Time-dependent Radiative Transfer Research Articles

The Asymptotic Telegrapher’s Equation (P1) Approximation for Time-Dependent, Thermal Radiative Transfer

We develop the asymptotic P 1 approximation for the time-dependent thermal radiative transfer equation for a multidimensional general geometry. Careful derivation of the asymptotic P 1 equations, directly from the time-dependent Boltzmann equation, yields a particle velocity that is closer (v≈0.91c) to the exact value of c but is based on an asymptotic analysis rather than diffusion theory (infinite velocity) or conventional P 1 theory (which gives rise to the Telegrapher’s equation, ). While this approach does not match the exact value of c as does the P 1/3 method, the latter method is an ad hoc approach that has not been justified on theoretical grounds. This article provides the theoretical justification for the almost-correct value of c that yields improved results for the well-known (one-dimensional) Su-Olson benchmark for radiative transfer, for which we obtain a semi-analytic solution in the case of local thermodynamic equilibrium. We found that the asymptotic P 1 approximation yields a better solution than the diffusion, the classic P 1, and the P 1/3 approximations, yielding the correct steady-state behavior for the energy density and the (almost) correct particle velocity.

A Consistent, Moment-Based, Multiscale Solution Approach for Thermal Radiative Transfer Problems

We present an efficient numerical algorithm for solving the time-dependent grey thermal radiative transfer (TRT) equations. The algorithm utilizes the first two angular moments of the TRT equations (Quasi-diffusion (QD)) together with the material temperature equation to form a nonlinear low-order (LO) system. The LO system is solved via the Jacobian-free Newton-Krylov method. The combined high-order (HO) TRT and LO-QD system is used to bridge the diffusion and transport scales. In addition, a “consistency” term is introduced to make the truncation error in the LO system identical to the truncation error in the HO equation. The derivation of the consistency term is rather general; therefore, it can be extended to a variety of spatial and temporal discretizations.

Infinite space Green’s function of the time-dependent radiative transfer equation

This study contains the derivation of an infinite space Green’s function of the time-dependent radiative transfer equation in an anisotropically scattering medium based on analytical approaches. The final solutions are analytical regarding the time variable and given by a superposition of real and complex exponential functions. The obtained expressions were successfully validated with Monte Carlo simulations.

Time-dependent simulations of multiwavelength variability of the blazar Mrk 421 with a Monte Carlo multizone code

(abridged) We present a new time-dependent multi-zone radiative transfer code and its application to study the SSC emission of Mrk 421. The code couples Fokker-Planck and Monte Carlo methods, in a 2D geometry. For the first time all the light travel time effects (LCTE) are fully considered, along with a proper treatment of Compton cooling, which depends on them. We study a set of simple scenarios where the variability is produced by injection of relativistic electrons as a `shock front' crosses the emission region. We consider emission from two components, with the second one either being pre-existing and co-spatial and participating in the evolution of the active region, or spatially separated and independent, only diluting the observed variability. Temporal and spectral results of the simulation are compared to the multiwavelength observations of Mrk 421 in March 2001. We find parameters that can adequately fit the observed SEDs and multiwavelength light curves and correlations. There remain however a few open issues, most notably: i) systematic soft intra-band X-ray lags. ii) The quadratic correlation between the TeV and X-ray flux during the flare decay has not been reproduced. These features are among those affected by the spatial extent and geometry of the source. The difficulty of producing hard X-ray lags is exacerbated by a bias towards soft lags caused by the combination of energy dependent radiative cooling time-scales and LCTE. About the second emission component, our results strongly favor the scenario where it is co-spatial and it participates in the flare evolution, suggesting that different phases of activity may occur in the same region. The cases presented in this paper represent only an initial study, and despite their limited scope they make a strong case for the need of true time-dependent and multi-zone modeling.

An algorithm for Monte Carlo time-dependent radiation transfer

A new Monte-Carlo algorithm for calculating time-dependent radiative-transfer under the assumption of LTE is presented. Unlike flux-limited diffusion the method is polychromatic, includes scattering, and is able to treat the optically thick and free-streaming regimes simultaneously. The algorithm is tested on a variety of 1-d and 2-d problems, and good agreement with benchmark solutions is found. The method is used to calculate the time-varying spectral energy distribution from a circumstellar disc illuminated by a protostar whose accretion luminosity is varying. It is shown that the time lag between the optical variability and the infrared variability results from a combination of the photon travel time and the thermal response in the disc, and that the lag is an approximately linear function of wavelength.

Investigation of the recombination of the retarded shell of “born-again” CSPNe by time-dependent radiative transfer models

AbstractA standard planetary nebula stays more than 10 000 years in the state of a photoionized nebula. As long as the timescales of the most important ionizing processes are much smaller, the ionization state can be characterized by a static photoionization model and simulated with codes like CLOUDY (Ferland et al. 1998). When the star exhibits a late helium flash, however, its ionizing flux stops within a very short period. The star then re-appears from its opaque shell after a few years (or centuries) as a cold giant star without any hard ionizing photons. Describing the physics of such behavior requires a fully time-dependent radiative transfer model. Pollacco (1999), Kerber et al. (1999) and Lechner & Kimeswenger (2004) used data of the old nebulae around V605 Aql and V4334 Sgr to derive a model of the pre-outburst state of the CSPN in a static model. Their argument was the long recombination time scale for such thin media. With regard to these models Schönberner (2008) critically raised the question whether a significant change in the ionization state (and thus the spectrum) has to be expected after a time of up to 80 years, and whether static models are applicable at all.

The bi-stability jump as the origin for multiple P-Cygni absorption components in luminous blue variables

Luminous Blue Variables (LBVs) oftentimes show double-troughed absorption in their strong Halpha lines, which are as yet not understood. Intriguingly, the feature has also been seen in the interacting supernova SN 2005gj, which was for this reason suggested to have an LBV progenitor. Our aims are to understand the double-troughed absorption feature in LBVs and investigate whether this phenomenon is related to wind variability. To this purpose, we perform time-dependent radiative transfer modeling using CMFGEN. We find that abrupt changes in the wind-terminal velocity - as expected from the bi-stability jump - are required to explain the double-troughed absorption profiles in LBVs. This strengthens scenarios that discuss the link between LBVs and SNe utilizing the progenitor's wind variability resulting from the bi-stability jump. We also discuss why the presence of double-troughed P-Cygni components may become an efficient tool to detect extra-galactic LBVs and how to analyze their mass-loss history on the basis of just one single epoch of spectral observations.

Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions

An original well-balanced (WB) Godunov scheme relying on an exact Riemann solver involving a non-conservative (NC) product is developed. It is meant to solve accurately the time-dependent one-dimensional radiative transfer equation in the discrete ordinates approximation with an arbitrary even number of velocities. The collision term is thus concentrated onto a discrete lattice by means of Dirac masses; this induces steady contact discontinuities which are integral curves of the stationary problem. One solves it by taking advantage of the method of elementary solutions mainly developed by Case, Zweifel and Cercignani. This approach produces a rather simple scheme that compares advantageously to standard existing upwind schemes, especially for the decay in time toward a Maxwellian distribution. It is possible to reformulate this scheme in order to handle properly the parabolic scaling in order to generate a so-called asymptotic-preserving (AP) discretization. Consistency with the diffusive approximation holds independently of the computational grid. Several numerical results are displayed to show the realizability and the efficiency of the method.

A COMPARISON OF METHODS FOR DETERMINING THE MOLECULAR CONTENT OF MODEL GALAXIES

Recent observations indicate that star formation occurs only in the molecular phase of a galaxy's interstellar medium. A realistic treatment of star formation in simulations and analytic models of galaxies therefore requires that one determine where the transition from the atomic to molecular gas occurs. In this paper we compare two methods for making this determination in cosmological simulations where the internal structures of molecular clouds are unresolved: a complex time-dependent chemistry network coupled to a radiative transfer calculation of the dissociating ultraviolet (UV) radiation field, and a simple time-independent analytic approximation. We show that these two methods produce excellent agreement at all metallicities >~10^-2 of the Milky Way value across a very wide range of UV fields. At lower metallicities the agreement is worse, likely because time-dependent effects become important; however, there are no observational calibrations of molecular gas content at such low metallicities, so it is unclear if either method is accurate. The comparison suggests that, in many but not all applications, the analytic approximation provides a viable and nearly cost-free alternative to full time-dependent chemistry and radiative transfer.

Determining the parameters of the laser radiation intensity limiter based on a time-dependent equation of radiative transfer in a nonlinear medium

The effect of the radiation intensity limitation is described using the time-dependent radiative transfer equation taking into account the nonlinearity of the working material of the limiter, which makes it possible to abstract from specific microscopic mechanisms of interaction of radiation with matter. An expression is presented which describes both the deformation of the shape of the laser pulse propagating through a nonlinear medium and the output characteristic (dependence of the output energy on the energy of incident radiation) of the limiter at a known dependence of the absorption coefficient on the laser pulsed radiation intensity with a given pulse shape. A functional equation is derived to determine the dependence of the absorption coefficient on the intensity from the experimental output characteristic, which allows one to predict the limiter properties for different thicknesses of the working medium, as well as to effectively compare limiters of different types.

The atomic physics underlying the spectroscopic analysis of massive stars and supernovae

We have developed a radiative transfer code, CMFGEN, which allows us to model the spectra of massive stars and supernovae. Using CMFGEN we can derive fundamental parameters such as effective temperatures and surface gravities, derive abundances, and place constraints on stellar wind properties. The last of these is important since all massive stars are losing mass via a stellar wind that is driven from the star by radiation pressure, and this mass loss can substantially influence the spectral appearance and evolution of the star. Recently we have extended CMFGEN to allow us to undertake time-dependent radiative transfer calculations of supernovae. Such calculations will be used to place constraints on the supernova progenitor, to place constraints on the supernova explosion and nucleosynthesis, and to derive distances using a physical approach called the "Expanding Photosphere Method". We describe the assumptions underlying the code and the atomic processes involved. A crucial ingredient in the code is the atomic data. For the modeling we require accurate transition wavelengths, oscillator strengths, photoionization cross-sections, collision strengths, autoionization rates, and charge exchange rates for virtually all species up to, and including, cobalt. Presently, the available atomic data varies substantially in both quantity and quality.

TIME-DEPENDENT RADIATION TRANSFER IN THE INTERNAL SHOCK MODEL SCENARIO FOR BLAZAR JETS

We describe the time-dependent radiation transfer in blazar jets, within the internal shock model. We assume that the central engine, which consists of a black hole and an accretion disk, spews out relativistic shells of plasma with different velocity, mass, and energy. We consider a single inelastic collision between a faster (inner) and a slower (outer) moving shell. We study the dynamics of the collision and evaluate the subsequent emission of radiation via the synchrotron and synchrotron self Compton (SSC) processes after the interaction between the two shells has begun. The collision results in the formation of a forward shock (FS) and a reverse shock (RS) that convert the ordered bulk kinetic energy of the shells into magnetic field energy and accelerate the particles, which then radiate. We assume a cylindrical geometry for the emission region of the jet. We treat the self-consistent radiative transfer by taking into account the inhomogeneity in the photon density throughout the region. In this paper, we focus on understanding the effects of varying relevant input parameters on the simulated spectral energy distribution (SED) and spectral variability patterns.

Angular smoothing and spatial diffusion from the Feynman path integral representation of radiative transfer

The propagation kernel for time dependent radiative transfer is represented by a Feynman path integral (FPI). The FPI is approximately evaluated in the spatial-Fourier domain. Spatial diffusion is exhibited in the kernel when the approximations lead to a Gaussian dependence on the Fourier domain wave vector. The approximations provide an explicit expression for the diffusion matrix. They also provide an asymptotic criterion for the self-consistency of the diffusion approximation. The criterion is weakly violated in the limit of large numbers of scattering lengths. Additional expansion of higher-order terms may resolve whether this weak violation is significant.

Time-dependent multi-zone radiation transfer modeling of fast blazar variability

AbstractWe present the first applications of a new time-dependent multi-zone jet radiation transfer code to the study the multiwavelength emission of the TeV Blazar Mrk 421. The code couples Fokker-Planck and Monte Carlo methods. For the first time all light travel time effects are fully considered as well as proper self-consistent treatment of Compton cooling, which depends on them. The first tests focus on the March 2001 observations of Mrk 421, still one of the best datasets available for phenomenology and X-ray/TeV data coverage. We summarize the results of scenarios of variability induced by injection of relativistic electrons in a blob encountering a shock, and with different combinations with a second component, either co-spatial or independent from the active region.

Mathematical modelling of ground temperature with suction velocity and radiation

This paper investigates the problem of ground temperature under a vegetative cover with time dependent suction and radiative heat transfer. By assuming that the ground radiates heat from its’ surface and that the radiative heat flux takes a differential form which follows that of Tarkhar et al [1], the problem was formulated. By imposing a regular time dependent perturbation, the problem was solved. The numerical results and graphs were computed using the software, Mathematica. Results revealed that individual and simultaneous increase in flow parameters caused decrease in the temperature.

Change of gain characters for short pulses with third order dispersion in a fiber amplifier

The effect of the third order dispersion (TOD) induced into pulses before entering an Yb-doped fiber amplifier on their power gain characters is investigated using a model of which the pulse amplification process is described by nonlinear time-dependent radiation transfer equations. Numerical results show that peak power gain for the pulse with positive TOD is larger than that with zero and without TOD while the improvement of the peak power gain varies as the obtained output pulse energy changes. The improvement also depends on the ratio of dispersion lengths for group velocity dispersion and TOD.

DOUBLE-DETONATION SUB-CHANDRASEKHAR SUPERNOVAE: SYNTHETIC OBSERVABLES FOR MINIMUM HELIUM SHELL MASS MODELS

Abridged. In the double detonation scenario for Type Ia supernovae (SNe Ia) a detonation initiates in a shell of He-rich material accreted from a companion star by a sub-Chandrasekhar-mass White Dwarf (WD). This shell detonation drives a shock front into the carbon-oxygen (C/O) WD that triggers a secondary detonation in the core. The core detonation results in a complete disruption of the WD. Earlier studies concluded that this scenario has difficulties in accounting for the observed properties of SNe Ia since the explosion ejecta are surrounded by the products of explosive He burning in the shell. Recently, it was proposed that detonations might be possible for much less massive He shells than previously assumed. Moreover, it was shown that even detonations of these minimum He shell masses robustly trigger detonations of the C/O core. Here we present time-dependent multi-wavelength radiative transfer calculations for models with minimum He shell mass and derive synthetic observables for both the optical and {\gamma}-ray spectral regions. These differ strongly from those found in earlier simulations of sub-Chandrasekhar-mass explosions in which more massive He shells were considered. Our models predict light curves which cover both the range of brightnesses and the rise and decline times of observed SNe Ia. However, their colours and spectra do not match the observations. In particular, their B-V colours are generally too red. We show that this discrepancy is mainly due to the composition of the burning products of the He shell of our models which contain significant amounts of Ti and Cr. Using a toy model, we also show that the burning products of the He shell depend crucially on its initial composition. This leads us to conclude that good agreement between sub-Chandrasekhar-mass explosions and observed SNe Ia may still be feasible but further study of the shell properties is required.

Computational Laser Spectroscopy in a Biological Tissue

We present a numerical spectroscopic study of visible and infrared laser radiation in a biological tissue. We derive a solution of a general two-dimensional time dependent radiative transfer equation in a tissue-like medium. The used model is suitable for many situations especially when the external source is time-dependent or continuous. We use a control volume-discrete ordinate method associated with an implicit three-level second-order time differencing scheme. We consider a very thin rectangular biological-tissue-like medium submitted to a visible or a near infrared light sources. The RTE is solved for a set of different wavelength source. All sources are assumed to be monochromatic and collimated. The energetic fluence rate is computed at a set of detector points on the boundaries. According to the source type, we investigate either the steady-state or transient response of the medium. The used model is validated in the case of a heterogeneous tissue-like medium using referencing experimental results from the literature. Also, the developed model is used to study changes on transmitted light in a rat-liver tissue-like medium. Optical properties depend on the source wavelength and they are taken from the literature. In particular, light-transmission in the medium is studied for continuous wave and for short pulse.

Application of the RBF meshless method to the solution of the radiative transport equation

In this paper, we introduce a radial basis function collocation method for computing solutions to the time-dependent radiative transfer equation. For these computations, we use finite differences to discretize the time coordinate, a discrete ordinate method to discretize the directional variable, and an expansion in radial basis functions to approximate the spatial dependence of the solution. The main advantages of the RBF method are that it does not require any mesh or grid, achieves spectral accuracy in multi-dimensions for arbitrary node layouts, and it is extremely simple to implement.

MILLIMETER FLARES AND VLBI VISIBILITIES FROM RELATIVISTIC SIMULATIONS OF MAGNETIZED ACCRETION ONTO THE GALACTIC CENTER BLACK HOLE

The recent VLBI observation of the Galactic center black hole candidate Sgr A* at 1.3mm shows source structure on event-horizon scales. This detection enables a direct comparison of the emission region with models of the accretion flow onto the black hole. We present the first results from time-dependent radiative transfer of general relativistic MHD simulation data, and compare simulated synchrotron images at black hole spin a=0.9 with the VLBI measurements. After tuning the accretion rate to match the millimeter flux, we find excellent agreement between predicted and observed visibilities, even when viewed face-on (i < 30 degrees). VLBI measurements on 2000-3000km baselines should constrain the inclination. The data constrain the accretion rate to be (1.0-2.3)x10^-9 M_sun / yr with 99% confidence, consistent with but independent of prior estimates derived from spectroscopic and polarimetric measurements. Finally, we compute light curves, which show that magnetic turbulence can directly produce flaring events with .5 hour rise times, 2-3.5 hour durations and 40-50% flux modulation, in agreement with observations of Sgr A* at millimeter wavelengths.