Radiative Transfer Method Research Articles

Numerical problems in coupling photon momentum (radiation pressure) to gas

Radiation pressure (RP; or photon momentum absorbed by gas) is important in a tremendous range of astrophysical systems. But we show the usual method for assigning absorbed photon momentum to gas in numerical radiation-hydrodynamics simulations (integrating over cell volumes or evaluating at cell centers) can severely under-estimate the RP force in the immediate vicinity around un-resolved (point/discrete) sources (and subsequently under-estimate its effects on bulk gas properties), unless photon mean-free-paths are highly-resolved in the fluid grid. The existence of this error is independent of the numerical radiation transfer (RT) method (even in exact ray-tracing/Monte-Carlo methods), because it depends on how the RT solution is interpolated back onto fluid elements. Brute-force convergence (resolving mean-free paths) is impossible in many cases (especially where UV/ionizing photons are involved). Instead, we show a 'face-integrated' method -- integrating and applying the momentum fluxes at interfaces between fluid elements -- better approximates the correct solution at all resolution levels. The 'fix' is simple and we provide example implementations for ray-tracing, Monte-Carlo, and moments RT methods in both grid and mesh-free fluid schemes. We consider an example of star formation in a molecular cloud with UV/ionizing RP. At state-of-the-art resolution, cell-integrated methods under-estimate the net effects of RP by an order of magnitude, leading (incorrectly) to the conclusion that RP is unimportant, while face-integrated methods predict strong self-regulation of star formation and cloud destruction via RP.

Fast and slow winds from supergiants and luminous blue variables

We predict quantitative mass-loss rates and terminal wind velocities for early-type supergiants and luminous blue variables (LBVs) using a dynamical version of the Monte Carlo radiative transfer method. First, the observed drop in terminal wind velocity around spectral type B1 is confirmed by the Monte Carlo method at the correct effective temperature of about 21 000 K. This drop in wind velocity is much steeper than would be expected from the drop in escape speed for cooler stars. The results may be particularly relevant for slow winds inferred for some high-mass X-ray binaries. Second, the strength of the mass-loss bi-stability jump is found to be significantly greater than previously assumed. This could this make bi-stability braking more efficient in massive star evolution; in addition, a rotationally induced version of the bi-stability mechanism may now be capable of producing the correct density of outflowing disks around B[e] supergiants, although multi-dimensional modelling including the disk velocity structure is still needed. For LBVs we find that the bi-stability jump becomes larger at higher metallicities, but perhaps surprisingly also larger at lower Eddington parameters. This may have consequences for the role of LBVs in the evolution of massive stars at different metallicities and cosmic epochs. Finally, our predicted low wind velocities may be important for explaining the slow outflow speeds of supernova type IIb/IIn progenitors, for which the direct LBV-SN link was first introduced.

Simulating an isolated dwarf galaxy with multichannel feedback and chemical yields from individual stars

In order to better understand the relationship between feedback and galactic chemical evolution, we have developed a new model for stellar feedback at grid resolutions of only a few parsecs in global disk simulations, using the adaptive mesh refinement hydrodynamics code Enzo. For the first time in galaxy scale simulations, we simulate detailed stellar feedback from individual stars including asymptotic giant branch winds, photoelectric heating, Lyman-Werner radiation, ionizing radiation tracked through an adaptive ray-tracing radiative transfer method, and core collapse and Type Ia supernovae. We furthermore follow the star-by-star chemical yields using tracer fields for 15 metal species: C, N, O, Na, Mg, Si, S, Ca, Mn, Fe, Ni, As, Sr, Y, and Ba. We include the yields ejected in massive stellar winds, but greatly reduce the winds' velocities due to computational constraints. We describe these methods in detail in this work and present the first results from 500~Myr of evolution of an isolated dwarf galaxy with properties similar to a Local Group, low-mass dwarf galaxy. We demonstrate that our physics and feedback model is capable of producing a dwarf galaxy whose evolution is consistent with observations in both the Kennicutt-Schmidt relationship and extended Schmidt relationship. Effective feedback drives outflows with a greater metallicity than the ISM, leading to low metal retention fractions consistent with observations. Finally, we demonstrate that these simulations yield valuable information on the variation in mixing behavior of individual metal species within the multi-phase interstellar medium.

An empirical recipe for inelastic hydrogen-atom collisions in non-LTE calculations

Context. Determination of high-precision abundances of late-type stars has been and always will be an important goal of spectroscopic studies, which requires accurate modeling of their stellar spectra with non-local thermodynamic equilibrium (NLTE) radiative transfer methods. This entails using up-to-date atomic data of the elements under study, which are still subject to large uncertainties.Aims. We investigate the role of hydrogen collisions in NLTE spectral line synthesis, and introduce a new general empirical recipe to determine inelastic charge transfer (CT) and bound-bound hydrogen collisional rates. This recipe is based on fitting the energy functional dependence of published quantum collisional rate coefficients of several neutral elements (BeI, Na I, Mg I, Al I, Si Iand Ca I) using simple polynomial equations.Methods. We perform thorough NLTE abundance calculation tests using our method for four different atoms, Na, Mg, Al and Si, for a broad range of stellar parameters. We then compare the results to calculations computed using the published quantum rates for all the corresponding elements. We also compare to results computed using excitation collisional rates via the commonly used Drawin equation for different fudge factors,SMH, applied.Results. We demonstrate that our proposed method is able to reproduce the NLTE abundance corrections performed with the quantum rates for different spectral types and metallicities for representative Na Iand Al Ilines to within ≤0.05 dex and ≤0.03 dex, respectively. For Mg Iand Si Ilines, the method performs better for the cool giants and dwarfs, while larger discrepancies up to 0.2 dex could be obtained for some lines for the subgiants and warm dwarfs. We obtained larger NLTE correction differences between models incorporating Drawin rates relative to the quantum models by up to 0.4 dex. These large discrepancies are potentially due to ignoring either or both CT and ionization collisional processes by hydrogen in our Drawin models.Conclusions. Our general empirical fitting method (EFM) for estimating hydrogen collision rates performs well in its ability to reproduce, within narrow uncertainties, the abundance corrections computed with models incorporating quantum collisional rates. It performs generally best for the cool and warm dwarfs, with slightly larger discrepancies obtained for the giants and subgiants. It could possibly be extended in the future to transitions of the same elements for which quantum calculations do not exist, or, in the absence of published quantum calculations, to other elements as well.

Stellar Radiation Is Critical for Regulating Star Formation and Driving Outflows in Low-mass Dwarf Galaxies

Abstract Effective stellar feedback is used in models of galaxy formation to drive realistic galaxy evolution. Models typically include energy injection from supernovae (SNe) as the dominant form of stellar feedback, often in some form of sub-grid recipe. However, it has been recently suggested that pre-SN feedback (stellar winds or radiation) is necessary in high-resolution simulations of galaxy evolution to properly regulate star formation and properties of the interstellar medium (ISM). Following these processes is computationally challenging, so many prescriptions model this feedback approximately, accounting for the local destruction of dense gas clouds around newly formed stars in lieu of a full radiative transfer calculation. In this Letter we examine high-resolution simulations (1.8 pc) of an isolated dwarf galaxy with detailed stellar feedback tracked on a star-by-star basis. By following stellar ionizing radiation with an adaptive ray-tracing radiative transfer method, we test its importance in regulating star formation and driving outflows in this galaxy. We find that including ionizing radiation reduces the star formation rate (SFR) by over a factor of 5, and is necessary to produce the ISM conditions needed for SNe to drive significant outflows. We find that a localized approximation for radiation feedback is sufficient to regulate the SFR on short timescales, but does not allow significant outflows. Short- and long-range radiation effects are both important in driving the evolution of our low-metallicity, low-mass dwarf galaxy. Generalizing these results to more massive galaxies would be a valuable avenue of future research.

Surface waves in protoplanetary disks induced by outbursts: Concentric rings in scattered light

Context. Vertically hydrostatic protoplanetary disk models are based on the assumption that the main heating source, stellar irradiation, does not vary much with time. However, it is known that accreting young stars are variable sources of radiation. This is particularly evident for outbursting sources such as EX Lupi and FU Orionis stars. Aims. We investigate how such outbursts affect the vertical structure of the outer regions of the protoplanetary disk, in particular their appearance in scattered light at optical and near-infrared wavelengths. Methods. We employ the 3D FARGOCA radiation-hydrodynamics code, in polar coordinates, to compute the time-dependent behavior of the axisymmetric disk structure. The temperature is computed self-consistently and time-dependently from the irradiation flux using a two-stage radiative transfer method: first the direct illumination is computed; then the diffuse radiation is treated with the flux-limited diffusion method. The outbursting inner disk region is not included explicitly. Instead, its luminosity is added to the stellar luminosity and is thus included in the irradiation of the outer disk regions. For time snapshots of interest we insert the density structure into the RADMC-3D radiative transfer code and compute the appearance of the disk at optical/near-infrared wavelengths, where we observe stellar light that is scattered off the surface of the disk. Results. We find that, depending on the amplitude of the outbursts, the vertical structure of the disk can become highly dynamic, featuring circular surface waves of considerable amplitude. These “hills” and “valleys” on the disk’s surface show up in the scattered light images as bright and dark concentric rings. Initially these rings are small and act as standing waves, but they subsequently lead to outward propagating waves, like the waves produced by a stone thrown into a pond. These waves continue long after the actual outburst has died out. Conclusions. Single, periodic, or quasiperiodic outbursts of the innermost regions of protoplanetary disks will necessarily lead to wavy structures on the surface of these disks at all radii. We propose that some of the multi-ringed structures seen in optical/infrared images of several protoplanetary disks may have their origin in outbursts that occurred decades or centuries ago. However, the multiple rings seen at (sub-)millimeter wavelengths in HL Tau and several other disks are not expected to be related to such waves.

Multi-grid Monte Carlo method for radiative transfer in multi-dimensional graded index media with diffuse-specular-gray boundaries

The Monte Carlo method, based on the radiative energy absorption distribution, is performed to simulate the radiative transfer in general multi-dimensional absorbing-emitting-scattering graded index media with variable refractive index and diffuse-specular-gray boundaries. A cell-based ray tracing approach is used to track the curve paths of energy particles throughout the medium. In order to accurate curved ray tracing, a multi-grid approach is used in which the domain of interest is discretized into a secondary finer grid for ray tracing, which is embedded into a primary coarser grid to simulate the radiation field. The multi-dimensional media with complex geometries are simulated by blocked-off approach. The accuracy of the present method is verified by comparing its result with benchmarks. Some examples are presented to show the performance of the present method for solving the radiative transfer in multi-dimensional graded index media with irregular geometries and diffuse-specular-gray boundaries.

Analysis of four-flux parameters of TiO2 films commonly used in DSSCs

The four-flux radiative transfer method describes light propagation through a light scattering media for a low computational cost by using experimentally extracted input parameters. The method can be used to aid the optimal design of the light harvesting layers in solar technologies such as the porous TiO2 films used dye-sensitized solar cells (DSSCs). The results of these light propagation simulations are sensitive to the input parameters, particularly the simulation results of systems involving mesoporous scattering centres. Here we present the four-flux parameters to simulate common mesoporous TiO2 film architectures. The parameters are highly interdependent, and there is no linear relationship between the independent scattering centres used in the films and experimentally extracted parameters. The results suggest that theoretically evaluated four flux parameters cannot be used in place of experimentally extracted parameters for four-flux simulation of DSSCs.

The Failure of Monte Carlo Radiative Transfer at Medium to High Optical Depths

Abstract Computer simulations of photon transport through an absorbing and/or scattering medium form an important research tool in astrophysics. Nearly all software codes performing such simulations for three-dimensional geometries employ the Monte Carlo (MC) radiative transfer (RT) method, including various forms of biasing to accelerate the calculations. Because of the probabilistic nature of the MC technique, the outputs are inherently noisy, but it is often assumed that the average values provide the physically correct result. We show that this assumption is not always justified. Specifically, we study the intensity of radiation penetrating an infinite, uniform slab of material that absorbs and scatters the radiation with equal probability. The basic MCRT method, without any biasing mechanisms, starts to break down for transverse optical depths τ ≳ 20 because so few of the simulated photon packets reach the other side of the slab. When including biasing techniques such as absorption/scattering splitting and path length stretching, the simulated photon packets do reach the other side of the slab but the biased weights do not necessarily add up to the correct solution. While the noise levels seem to be acceptable, the average values sometimes severely underestimate the correct solution. Detecting these anomalies requires the judicious application of statistical tests, similar to those used in the field of nuclear particle transport, possibly in combination with convergence tests employing consecutively larger numbers of photon packets. In any case, for transverse optical depths τ ≳ 75 the MC methods used in our study fail to solve the one-dimensional slab problem, implying the need for approximations such as a modified random walk.

Three-dimensional heuristic radiation transfer method based on enumeration using the directions grid

Abstract In this study, we present an effective three-dimensional dust radiation transfer method based on enumeration using the directions grid and the law of total probability. The method is suitable for several simple problems. These problems are solved with different grades of direction sampling and by considering various photon scattering numbers. The polarization is calculated together with the photometry. The results obtained with the proposed method were compared with those produced by Monte-Carlo modeling. Both methods use the same physical and geometrical models. The results were obtained with one testing program, which allowed detailed comparisons of the results in terms of their quality and time rate. The analysis of multiple scattering was conducted carefully because considering all of the scattering results may be similar to the results obtained by single scattering modeling.

Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere.

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles

A discontinuous Galerkin finite element method (DGFEM) with unstructured meshes is presented to solve the radiative transfer equation (RTE) in two-dimensional media with inner obstacles. The computation domain is discretized into a tessellation of unstructured elements and the elements are assumed to be discontinuous on the inner-element boundaries. The shape functions are constructed on each element and the continuity of the computation domain is maintained by modeling an up-winding numerical flux across the inner boundaries, which makes the DGFEM suitable and numerical stable for radiative transfer problems involved with strong non-uniformity and discontinuity induced by ray effects. The DGFEM discretization for RTE is presented and the accuracy of DGFEM is verified. Radiative transfer problems in square and irregular media with inner obstacles are investigated, the influence of medium parameters and the obstacle shielding effects are discussed.

Efficient radiative transfer techniques in hydrodynamic simulations

Radiative transfer is an important component of hydrodynamic simulations as it determines the thermal properties of a physical system. It is especially important in cases where heating and cooling regulate significant processes, such as in the collapse of molecular clouds, the development of gravitational instabilities in protostellar discs, disc-planet interactions, and planet migration. We compare two approximate radiative transfer methods which indirectly estimate optical depths within hydrodynamic simulations using two different metrics: (i) the gravitational potential and density of the gas (Stamatellos et al.), and (ii) the pressure scale-height (Lombardi et al.). We find that both methods are accurate for spherical configurations e.g. in collapsing molecular clouds and within clumps that form in protostellar discs. However, the pressure scale-height approach is more accurate in protostellar discs (low and high-mass discs, discs with spiral features, discs with embedded planets). We also investigate the $\beta$-cooling approximation which is commonly used when simulating protostellar discs, and in which the cooling time is proportional to the orbital period of the gas. We demonstrate that the use of a constant $\beta$ cannot capture the wide range of spatial and temporal variations of cooling in protostellar discs, which may affect the development of gravitational instabilities, planet migration, planet mass growth, and the orbital properties of planets.

Modeling scattering and intrinsic attenuation of crosswell seismic data in the Michigan Basin

Scattering of seismic waves may cause a significant amount of apparent attenuation, particularly in regions having strong heterogeneity. We find that the radiative transfer method can be used to model intrinsic and scattering attenuation of crosswell seismic data. To solve the radiative transfer equation, we assume that the medium can be described by a stationary random field and then we use a particle-based method in which millions of particles are released from a source and scatter through the random medium according to the Born scattering coefficients. Intrinsic attenuation is modeled by assigning probabilities that the particles will die off on the path between scattering events. We apply the seismic particle method to crosswell seismic data acquired on a carbonate reef in the Michigan Basin, and we use a grid search to estimate a set of model parameters that fit the data. Based on well logs, we fixed the velocity heterogeneity strength at 3% and estimated the medium velocity correlation length to be between [Formula: see text] and [Formula: see text]. Our results also indicate that scattering attenuation is one order of magnitude larger than intrinsic attenuation, a result that we interpret as a consequence of the highly heterogeneous nature of the rock comprising the reservoir. Contrary to other attenuation estimation methods, such as the centroid method, in which scattering attenuation is neglected by invoking that the reservoir is homogeneous, our results indicate that accounting for scattering attenuation is of paramount importance in attenuation estimation using crosswell seismic data.

Understanding the temporal dimension of the red-edge spectral region for forest decline detection using high-resolution hyperspectral and Sentinel-2a imagery

The operational monitoring of forest decline requires the development of remote sensing methods that are sensitive to the spatiotemporal variations of pigment degradation and canopy defoliation. In this context, the red-edge spectral region (RESR) was proposed in the past due to its combined sensitivity to chlorophyll content and leaf area variation. In this study, the temporal dimension of the RESR was evaluated as a function of forest decline using a radiative transfer method with the PROSPECT and 3D FLIGHT models. These models were used to generate synthetic pine stands simulating decline and recovery processes over time and explore the temporal rate of change of the red-edge chlorophyll index (CI) as compared to the trajectories obtained for the structure-related Normalized Difference Vegetation Index (NDVI). The temporal trend method proposed here consisted of using synthetic spectra to calculate the theoretical boundaries of the subspace for healthy and declining pine trees in the temporal domain, defined by CItime=n/CItime=n+1 vs. NDVItime=n/NDVItime=n+1. Within these boundaries, trees undergoing decline and recovery processes showed different trajectories through this subspace. The method was then validated using three high-resolution airborne hyperspectral images acquired at 40 cm resolution and 260 spectral bands of 6.5 nm full-width half-maximum (FWHM) over a forest with widespread tree decline, along with field-based monitoring of chlorosis and defoliation (i.e., ‘decline’ status) in 663 trees between the years 2015 and 2016. The temporal rate of change of chlorophyll vs. structural indices, based on reflectance spectra extracted from the hyperspectral images, was different for trees undergoing decline, and aligned towards the decline baseline established using the radiative transfer models. By contrast, healthy trees over time aligned towards the theoretically obtained healthy baseline. The applicability of this temporal trend method to the red-edge bands of the MultiSpectral Imager (MSI) instrument on board Sentinel-2a for operational forest status monitoring was also explored by comparing the temporal rate of change of the Sentinel-2-derived CI over areas with declining and healthy trees. Results demonstrated that the Sentinel-2a red-edge region was sensitive to the temporal dimension of forest condition, as the relationships obtained for pixels in healthy condition deviated from those of pixels undergoing decline.

A New Numerical Scheme for Cosmic-Ray Transport

Abstract Numerical solutions of the cosmic-ray (CR) magnetohydrodynamic equations are dogged by a powerful numerical instability, which arises from the constraint that CRs can only stream down their gradient. The standard cure is to regularize by adding artificial diffusion. Besides introducing ad hoc smoothing, this has a significant negative impact on either computational cost or complexity and parallel scalings. We describe a new numerical algorithm for CR transport, with close parallels to two-moment methods for radiative transfer under the reduced speed of light approximation. It stably and robustly handles CR streaming without any artificial diffusion. It allows for both isotropic and field-aligned CR streaming and diffusion, with arbitrary streaming and diffusion coefficients. CR transport is handled explicitly, while source terms are handled implicitly. The overall time step scales linearly with resolution (even when computing CR diffusion) and has a perfect parallel scaling. It is given by the standard Courant condition with respect to a constant maximum velocity over the entire simulation domain. The computational cost is comparable to that of solving the ideal MHD equation. We demonstrate the accuracy and stability of this new scheme with a wide variety of tests, including anisotropic streaming and diffusion tests, CR-modified shocks, CR-driven blast waves, and CR transport in multiphase media. The new algorithm opens doors to much more ambitious and hitherto intractable calculations of CR physics in galaxies and galaxy clusters. It can also be applied to other physical processes with similar mathematical structure, such as saturated, anisotropic heat conduction.

Spatially Resolved Sub-millimeter Continuum Imaging of Neptune with ALMA

Abstract This paper reports the result of spatially resolved 646 GHz sub-millimeter imaging observation of Neptune obtained by the Atacama Large Millimeter and sub-millimeter Array. The observation was performed in 2012 August as the flux calibration and synthesized beam size were small enough to resolve Neptune’s disk at this time. This analysis aims to constrain the vertical structure of deep and upper-tropospheric South polar hot spot detected previously with mid-IR, millimeter, and centimeter wavelength. The probed atmospheric pressure region estimated by the radiative-transfer method was between 1.0 and 0.6 bar for the nadir and South pole views, respectively. The South polar hot spot was not detected clearly with an uncertainty of 2.1 K. The apparent discontinuity of tropospheric and stratospheric hot spot may be caused by the vertical wind shear of South polar zonal jet.

Seurat: SPH scheme extended with ultraviolet line radiative transfer

We present a novel Lyman alpha (Ly$\alpha$) radiative transfer code, SEURAT, where line scatterings are solved adaptively with the resolution of the smoothed particle hydrodynamics (SPH). The radiative transfer method implemented in SEURAT is based on a Monte Carlo algorithm in which the scattering and absorption by dust are also incorporated. We perform standard test calculations to verify the validity of the code; (i) emergent spectra from a static uniform sphere, (ii) emergent spectra from an expanding uniform sphere, and (iii) escape fraction from a dusty slab. Thereby we demonstrate that our code solves the Ly$\alpha$ radiative transfer with sufficient accuracy. We emphasise that SEURAT can treat the transfer of Ly$\alpha$ photons even in highly complex systems that have significantly inhomogeneous density fields. The high adaptivity of SEURAT is desirable to solve the propagation of Ly$\alpha$ photons in the interstellar medium of young star-forming galaxies like Ly$\alpha$ emitters (LAEs). Thus, SEURAT provides a powerful tool to model the emergent spectra of Ly$\alpha$ emission, which can be compared to the observations of LAEs.

Design and optimization of nanoparticle-pigmented solar selective absorber coatings for high-temperature concentrating solar thermal systems

We present a systematic approach for the design and optimization of nanoparticle-pigmented solar selective absorbers for operation at 750 °C. Using the scattering and absorption cross-sections calculated by Lorenz-Mie scattering theory as input, we employ a four-flux radiative transfer method to investigate the solar selectivity mechanism and optimize the optical-to-thermal conversion efficiency (ηtherm) as a function of the metallic nanoparticle material, the nanoparticle diameter, the volume fraction, and the coating thickness. Among the nanoparticle material candidates in this study, C54-TiSi2 is the best option with an optimized ηtherm = 87.0% for a solar concentration ratio of C = 100 and ηtherm = 94.4% for C = 1000 at 750 °C. NiSi is also a promising candidate comparable to TiSi2 in thermal efficiency. Experimentally, an un-optimized 200 nm-diameter TiSi2 nanoparticle-silicone solar selective coating has already achieved ηtherm = 89.8% for C = 1000 at 750 °C. This performance is consistent with the theoretical model and close to the thermal efficiency of the commercial Pyromark 2500 coatings (90.1%). We also demonstrate that Ni/NiSi core-shell structures embedded in the SiO1.5 matrix is thermally stable at 750 °C for 1000 h in air. These results indicate that silicide cermet coatings are promising to achieve high optical performance and high temperature thermal stability simultaneously.

Sprai: coupling of radiative feedback and primordial chemistry in moving mesh hydrodynamics

In this paper we introduce a new radiative transfer code SPRAI (Simplex Photon Radiation in the Arepo Implementation) based on the SimpleX radiation transfer method. This method, originally used only for post-processing, is now directly integrated into the Arepo code and takes advantage of its adaptive unstructured mesh. Radiated photons are transferred from the sources through the series of Voronoi gas cells within a specific solid angle. From the photon attenuation we derive corresponding photon fluxes and ionization rates and feed them to a primordial chemistry module. This gives us a self-consistent method for studying dynamical and chemical processes caused by ionizing sources in primordial gas. Since the computational cost of the SimpleX method does not scale directly with the number of sources, it is convenient for studying systems such as primordial star-forming halos that may form multiple ionizing sources.