Time-dependent Radiative Transfer Research Articles

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

The accretion burst of the massive young stellar object G323.46-0.08

Accretion bursts from low-mass young stellar objects (YSOs) have been known for many decades. In recent years, the first accretion bursts of massive YSOs (MYSOs) have been observed. These phases of intense protostellar growth are of particular importance for studying massive star formation. Bursts of MYSOs are accompanied by flares of Class II methanol masers (hereafter masers), which are caused by an increase in exciting mid-infrared (MIR) emission. They can lead to long-lasting thermal afterglows of the dust continuum radiation visible at infrared (IR) and (sub)millimeter (hereafter (sub)mm) wavelengths. Furthermore, they might cause a scattered light echo. The G323.46$-$0.08 (hereafter G323) event, which shows all these features, extends the small sample of known MYSO bursts. Maser observations of the MYSO G323 show evidence of a flare, which was presumed to be caused by an accretion burst. This should be verified with IR data. We used time-dependent radiative transfer (TDRT) to characterize the heating and cooling timescales for eruptive MYSOs and to infer the main burst parameters. Burst light curves, as well as the pre-burst spectral energy distribution (SED) were established from archival IR data. The properties of the MYSO, including its circumstellar disk and envelope, were derived by using static radiative transfer modeling of pre-burst data. For the first time, TDRT was used to predict the temporal evolution of the SED. Observations with SOFIA/HAWC+ were performed to constrain the burst energy from the strength of the thermal afterglow. Image subtraction and ratioing were applied to reveal the light echo. The G323 accretion burst is confirmed. It reached its peak in late 2013/early 2014 with a $K_ s $-band increase of $ sim ts ts mag. Both $K_ s $-band and integrated maser flux densities follow an exponential decay. TDRT indicates that the duration of the thermal afterglow in the far-infrared (FIR) can exceed the burst duration by years. The latter was proved by SOFIA observations, which indicate a flux increase of $(14.2 at $70\ m$ and $(8.5 at ts in 2022 (2 years after the burst ended). A one-sided light echo emerged that was propagating into the interstellar medium. The burst origin of the G323 maser flare has been verified. TDRT simulations revealed the strong influence of the burst energetics and the local dust distribution on the strength and duration of the afterglow. The G323 burst is probably the most energetic MYSO burst that has been observed so far. Within $8.4 \, yrs$, an energy of $(0.9 erg$ was released. The short timescale points to the accretion of a compact body, while the burst energy corresponds to an accumulated mass of at least $(7 Jup $ and possibly even more if the protostar is bloated. In this case, the accretion event might have triggered protostellar pulsations, which give rise to the observed maser periodicity. The associated IR light echo is the second observed from a MYSO burst.

TeV pion bumps in the gamma-ray spectra of flaring blazars

Context. Very high-energy (VHE, E > 100 GeV) observations of the blazar Mrk 501 with MAGIC in 2014 provided evidence for an unusual narrow spectral feature at about 3 TeV during an extreme X-ray flaring activity. The one-zone synchrotron-self Compton scenario, widely used in blazar broadband spectral modeling, fails to explain the narrow TeV component. Aims. Motivated by this rare observation, we propose an alternative model for the production of narrow features in the VHE spectra of flaring blazars. These spectral features may result from the decay of neutral pions (π0 bumps) that are in turn produced via interactions of protons (of tens of TeV energy) with energetic photons, whose density increases during hard X-ray flares. Methods. We explored the conditions needed for the emergence of narrow π0 bumps in VHE blazar spectra during X-ray flares reaching synchrotron energies ∼100 keV using time-dependent radiative transfer calculations. We focused on high-synchrotron peaked (HSP) blazars, which comprise the majority of VHE-detected extragalactic sources. Results. We find that synchrotron-dominated flares with peak energies ≳100 keV can be ideal periods for the search of π0 bumps in the VHE spectra of HSP blazars. The flaring region is optically thin to photopion production, its energy content is dominated by the relativistic proton population, and the inferred jet power is highly super-Eddington. Application of the model to the spectral energy distribution of Mrk 501 on MJD 56857.98 shows that the VHE spectrum of the flare is described well by the sum of a synchrotron self-Compton (SSC) component and a distinct π0 bump centered at 3 TeV. Spectral fitting of simulated SSC+π0 spectra for the Cherenkov Telescope Array (CTA) show that a π0 bump could be detected at a 5σ significance level with a 30-min exposure. Conclusions. A harder VHE γ-ray spectrum than the usual SSC prediction or, more occasionally, a distinct narrow bump at VHE energies during hard X-ray flares, can be suggestive of a relativistic hadronic component in blazar jets that otherwise would remain hidden. The production of narrow features or spectral hardenings due to π0 decay in the VHE spectra of blazars is testable with the advent of CTA.

The evolution of continuum polarization in type II supernovae as a diagnostic of ejecta morphology

The linear polarization of the optical continuum of type II supernovae (SNe), together with its temporal evolution is a promising source of information about the large-scale geometry of their ejecta. To help access this information, we undertook 2D polarized radiative transfer calculations to map the possible landscape of type II SN continuum polarization (Pcont) from 20 to 300 days after explosion. Our simulations were based on crafted 2D axisymmetric ejecta constructed from 1D nonlocal thermodynamic equilibrium time-dependent radiative transfer calculations for the explosion of a red supergiant star. Following the approach used in our previous work on SN 2012aw, we considered a variety of bipolar explosions in which spherical symmetry is broken by material within ~30° of the poles that has a higher kinetic energy (up to a factor of two) and higher 56Ni abundance (up to a factor of about five, allowing for 56Ni at high velocity). Our set of eight 2D ejecta configurations produced considerable diversity in Pcont (λ ~ 7000 Å), although its maximum of 1–4% systematically occurs around the transition to the nebular phase. Before and after this transition, Pcont may be null, constant, rising, or decreasing, which is caused by the complex geometry of the depth-dependent density and ionization and also by optical depth effects. Our modest angle-dependent explosion energy can yield a Pcont of 0.5–1% at early times. Residual optical-depth effects can yield an angle-dependent SN brightness and constant polarization at nebular times. The observed values of Pcont tend to be lower than obtained here. This suggests that more complicated geometries with competing large-scale structures cancel the polarization. Extreme asymmetries seem to be excluded.

A theoretical Fourier-transformation model for the formation of X-ray time lags from black hole accretion discs

ABSTRACT X-ray emission from active galactic nuclei (AGNs) often displays complex and rapid variability, which may provide a glimpse into the detailed thermal and dynamical structure of the accreting gas near the event horizon of the central black hole. The observed variability can be analysed using Fourier transforms of the light curves in multiple energy channels, which can be used to generate Fourier phase lags, corresponding to lags in the time domain. The X-ray time lags may be either soft lags or hard lags, depending on whether the variability in the hard energy channel precedes that in the soft channel or vice versa. The physical explanation for the observed X-ray time lags from AGNs has been puzzling, and several scenarios have been proposed. In this paper, we explore the hypothesis that the X-ray time lags are produced as a result of the reprocessing of iron L-line and K-line seed photons generated via fluorescence, which is driven by a variable incident radiation field. The seed photons are reprocessed by a combination of thermal and bulk Comptonization and spatial reverberation. We assume that the inner region of the accretion flow can be approximated as a hot, geometrically thick ADAF disc. The outer radius of the ADAF region is equal to the shock formation radius, which is located just outside the centrifugal barrier. The time-dependent radiative transfer in the disc is analysed using a Fourier-transformed, vertically averaged transport equation in cylindrical coordinates. We demonstrate that the new model can successfully reproduce the complex X-ray variability data for the Seyfert 1 galaxies 1H 0707–495 and Ark 564.

A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation

A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation

A sequence of Type Ib, IIb, II-L, and II-P supernovae from binary-star progenitors with varying initial separations

Over the last decade, evidence has accumulated that massive stars do not typically evolve in isolation but instead follow a tumultuous journey with a companion star on their way to core collapse. While Roche-lobe overflow appears instrumental for the production of a large fraction of Type Ib and Ic supernovae (SNe), variations in the initial orbital period, $P_ init $, of massive interacting binaries may also produce a wide diversity of case B, BC, or C systems, with pre-SN stars endowed from minute to massive H-rich envelopes. Focusing here on the explosion of the primary donor star, originally 12.6\ we used radiation hydrodynamics and nonlocal thermodynamic equilibrium time-dependent radiative transfer to document the gas and radiation properties of such SNe, covering Types Ib, IIb, II-L, and II-P. Variations in init $ are the root cause of the wide diversity of our SN light curves, which present single-peak, double-peak, fast-declining, or plateau-like morphologies in the $V$ band. The different ejecta structures, expansion rates, and relative abundances (e.g., H, He, and can lead to a great deal of diversity in terms of spectral line shapes (absorption versus emission strength and width) and evolution. We emphasize that Halpha is a key tracer of these modulations, and that He is an enduring optical diagnostic for the presence of He. Our grid of simulations fares well against representative Type Ib, IIb, and II-P SNe, but interaction with circumstellar material, which is ignored in this work, is likely at the origin of the tension between our Type II-L SN models and observations (e.g., of SN\,2006Y). Remaining discrepancies in the rise time to bolometric maximum of our models call for a proper account of both small-scale and large-scale structures in core-collapse SN ejecta. Discrepant Type II-P SN models, with a high plateau brightness but small spectral line widths, can be fixed by adopting more compact red-supergiant star progenitors.

Light curves and spectra for theoretical models of high-velocity red-giant star collisions

High-velocity stellar collisions driven by a supermassive black hole (BH) or BH-driven disruptive collisions in dense, nuclear clusters can rival the energetics of supergiant star explosions following the gravitational collapse of their iron core. Starting from a sample of red-giant star collisions simulated with the hydrodynamics code AREPO, we generated photometric and spectroscopic observables using the nonlocal thermodynamic equilibrium time-dependent radiative transfer code CMFGEN. Collisions from more extended giants or more violent collisions (with higher velocities or smaller impact parameters) yield bolometric luminosities on the order of 1043 erg s−1 at 1 d, evolving on a timescale of a week to a bright plateau at ∼1041 erg s−1 before plunging precipitously after 20–40 d at the end of the optically thick phase. This luminosity falls primarily in the UV in the first few days, thus when it is at its maximum, and shifts to the optical thereafter. Collisions at lower velocities or from less extended stars produce ejecta that are fainter but can remain optically thick for up to 40 d if they have a low expansion rate. This collision debris shows a similar spectral evolution as that observed or modeled for Type II supernovae from blue-supergiant star explosions, differing only in the more rapid transition to the nebular phase. Such BH-driven disruptive collisions should be detectable by high-cadence surveys in the UV such as ULTRASAT.

Exact analytical solution of the one-dimensional time-dependent radiative transfer equation with linear scattering

The radiative transfer equation (RTE) is a cornerstone for describing the propagation of electromagnetic radiation in a medium, with applications spanning atmospheric science, astrophysics, remote sensing, and biomedical optics. Despite its importance, an exact analytical solution to the RTE has remained elusive, necessitating the use of numerical approximations such as Monte Carlo, discrete ordinate, and spherical harmonics methods. In this paper, we present an exact solution to the one-dimensional time-dependent RTE. We delve into the moments of the photon distribution, providing a clear view of the transition to the diffusion regime. This analysis offers a deeper understanding of light propagation in the medium. Furthermore, we demonstrate that the one-dimensional RTE is equivalent to the Klein-Gordon equation with an imaginary mass term determined by the inverse reduced scattering length. Contrary to naive expectations of superluminal solutions, we find that our solution is strictly causal under appropriate boundary conditions, determined by the light transport problem. We validate the found solution using Monte Carlo simulations and benchmark the performance of the latter. Our analysis reveals that even for highly forward scattering, dozens of random light scatterings are required for an accurate estimate, underscoring the complexity of the problem. Moreover, we propose a method for faster convergence by adjusting the parameters of Monte Carlo sampling. We show that a Monte Carlo method sampling photon scatterings with input parameters (μs,g), where μs is the inverse scattering length and g is the scattering anisotropy parameter, is equivalent to that with (μs(1−g)/2,−1). This equivalence leads to a significantly faster convergence to the exact solution, offering a substantial improvement of the Monte Carlo method for the one-dimensional RTE. Our findings not only contribute to the theoretical understanding of the RTE but also have potential implications for improving the numerical methods used to approximate it.

A benchmark study of atomic models for the transition region against quiet Sun observations

ABSTRACT The use of the coronal approximation to model line emission from the solar transition region has led to discrepancies with observations over many years, particularly for Li- and Na-like ions. Studies have shown that a number of atomic processes are required to improve the modelling for this region, including the effects of high densities, solar radiation, and charge transfer on ion formation. Other non-equilibrium processes, such as time-dependent ionization and radiative transfer, are also expected to play a role. A set of models which include the three relevant atomic processes listed above in ionization equilibrium has recently been built. These new results cover the main elements observed in the transition region. To assess the effectiveness of the results, this work predicts spectral line intensities using differential emission measure modelling. Although limited in some respects, this differential emission measure modelling does give a good indication of the impact of the new atomic calculations. The results are compared to predictions of the coronal approximation and to observations of the average, quiet Sun from published literature. Significant improvements are seen for the line emission from Li- and Na-like ions, intercombination lines, and many other lines. From this study, an assessment is made of how far down into the solar atmosphere the coronal approximation can be applied, and the range over which the new atomic models are valid.

Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients

Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients

A comparison of numerical methods for computing the reionization of intergalactic hydrogen and helium by a central radiating source

ABSTRACT We compare numerical methods for solving the radiative transfer equation in the context of the photoionization of intergalactic gaseous hydrogen and helium by a central radiating source. Direct integration of the radiative transfer equation and solutions using photon packets are examined, both for solutions to the time-dependent radiative transfer equation and in the infinite-speed-of-light approximation. The photon packet schemes are found to be more generally computationally efficient than a direct integration scheme. While all codes accurately describe the growth rate of hydrogen and helium ionization zones, it is shown that a fully time-dependent method is required to capture the gas temperature and ionization structure in the near zone of a source when an ionization front expands at a speed close to the speed of light. Applied to quasi-stellar objects in the Epoch of Reionization (EoR), temperature differences as high as 5 × 104 K result in the near zone for solutions of the time-dependent radiative transfer equation compared with solutions in the infinite-speed-of-light approximation. Smaller temperature differences are found following the nearly full photoionization of helium in gas in which the hydrogen was already ionized and the helium was singly ionized. Variations found in the temperature and ionization structure far from the source, where the gas is predominantly neutral, may affect some predictions for 21-cm EoR experiments.

Time-dependent Monte Carlo continuum radiative transfer

Context. Variability is a characteristic feature of young stellar objects that is caused by various underlying physical processes. Multi-epoch observations in the optical and infrared combined with radiative transfer simulations are key to study these processes in detail. Aims. We present an implementation of an algorithm for 3D time-dependent Monte Carlo radiative transfer. It allows one to simulate temperature distributions as well as images and spectral energy distributions of the scattered light and thermal reemission radiation for variable illuminating and heating sources embedded in dust distributions, such as circumstellar disks and dust shells on time scales up to weeks. Methods. We extended the publicly available 3D Monte Carlo radiative transfer code POLARIS with efficient methods for the simulation of temperature distributions, scattering, and thermal reemission of dust distributions illuminated by temporally variable radiation sources. The influence of the chosen temporal step width and the number of photon packages per time step as key parameters for a given configuration is shown by simulating the temperature distribution in a spherical envelope around an embedded central star. The effect of the optical depth on the temperature simulation is discussed for the spherical envelope as well as for a model of a circumstellar disk with an embedded star. Finally, we present simulations of an outburst of a star surrounded by a circumstellar disk. Results. The presented algorithm for time-dependent 3D continuum Monte Carlo radiative transfer is a valuable basis for preparatory studies as well as for the analysis of continuum observations of the dusty environment around variable sources, such as accreting young stellar objects. In particular, the combined study of light echos in the optical and near-infrared wavelength range and the corresponding time-dependent thermal reemission observables of variable, for example outbursting sources, becomes possible on all involved spatial scales.

Modeling Effect of Bubbles on Time-Dependent Radiation Transfer of Microalgae in a Photobioreactor for Carbon Dioxide Fixation

Microalgae are considered one of the most efficient and environmentally friendly ways for carbon dioxide fixation. The bubbles play an important role in analyzing the radiation transfer in photobioreactors during microalgae growth. Herein, Chlorella sp. and Scenedesmus obliquus were cultured in the airlift flat plate photobioreactor and evaluated for the temporal evolution of radiation characteristics. A one-dimensional model of bubbles on time-dependent radiation transfer in a photobioreactor was proposed, and it was well verified with the experimental result. The results indicated that with the increase of bubble volume fraction or the decrease of bubble radius, the local irradiance increased at the illuminated surface of the microalgal culture and was attenuated more rapidly along with the radiation transfer. The average specific growth rate of microalgae decreases as bubble volume fraction increases or bubble radius decreases. The volume fraction of 0.003 and a radius of 3.5 mm are the optimal operating conditions in this study for microalgae growth and carbon dioxide fixation. The presented analysis would facilitate the design and optimization of the optical and aeration configurations of photobioreactors for carbon dioxide fixation.

The initial magnetic criticality of pre-stellar cores

ABSTRACT Direct observational measurements of the magnetic field strength in pre-stellar cores typically find supercritical mass-to-flux ratios, suggesting that the magnetic field is insufficient to prevent gravitational collapse. These measurements suffer from significant uncertainties; an alternative approach is to utilize the sensitivity of pre-stellar chemistry to the evolutionary history, and indirectly constrain the degree of magnetic support. We combine non-ideal magnetohydrodynamic simulations of pre-stellar cores with time-dependent chemistry and radiative transfer modelling, producing synthetic observations of the model cores in several commonly observed molecular lines. We find that molecules strongly affected by freeze-out, such as CS and HCN, typically have much lower line intensities in magnetically subcritical models compared to supercritical ones, due to the longer collapse time-scales. Subcritical models also produce much narrower lines for all species investigated. Accounting for a range of core properties, ages, and viewing angles, we find that supercritical models are unable to reproduce the distribution of CS and N2H+ line strengths and widths seen in an observational sample, whereas subcritical models are in good agreement with the available data. This suggests that despite presently having supercritical mass-to-flux ratios, pre-stellar cores form as magnetically subcritical objects.

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [6,55], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely Δt=O(h) in the transport regime where Δt and h are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction Δt=O(h2) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.

Emission from HCN and CH3OH in comets

Aims. The aim of this work is to characterise HCN and CH3OH emission from recent comets. Methods. We used the Onsala 20-m telescope to search for millimetre transitions of HCN towards a sample of 11 recent and mostly bright comets in the period from December 2016 to November 2019. Also, CH3OH was searched for in two comets. The HCN sample includes the interstellar comet 2I/Borisov. For the short-period comet 46P/Wirtanen, we were able to monitor the variation of HCN emission over a time-span of about one month. We performed radiative transfer modelling for the observed molecular emission by also including time-dependent effects due to the outgassing of molecules. Results. HCN was detected in six comets. Two of these are short-period comets and four are long-period. Six methanol transitions were detected in 46P/Wirtanen, enabling us to determine the gas kinetic temperature. From the observations, we determined the molecular production rates using time-dependent radiative transfer modelling. For five comets, we were able to determine that the HCN mixing ratios lie near 0.1% using contemporary water production rates, ${Q_{{{\rm{H}}_2}{\rm{O}}}}$, taken from other studies. This HCN mixing ratio was also found to be typical in our monitoring observations of 46P/Wirtanen but here we notice deviations of up to 0.2% on a daily timescale which could indicate short-time changes in outgassing activity. From our radiative transfer modelling of cometary comae, we find that time-dependent effects on the HCN level populations are of the order of 5–15% when ${Q_{{{\rm{H}}_2}{\rm{O}}}}$ is around 2 × 1028 mol s−1. The effects may be stronger for comets with lower ${Q_{{{\rm{H}}_2}{\rm{O}}}}$. The exact details of the time-dependent effects depend on the amount of neutral and electron collisions, radiative pumping, and molecular parameters such as the spontaneous rate coefficient.

General Relativistic Radiation Magnetohydrodynamics Simulations of Black Hole Accretion Disks: Comparison of Methods Based on Variable Eddington Tensor and Based on M1 Closure

We perform 2.5-dimensional general relativistic radiation magnetohydrodynamics simulations of black hole accretion disks and disk winds in the range of mass accretion rate from 0.1 to 104.5 times the Eddington limit. In this paper, we compare the results of the INAZUMA code, in which the frequency-integrated time-dependent radiation transfer equation is solved in order to evaluate the Eddington tensor, with those of the first momentum (M1) approximation method. In both methods, accretion disks and disk winds appear, and there is no remarkable difference in accretion rate, outflow rate, or luminosity. However, the significant difference in the radiation field appears around the rotation axis. In the M1 method, the radial component of the radiation flux tends to be amplified owing to unphysical radiation collisions. Such an enhancement of the outward radiation flux does not appear in INAZUMA. Also, the problem of radiation not reaching the rotation axis occurs with M1, but not with INAZUMA. Our results indicate that the radiation transfer equation should be solved to obtain the accurate radiation field in the optically thin region around the rotation axis.

Conservation Laws in Time-Dependent Radiative Transfer Problems

Conservation Laws in Time-Dependent Radiative Transfer Problems

Time Dependent Radiative Transfer Problems in a One-Dimensional Medium

Time Dependent Radiative Transfer Problems in a One-Dimensional Medium