A Novel Solution for Resonant Scattering Using Self-Consistent Boundary Conditions

Abstract

We present two novel additions to the semi-analytic solution of Lyman $\alpha$ (Ly$\alpha$) radiative transfer in spherical geometry: (1) implementation of the correct boundary condition for a steady source, and (2) solution of the time-dependent problem for an impulsive source. For the steady-state problem, the solution can be represented as a sum of two terms: a previously-known analytic solution of the equation with mean intensity $J=0$ at the surface, and a novel, semi-analytic solution which enforces the correct boundary condition of zero-ingoing intensity at the surface. This solution is compared to that of the Monte Carlo method, which is valid at arbitrary optical depth. It is shown that the size of the correction is of order unity when the spectral peaks approach the Doppler core and decreases slowly with line center optical depth, specifically as $(a{\tau }_0{)}^{-1/3}$, which may explain discrepancies seen in previous studies. For the impulsive problem, the time, spatial, and frequency dependence of the solution are expressed using an eigenfunction expansion in order to characterize the escape time distribution and emergent spectra of photons. It is shown that the lowest-order eigenfrequency agrees well with the decay rate found in the Monte Carlo escape time distribution at sufficiently large line-center optical depths. The characterization of the escape-time distribution highlights the potential for a Monte Carlo acceleration method, which would sample photon escape properties from distributions rather than calculating every photon scattering, thereby reducing computational demand.

arXiv comment23 pages, 10 figures, 2 appendices, resubmitted to ApJ after review

References in corpus (28)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Array Programming with NumPy
• A giant comet-like cloud of hydrogen escaping the warm Neptune-mass exoplanet GJ 436b
• Lyman Alpha Radiation From Collapsing Protogalaxies I: Characteristics of the Emergent Spectrum
• Radiative transfer in very optically thick circumstellar disks
• Temporal variations in the evaporating atmosphere of the exoplanet HD 189733b
• Hubble PanCET: An extended upper atmosphere of neutral hydrogen around the warm Neptune GJ 3470 b
• Hint of a transiting extended atmosphere on 55 Cancri b
• Ly-alpha Radiative Transfer in Cosmological Simulations and Application to a z~8 Emitter
• Reconnaissance of the TRAPPIST-1 exoplanet system in the Lyman-$\alpha$ line
• The Lyman-alpha signature of the first galaxies
• Lyman alpha line formation in starbursting galaxies II. Extremely Thick, Dustless, and Static HI Media
• The Time Development of a Resonance Line in the Expanding Universe
• A Model of the H$\alpha$ and Na Transmission Spectrum of HD 189733b
• No hydrogen exosphere detected around the super-Earth HD97658 b
• Lyman-alpha radiation hydrodynamics of galactic winds before cosmic reionization
• Is pi Men c's atmosphere hydrogen-dominated? Insights from a non-detecton of Hy Ly-alpha absorption
• On the Modified Random Walk for Monte-Carlo Radiation Transfer
• Strong HI Lyman-$\alpha$ variations from the 11 Gyr-old host star Kepler-444: a planetary origin ?
• Ly-alpha Radiative Transfer: Monte-Carlo Simulation of the Wouthuysen-Field Effect
• Lyman-alpha in the GJ 1132 System: Stellar Emission and Planetary Atmospheric Evolution
• Discrete diffusion Lyman-alpha radiative transfer
• Lyman-alpha radiation pressure: an analytical exploration
• Resonant-line radiative transfer within power-law density profiles
• Atmosphere escape inferred from modelling the H$\alpha$ transmission spectrum of WASP-121b
• Resonant Scattering and Ly-alpha Radiation Emergent from Neutral Hydrogen Halos
• The Hubble PanCET program: Long-term chromospheric evolution and flaring activity of the M dwarf host GJ 3470
• Time-Dependent Behavior of Lyman$\alpha$ Photon Transfer in High Redshift Optically Thick Medium