the diffusive properties of LyA photons and demonstrate anomalous spatial diffusion behavior with fat-tailed distributions. This work deepens our understanding of diffusion in resonant-line transfer and identifies areas where simulations or analytics may be failing and how these failures may be resolved.

Figures (20)

• Figure 1: The relative percent error in the fitting formula given by Eq. (\ref{['eq:xcwast_fit']}) is shown in green. The bound on the error is very low for the parameter range of interest. In the inset plot, the exact solution for largest solution to Eq. (\ref{['eq:force_core_wing_frequency']}) found with a root finder is shown in black while the red line indicates $x_{\text{cw}}$ defined as the frequency where $H(a,x)$ transitions from a Gaussian to a Lorentzian distribution, which is important for core-skipping algorithms.
• Figure 1: The first five angular-averaged moments of the $R_\text{II}$ redistribution function $\mathcal{M}_k$. The solid lines show the moments assuming the isotropic scattering case and the dotted lines show the dipole scattering case.
• Figure 1: The relative error in the number of scatterings a photon takes to return to the core given an initial frequency $x_\text{init}$ when compared to the analytical estimate from Eq. (\ref{['eq:n_scat_to_core_analytical']}). The error starts out high but decreases with a power-law scaling until the error $\leq 1 \%$. The MCRT simulations are with $5 \times 10^7$ photons for each initial frequency with 40 initial frequencies ranging from $x = 5$ to $x = 80$. The temperature is held fixed at $T = 10^4 \,\text{K}$
• Figure 2: A comparison between analytical trapping time (thin lines) and MCRT simulations (thick lines) as defined in Eq. (\ref{['eq:general_spectral_trapping_time']}). Bottom Panel: The analytical $t_{\text{trap},x}$ from Eq. (\ref{['eq:spherical_point_spectral_t_trap']}) evaluated exactly with numerical integration for various values of $\tau_0$ with a fixed temperature of $T = 10^4 \,\text{K}$ compared with binned contributions from MCRT simulations. The vertical line shows $x_\text{cw}^\ast$ at this temperature. Top Panel: The cumulative fraction of $t_\text{trap}$ from frequencies $< |x|$ ($f_\text{trap}$) as defined in Eq. (\ref{['eq:fractional_contribution']}). The agreement here is almost perfect once the diffusion approximation is valid.
• Figure 2: The Error in the redistribution moment approximation for various temperatures. We plot only the isotropic case since the dipole case is nearly identical for $\mathcal{M}_2$. The dots represent the value of $x_\text{cw}^\ast$ corresponding to the temperature plotted, which varies slightly over this temperature range. The agreement is only good in the wing. Bottom Panel: The absolute difference between the second moment and zeroth moments $|\mathcal{M}_2 - \mathcal{M}_0|$. Top Panel: the relative difference between the first and second moments $|\mathcal{M}_2 - \mathcal{M}_0|/ \mathcal{M}_0$.