Monte Carlo radiative transfer
Abstract
The theory and numerical modelling of radiation processes and radiative transfer play a key role in astrophysics: they provide the link between the physical properties of an object and the radiation it emits. In the modern era of increasingly highquality observational data and sophisticated physical theories, development and exploitation of a variety of approaches to the modelling of radiative transfer is needed. In this article, we focus on one remarkably versatile approach: Monte Carlo radiative transfer (MCRT). We describe the principles behind this approach, and highlight the relative ease with which they can (and have) been implemented for application to a range of astrophysical problems. All MCRT methods have in common a need to consider the adverse consequences of Monte Carlo noise in simulation results. We overview a range of methods used to suppress this noise and comment on their relative merits for a variety of applications. We conclude with a brief review of specific applications for which MCRT methods are currently popular and comment on the prospects for future developments.
Keywords
Monte Carlo methods Radiative transfer Methods: numerical Radiation: dynamicsAbbreviations
 CDF
Cumulative distribution function
 CMF
Comoving frame
 DDMC
Discrete diffusion Monte Carlo
 IMC
Implicit Monte Carlo
 IMD
Implicit Monte Carlo diffusion
 LF
Laboratory frame
 LHS
Left hand side
 LTE
Local thermodynamic equilibrium
 MC
Monte Carlo
 MCRT
Monte Carlo radiative transfer
 MRW
Modified random walk
 NLTE
Local thermodynamic nonequilibrium
Probability density function
 RE
Radiative equilibrium
 RH
Radiation hydrodynamics
 RHS
Right hand side
 RNG
Random number generator
 RT
Radiative transfer
 RW
Random walk
 SIMC
Symbolic implicit Monte Carlo
 SN
Supernova
 SN Ia
Type Ia supernova
 SPH
Smoothed particle hydrodynamics
 TE
Thermal equilibrium
 TRT
Thermal radiative transfer
1 Introduction
1.1 The role of radiative transfer in astrophysics
Much of astrophysics is at a disadvantage compared to other fields of physics. While normally theories can be tested and phenomena studied by performing repeatable experiments in the controlled environment of a lab, astrophysics generally lacks this luxury. Instead, researchers have to mainly rely on observations of very distant objects and phenomena over which they have no control. The vast majority of information about astrophysical systems is gathered by observing their emitted radiation over the electromagnetic spectrum. Other messengers, such as neutrinos, charged particles and recently gravitational waves, are also used but typically restricted to specific astrophysical phenomena.
Given that the observation and interpretation of electromagnetic radiation is therefore the cornerstone of astrophysical research, a firm understanding of how the observed signal forms and propagates is crucial. The framework of radiative transfer (RT) builds the theoretical foundation for this problem. It combines concepts from kinetic theory, atomic physics, special relativity and quantum mechanics, and provides a formalism to describe how the radiation field is shaped by the interactions with the ambient medium.
Finding an analytic solution for RT problems is usually very challenging, a process that typically requires approximations and quickly reaches its limits as the complexity of the problem increases. Thus, numerical methods are normally employed instead. In such cases, one considers a discretized version of the transfer equation, e.g. by replacing differentials with finite differences, and uses sophisticated solution schemes to minimize the inevitably introduced numerical errors. While being an established approach, this often leads to very complex numerical schemes and faces some particular challenges when scattering interactions have to be included or when problems without internal symmetries require a fully multidimensional treatment.
MC methods offer a completely different approach to RT problems. Instead of discretizing the RT equations, the underlying RT process is “simulated” by introducing a large number of “test particles” (later referred to as “packets” in this article). These test particles behave in a manner similar to their physical counterparts, namely real photons. In particular, particles move, can scatter or be absorbed during a MC calculation. In the simulations, decisions about the propagation behaviour of a particular test particle, e.g. when, where and how it interacts, are taken stochastically. Seemingly, this leads to a random propagation behaviour of the individual particles. However, as an ensemble, the particle population can provide an accurate representation of the transfer process and the evolution of the radiation field, provided that the sample size is chosen sufficiently large.
Given its design, the MC approach to RT offers a number of compelling benefits. Due to its inspiration from the microphysical interpretation of the RT process, devising a MC RT scheme is very intuitive and conceptually simple. This often leads to comparably simple computer programs and involves moderate coding efforts: basic MCRT routines to solve simple RT problems can be coded in only a few lines that combine a random number generator with a number of basic loops (we provide a number of simple examples of how this can be done as part of our discussions later in this article). From a physical standpoint, a significant advantage of MC methods is the ease with which scattering processes are incorporated, a task which proves much more challenging for traditional, deterministic solution approaches to RT problems. In addition, MCRT calculations can be generalised with little effort from problems with internal symmetries to problems with arbitrarily complex geometrical characteristics. This feature makes MCRT techniques often the preferred choice for multidimensional RT calculations. Finally, the MCRT treatment is often referred to as “embarrassingly parallel” to describe its ideal suitability for modern high performance computing in which the workload is distributed on a multitude of processing units. Just as the photons they represent, the individual MC particles are completely decoupled and propagate independently of each other. Thus, each processing unit can simply treat a subset of the entire particle population without the need for much communication.^{1}
Of course, the MC approach is not without its downsides. The most severe disadvantage is a direct consequence of the probabilistic nature of MC techniques: inevitably, any physical quantity extracted from MC calculations will be subject to stochastic fluctuations. This MC noise can be decreased by increasing the number of particles, which naturally requires more computational resources. Consequently, MC calculations are often computationally expensive. These costs further increase if the optical thickness of the simulated environment is high. Since the propagation of each particle has to be followed explicitly, the efficiency of conventional MCRT schemes suffers greatly if the number of interactions the particles experience increases. Consequently, MCRT approaches are typically illsuited for RT problems in the diffusion regime. Furthermore, as pointed out by Camps and Baes (2018), care has to be taken when interpreting results of MCRT simulations applied to environments with intermediate to high optical depth. The need to explicitly follow the propagation of the individual MC particles is the cause for yet another drawback of MCRT approaches. In deterministic solution strategies to RT, implicit timestepping is often used to improve numerical stability in situations with short characteristic time scales. By design, conventional MCRT schemes require following the propagation of the individual particles in a timeexplicit fashion. It is thus very challenging to devise truly implicit MCRT approaches to overcome numerical stability problems. In the course of this review, we will highlight a variety of different techniques which have been devised to address and alleviate each of these drawbacks.
1.2 Scope of this review
MC techniques have become a popular and widelyused approach to address RT problems in many disciplines of physical and engineering research. Covering all the different aspects and applications of MCRT is beyond the scope of this article and we refer readers to existing surveys of the respective fields. Among these, we highlight the recent overview of MCRT in atmospheric physics by Mayer (2009), the seminal report by Carter and Cashwell (1975) and the book by Dupree and Fraley (2002), which both discuss MCRT techniques to solving neutron transport problems, and to the article by Rogers (2006), who reviews MCRT methods in the field of medical physics. In this article, we aim to provide an introduction to MC techniques used in astrophysics to mainly address photon transport problems. While attempting to provide a general and comprehensive overview, we take the liberty to put some emphasis on specific techniques used in our own field of research, namely RT in fast outflows, i.e. supernova (SN) ejecta, accretion disc and stellar winds. We feel that this approach is appropriate given that dedicated overviews of MCRT methods for specific fields of astrophysical research already exist. In particular, we refer the reader to the reviews by Whitney (2011) and Steinacker et al. (2013) on MCRT for astrophysical dust RT problems.
1.3 Structure of this review
We have structured this review as follows: in Sect. 2 we briefly review some fundamentals of radiative transfer theory that are relevant for our presentation. We begin the actual discussion of MCRT methods with a brief look at their history and review of their astrophysical applications in Sect. 3, and by introducing the basic concepts of a random number generator and random sampling in Sect. 4. In the following part, Sect. 5, the basic discretization into MC quanta or packets will be introduced before their propagation procedure is explained in Sect. 6. In Sect. 7, we discuss how emissivity by thermal and/or fluorescent processes can be incorporated in MCRT simulations.
Having introduced the basic MCRT principles, the complications arising in moving media, in particular the need to distinguish reference frames, are discussed in Sect. 8. In Sect. 9 we detail various techniques to reconstruct important radiation field quantities from the ensemble of MC packet trajectories and interaction histories. Here, particular emphasis is put on methods that reduce the inherent stochastic fluctuations in the reconstructed quantities, such as biasing and volumebased estimators. In Sect. 10 advanced MC techniques, such as Implicit Monte Carlo (IMC) and Discrete Diffusion Monte Carlo (DDMC), are described which can be used to improve the numerical stability of MCRT calculations and their efficiency in optically thick environments. We conclude this review by touching upon the challenge of coupling MCRT to hydrodynamical calculations in Sect. 11 and by presenting a handson example of applying MCRT to SN ejecta in Sect. 12.
2 Radiative transfer background
Before turning to the main focus of this review, a brief overview of the fundamentals of RT is in order to introduce the necessary nomenclature and to define the basic physical concepts underlying MCRT calculations. We assume the reader is already familiar with the principles of RT and so will not present a complete derivation. More rigorous presentation of these principles are available in the literature, for example in the books by Chandrasekhar (1960), Mihalas (1978), Rybicki and Lightman (1979), Mihalas and Mihalas (1984) and Hubeny and Mihalas (2014).
Scatterings can be incorporated into this description by formally splitting the scattering process into an absorption which is immediately followed by an emission. It should be noted, however, that the RT problem is often significantly complicated by the presence of scattering interactions since these processes redistribute radiation in both frequency and direction and introduce a nonlocal coupling to the ambient material.
3 Historical sketch of the Monte Carlo method
When Nicholas Metropolis suggested a name for the statistical method just invented to study neutron transport through fissionable material (Metropolis 1987), he clearly drew inspiration from the game of chance which is always played at the heart of MC calculations. From a historical perspective, GeorgesLouis Leclerc, Comte de Buffon, is commonly credited as having devised the first MC experiment (cf. House and Avery 1968; Dupree and Fraley 2002; Kalos and Whitlock 2008). He considered a plane with a superimposed grid of parallel lines and was interested in the probability that a needle which is tossed onto the plane intersects one of the lines (Buffon 1777). It was later suggested, by Laplace, that such a scenario may be used to experimentally determine the value of \(\pi \) (Laplace 1812). In 1873, the astronomer Asaph Hall reports in a short note to the Messenger of Mathematics the successful execution of this experiment, carried out in 1864 by his friend Captain O. C. Fox (Hall 1873). A detailed description of what is known today as “Buffon’s needle problem” is for example provided by Dupree and Fraley (2002) or Kalos and Whitlock (2008).
Notwithstanding these early rudimentary applications, the MC method in its modern form to solve transport problems has been established and shaped in the 1940s, mostly by Stanisław Ulam and John von Neumann (see e.g. Metropolis 1987). Recognising the immense potential and utility of the first largescale electronic computers, which became operational at the time, they harnessed the mathematical concept of “statistical sampling” to solve the neutron transport problems in fissionable material, thus launching the MC method.^{4}
With the growing availability of computational resources, which accompanied the rapid development of computers, MC methods became increasingly popular and their application spread across many different scientific disciplines. In the late 1960s, MC calculations finally entered the astrophysics stage, for example with the works by Auer (1968), Avery and House (1968) and Magnan (1968, 1970). House and Avery (1968) review the status of these early MCbased RT investigations. In the time since, MC methods have become established, successful and reliable tools for the study of a variety of astrophysical RT phenomena. These range all the way from planetary atmospheres (e.g. Lee et al. 2017) to cosmological simulations of reionization (e.g. Ciardi et al. 2001; Baek et al. 2009; Maselli et al. 2009; Graziani et al. 2013). The wide range of applications indicates the broad utility of MC methods for astrophysical applications. Many of these fields have in common needs that involve a sophisticated treatment of scattering, complex (i.e. nonspherical) geometries and/or complicated opacities. For example, many astrophysical MCRT studies involve stellar winds (e.g. Lucy 1983, 2007, 2010, 2012a, b, 2015; Abbott and Lucy 1985; Lucy and Perinotto 1987; Hillier 1991; Lucy and Abbott 1993; Schmutz 1997; Vink et al. 1999, 2000, 2011; Harries 2000; Sim 2004; Watanabe et al. 2006; Müller and Vink 2008, 2014; Muijres et al. 2012a, b; Šurlan et al. 2012, 2013; Noebauer and Sim 2015; Vink 2018), mass outflows from disks (e.g. Knigge et al. 1995; Knigge and Drew 1997; Long and Knigge 2002; Sim 2005; Sim et al. 2005, 2008, 2010, 2012; Noebauer et al. 2010; Odaka et al. 2011; Higginbottom et al. 2013; Kusterer et al. 2014; Hagino et al. 2015; Matthews et al. 2015, 2016, 2017; Tomaru et al. 2018), or supernovae (e.g. Lucy 1987, 1999b, 2005; Janka and Hillebrandt 1989; Mazzali and Lucy 1993; Mazzali 2000; Stehle et al. 2005; Kasen et al. 2006; Sim 2007; Kromer and Sim 2009; Jerkstrand et al. 2011, 2012, 2015; Abdikamalov et al. 2012; Wollaeger et al. 2013; Kerzendorf and Sim 2014; Wollaeger and van Rossum 2014; Bulla et al. 2015; Fransson and Jerkstrand 2015; Roth and Kasen 2015; Botyánszki et al. 2018; Ergon et al. 2018; Sand et al. 2018). In these environments a treatment of multiply overlapping spectral lines in highvelocity gradient flows are crucial. Others depend on accurate simulations of scattering, be it for highenergy processes (e.g. Pozdnyakov et al. 1983; Stern et al. 1995; Molnar and Birkinshaw 1999; Cullen 2001; Yao et al. 2005; Dolence et al. 2009; Ghosh et al. 2009, 2010; Schnittman and Krolik 2010; Tamborra et al. 2018) or from dustrich structures (e.g. Witt 1977; YusefZadeh et al. 1984; Dullemond and Turolla 2000; Bjorkman and Wood 2001; Gordon et al. 2001; Misselt et al. 2001; Juvela and Padoan 2003; Niccolini et al. 2003; Jonsson 2006; Pinte et al. 2006, 2009; Bianchi 2008; Jonsson et al. 2010; Baes et al. 2011; Robitaille 2011; Whitney 2011; Lunttila and Juvela 2012; Camps et al. 2013; Camps and Baes 2015; Gordon et al. 2017; Verstocken et al. 2017). Many of the applications primarily aim to calculate synthetic observables but MCRT methods are also used to determine physical and/or dynamical conditions in complex multidimensional geometries, such as star forming environments, disclike structures, nebulae or circumstellar material configurations (e.g. Wood et al. 1996; Och et al. 1998; Bjorkman and Wood 2001; Ercolano et al. 2003, 2005, 2008; Kurosawa et al. 2004; Carciofi and Bjorkman 2006, 2008; Altay et al. 2008; Pinte et al. 2009; Harries 2011, 2015; Haworth and Harries 2012; Hubber et al. 2016; Lomax and Whitworth 2016; Harries et al. 2017). MCRT schemes have also found use in astrophysical problems that require a general relativistic treatment of radiative transfer processes (e.g. Zink 2008; Dolence et al. 2009; Ryan et al. 2015).
4 Monte Carlo basics
At the heart of MCRT techniques lies a large sequence of decisions about the fate of the simulated quanta. These decisions reflect the underlying physical processes and, as an ensemble, provide a representative description of the transport process. On an individual level, this is realised by selecting from the pool of possible outcomes based on a set of probabilities that encode the underlying physics. This procedure is typically referred to as “random sampling” and will be discussed below.
4.1 Random number generation
4.2 Random sampling
With the help of RNGs, random numbers^{7} can be rapidly produced on the computer and used for sampling physical processes by mapping them onto the target probability distribution. To illustrate the different sampling schemes, we first introduce some basic concepts from statistics. For the sake of brevity, we again refrain from a rigorous mathematical presentation and instead refer the interested reader to the standard literature on the topic, e.g. Kalos and Whitlock (2008).
4.2.1 Sampling from an inverse transformation
4.2.2 Alternative sampling techniques
5 Monte Carlo quanta
Unlike traditional approaches to RT problems, MCRT calculations do not attempt to solve the RT equation directly. Instead, a simulation of the RT process is performed. Specifically, the radiation is discretized so that it may be represented by a large number of MC quanta. During the initialization of such a simulation, each quantum is assigned a position, an initial propagation direction, an energy and frequency, if desired, a polarization vector, and some measure of importance or weight. This last property essentially determines the contribution of the individual quanta to the final results. After the discretization and initialization, the quanta are propagated through the computational domain to simulate the RT process. In the following sections, we highlight two discretization paradigms, namely the photon packet and the energy packet scheme. These derive from different interpretations of what the quanta represent and provide different prescriptions for the choice and treatment of their weights. We then discuss packet initialization. The process of propagating packets during the simulation is described in Sect. 6.
5.1 Discretization into photon packets
Historically, MCRT applications drew inspiration from nature’s inherent discretization of radiation and thus interpreted the fundamental MC quanta as photons. Indeed, in many early MCRT studies performed in astrophysics, such as Auer (1968), Avery and House (1968) and Caroff et al. (1972), the quanta are simply referred to as “photons”. Although the number of MC photons that are introduced and considered is usually large in a statistical sense, it is completely insignificant compared to the actual number of real photons constituting the physical radiation field. Thus, it is inherent to this discretization scheme that the MC photons, or machine photons as they are sometimes called (cf. House and Avery 1968), actually represent a large number of physical photons instead of individual ones. As a consequence, the MC quanta are typically referred to as photon packets or simply packets.
From this discretization perspective, packet weights can be interpreted as encoding that the individual MC packets represent many physical photons. However, the weights are practically never assigned uniformly or held constant during the simulation in MCRT schemes that rely on the photon packet discretization approach. These manipulations of packet weights lead to an often dramatic reduction in variance (i.e. increase of statistics and reduction of MC noise) and belong to the more generic class of biasing techniques (see Sect. 9.4). Considering MCRT applications in astrophysics, the majority relies on the photon packet discretization scheme with nonuniform and variable packet weights. Prominent examples certainly include the many MCRT simulations performed in dust RT problems (see, e.g. reviews by Whitney 2011; Steinacker et al. 2013).
5.2 Energy packets and indivisibility of packets
The energy packet discretization approach has been mainly developed and shaped by L. Lucy. The basic interpretation was already given by Abbott and Lucy (1985), but it was only after extending the approach and applying it to radiative equilibrium (RE) calculations (Lucy 1999a), that its full potential and benefits were explored. The scheme was further generalized to include the possibility of nonresonant interactions and of realising statistical equilibrium (Lucy 1999b, 2002, 2003, see Sect. 7 for further details).
Compared to the photon packet scheme introduced above, the energy packet approach rests on a different interpretation of what MC quanta fundamentally represent: packets are now primarily thought of as parcels of radiant energy and the packet energy also acts as its weight. Again, these parcels of radiant energy represent many physical photons. At this point, the difference between the photon and energy packet schemes seems very subtle, almost semantic. However, the distinctiveness of this discretization scheme becomes apparent once the treatment of packet weights is included into the consideration.
The primary attraction of viewing the quanta as packets of radiation energy, rather than bundles of photons, is the ease (and accuracy) which with energy flows can be tracked during a simulation. For example, in RE problems, the combination of an energy packet discretization and an indivisible packet algorithm allow strict energy conservation to be imposed (Lucy 1999b). While all other packet properties, in particular its frequency, can change during the simulation, the packet energy, i.e. its weight, is strictly held constant after the initial assignment (i.e. the packets are indivisible, and also indestructible, excepting that they can exit through the boundaries of the computational domain). The indivisibility property can readily be applied to all interactions, even those that on first sight seem to require the splitting of packets or adjustment of weights. Instead of splitting, such events are handled by probabilistically sampling the different outcome channels (see the downbranching scheme by Lucy 1999b or the macro atom approach by Lucy 2002, 2003 which will both be described in detail in Sect. 7.4). In this process, a change in frequency assigned to the packets may occur, but the CMF energy will always stay constant. A noteworthy property of indivisible energy packet schemes is that a MC packet may represent a varying number of physical photons during its lifetime: the scheme does not enforce conservation of photon number (and nor should it: many physical radiation–matter processes e.g. recombination cascades or fluorescence do not conserve the number of photons).
Relying on this indivisible energy packet formalism offers a number of advantages as pointed out by Abbott and Lucy (1985) and Lucy (1999a). Most importantly, it enforces strict local energy conservation in RE applications by construction. However, we note that this energy conserving property does not restrict the scheme to RE problems: well posed sources and sinks of radiative energy can be readily incorporated while maintaining strict energy conservation (see Sect. 11). In addition, the packet indivisibility naturally controls the number of quanta tracked in an MCRT calculation and avoids the need to incorporate an elimination scheme for quanta with small weights which may otherwise accumulate and slowdown the calculation. The indivisible energy packet scheme has been widely used in MCRT calculations of RT in mass outflows (e.g. Abbott and Lucy 1985; Vink et al. 1999; Long and Knigge 2002; Sim 2004, 2005; Carciofi and Bjorkman 2006, 2008; Noebauer et al. 2010) and in SN ejecta (e.g. Lucy 2005; Kasen et al. 2006; Sim 2007; Kromer and Sim 2009; Noebauer et al. 2012; Kerzendorf and Sim 2014).
We note that many of the advantages of indivisible energy packet schemes can still be retained when strict indivisibility is relaxed. In particular, splitting of energy packets can be introduced in attempts to improve statistics (e.g. Harries 2015; Ergon et al. 2018) where strict energy conservation is retained (i.e. the algorithm is free to split an energy packet at any point, provided that the sum of the energies of the newly created packets matches that of the original unsplit packet). Similarly, there is no requirement of the scheme that all packets have the same energy as each other: the only rule is that the combined packet energies correctly sum to the total energy / energy flow of the process under consideration.
In a photon packet scheme, the manner in which this process can be simulated is obvious: whenever one of the MC photon packets undergoes such a Compton scattering event, the number of photons it represents remains fixed but the frequency of the photons represented by the packet is reduced (accordingly, the packet then represents less energy).
For an indivisibly energy packet scheme, the treatment is more subtle (Lucy 2005). Here we consider how energy flows through the problem: from the initial energy of the incoming photon population (\(\gamma _{\mathrm{i}}\)) to the combination of final photon population (\(\gamma _{\mathrm{f}}\)) and final electron kinetic energy (\(e^_{\mathrm{f}}\)). In particular, a fraction \(F_\gamma = E^\gamma _{\mathrm{f}} / E^\gamma _{\mathrm{i}}\) of the incident photon energy is passed to the outgoing photon while \(F_e = E^e_{\mathrm{f}} / E^\gamma _{\mathrm{i}} = 1  F_\gamma \) goes to the electron. Thus, adopting the indivisible energy packet principle, an initial MC (\(\gamma _{\mathrm{i}}\)) packet is converted to an outgoing \(\gamma _{\mathrm{f}}\) packet with probability \(F_\gamma \) or to a packet representing the electron kinetic energy with probability \(F_e\). In either case, the energy represented by the packet remains fixed (i.e. strict energy conservation), but the nature of the energy has changed: in the first case the energy is still being carried by photons, but now of lower photon frequency (in accordance with the \(\gamma _{\mathrm{f}}\) state); in the second case, the energy has been passed to the electron kinetic pool from where its role in powering further emission of the material can be followed using e.g. the kpacket formalism of Lucy (2002, see also Sect. 7.4).
This example primarily serves to illustrate the subtle difference between photonpacket and energypacket schemes but it is natural to wonder which scheme is better. In general, there is no absolute statement to be made: both approaches are valid and which is better suited will depend on the problem in question. However, the relative merits are clear and can be stated fairly simply for our example: the photon packet scheme will rigorously conserve photon number (as does the physical Compton scattering process) and is well suited if the aim of the simulation is to calculate the Comptonized photon spectrum (e.g. Pozdnyakov et al. 1983; Laurent and Titarchuk 1999), potentially following many scattering events. On the other hand, multiple scattering in the photon packet approach may lead to a proliferation of lowfrequency photon packets that carry very little energy, but still require the same computational effort per scattering to simulate. This may not be ideal for e.g. applications in which the primary interest in highenergy Compton scattering lies in its role as a heating process (such as the modelling of SN ejecta powered by radioactive decay, as discussed by Lucy 2005). For such a problem, the indivisible energy packet scheme provides a simple means to determine the rate at which energy flows into the electron pool with the computational effort being invested in proportion to the energy carried by the photons, rather than to the photon number.
5.3 Initialisation of packets
Closely related to the fundamental discretization of the radiation field into discrete MC packets is the initialization process. Here, the initial packet properties are assigned by drawing from the spatial, directional and spectral distribution of the radiation field by relying on the sampling techniques presented in Sect. 4.2. The instantaneous values of these properties,^{11} i.e. the position, direction, frequency,^{12} energy and weight,^{13} fully describe the packet state during the entire MC simulation. If the effect of polarization is included in MCRT simulations, packets are additionally assigned appropriate values for the Stokes vectors (see, e.g. Kasen et al. 2006; Whitney 2011; Bulla et al. 2015).
In the following, we briefly sketch the initialization process within the indivisible energy packet scheme. Note that the corresponding procedure is not fundamentally different within the photon packet scheme. In the following presentation, we distinguish between the initial assignment of properties for packets that represent the radiation field in the domain at the onset of the MC simulation and for packets that represent the inflow of radiation into the domain through the boundaries.
We conclude this description by noting that in MCRT applications packets may also be initialized to represent the continuous radiative cooling of the ambient material or to represent the emission from other sources within the domain (e.g. in radiative nonequilibrium applications). The basic initialization principles highlighted remain the same and can be simply transferred to these applications.
6 Propagation of quanta
The discretization paradigms and the initialization principles outlined above (see Sect. 5) provide rules for the creation and launching of MC packets. The bulk of the computational effort involved in a MCRT calculation is spent in tracing the movement of these packets through the ambient material to simulate the RT process. During the propagation, their trajectories are interrupted as the packets experience radiation–matter interactions. Depending on the nature of these interactions, the packet properties may change or the propagation may even be terminated. The MC packets are thus followed until certain termination conditions are met, e.g. the packet leaves the computational domain or has been active for a predefined time (this aspect will be treated in detail in Sect. 6.6). The propagation procedure of a MCRT simulation is complete when all packets representing the initial radiation field and the effects of sources of radiative energy (e.g. inflows through boundaries, internal sources in radiative nonequilibrium applications, etc.) have been processed this way. In the following, we first outline the fundamental propagation principles (Sect. 6.1) and then detail how basic absorption and scattering interactions are handled (Sect. 6.2–6.5) before finally turning to the inclusion of time dependence (Sect. 6.6).
6.1 Basic propagation principle
In the absence of general relativistic effects (which we neglect in this review), a MC packet propagates on a straight path in its propagation direction \(\mathbf n \). In the simplest version of a MC packet propagation algorithm, the packet properties do not change along these straightflight elements of its path: interactions with the medium are treated as discrete interaction events, and the primary aim of the MC algorithm is to determine where those interaction events occur.
For locating packet interactions using Eq. (45), we highlight an important property of the exponential distribution, namely its infinite divisibility (see for example Bose et al. 2002). Probability distributions with this property can be replaced by the “distribution of the sum of an arbitrary number of independent and identically distributed random variables”.^{16} For the purpose of MCRT, this implies that, as long as the packet has not interacted, one is at liberty to reset the comparison between accumulated and interaction optical depth arbitrarily often. I.e. one can opt to redraw the optical depth distance to the next interaction with Eq. (45) and reset the tracking of accumulated optical depth, Eq. (44), at the current packet location. This property is often used when performing MCRT simulations on numerical grids (see Sect. 6.3).
6.2 Absorption as continuous weight degradation
The general scheme outlined in the previous section can be applied to find discrete interaction events for any physical contribution to the opacity. It is particularly important (and widely used) for addressing scattering problems: an accurate treatment of scattering is most easily formulated as an ensemble of discrete interactions where the properties of the packets are changed at the interaction point in accordance with the physics of the scattering process. The scheme is also widely applied to true absorption processes, and this is particularly attractive in applications that aim to exploit the energyconserving qualities of radiative equilibrium problems (see Sect. 7).
There are several advantages for this approach to absorption compared to the discrete scheme outlined above. First, it can reduce the MC noise since it replaces the stochastic identification of specific interaction points with a smooth degradation of packet weight. This seems especially attractive if considering small contributions to the opacity (e.g. background continua): using the discrete event algorithm to simulate such interactions would be noisy since the number of events associated with a low opacity will be small.^{17} Second, it can greatly simplify the MC algorithm for applications in which pure absorption is dominant: in such cases, pure weight attenuation of packets on straight trajectories may be sufficient to solve the problem.^{18}
However, there are some limitations associated with continuous weight degradation. In particular, if the interaction processes is associated (at the microphysical level) with some radiative reemission process, such as effective scattering/fluorescence in atomic or molecular line transitions, this approach loses a direct connection between the absorption and reemission process. If important, the reemission must be simulated by injecting new MC packets to represent it (see Sect. 7.1). For this reason, MCRT applications that depend on simulating e.g. atomic line interactions have found it more practical to use a discrete interaction approach for this process (similar to e.g. Abbott and Lucy 1985). We note, however, that hybrid schemes have been successfully employed where the continuous attenuation approach is used for smooth continuum absorption opacity while a discrete interaction algorithm is applied for atomic line absorption and electron scattering (e.g. Long and Knigge 2002). Throughout most of this review we will focus on methods that adopt the discrete interaction approach for treating both scattering and absorption but note that many of the principles discussed in later chapters can be applied to simulations that employ a weightdegradation approach to absorption, or a combination of both.
6.3 Material properties and numerical discretization
To perform the packet propagation process, the local material state, such as velocity, density and temperature, has to be accessible. It sets the local opacity and thus determines the rate at which optical depth is accumulated along the propagation path [cf. Eq. (44)]. Moreover, the material state dictates the reemission characteristics in scattering interactions. Ideally, the material state is directly accessible in closed analytic form such that the optical depth integration can be performed analytically. In practise, however, the complex local dependence of the material properties calls for a numerical integration. In addition, the continuous material state is often not available but instead only a discrete representation. This could, for example, be the snapshot of a hydrodynamical simulation. Consequently, the packet propagation is typically realised by introducing a computational grid, dividing the domain into cells on which the matter state is discretely represented. Often, the material properties are approximated as constant throughout the grid cells (although interpolation can be used).
The packet propagation process can then proceed on the numerical grid along the basic principles outlined above. If the material state is assumed to be constant throughout the individual grid cells, determining the rate of accumulation of \(\tau \) is generally simple.^{19} However, one does need to track when packets cross over grid cell boundaries: at such points, quantities that depend on the material state, such as opacities, have to be recalculated. Some codes, for algorithmic convenience, also exploit the infinite indivisibility property of the exponential distribution and redraw the random optical depth from the usual sampling law, Eq. (45), when cell crossing occurs.
6.4 Absorption and scattering
Having outlined the principles of how packets can be propagated through the simulation domain, we now discuss how interactions are handled. In any real application, the details of how interaction events are to be processed will depend on the particulars of the radiation–matter physics being simulated. To illustrate the general principles here, we adopt a number of simplifications, namely that the medium is static and that all material functions are frequency independent and isotropic. We also restrict the presentation to basic absorption^{20} and coherent and isotropic scattering interactions. Lifting these simplifications, in particular, including more complex interaction processes, naturally complicates the individual steps of the propagation process but the basic structure of the procedure remains the same.
After the scattering event, the packet continues its propagation along the new direction. A new distance to the next interaction event is drawn from Eq. (45) and the tracking of the accumulated optical depth of Eq. (44) is reinitialized. In this manner, the flow of packets can be followed including multiple scatterings in arbitrary media. We emphasise that the ease with which scattering interactions can be treated is a major benefit of MCRT approaches.
6.5 Propagation example
 The MCRT simulation begins by initialising N packets and uniformly distributing them throughout the sphere. As this is a onedimensional problem and since we are only interested in the escape probability, the packet state is essentially described by r and \(\mu \). The initial location in the uniform sphere is assigned bywhere R is the outer radius of the sphere (code line 90), and the initial direction is chosen isotropically (code line 92)$$\begin{aligned} r = R \xi ^{\frac{1}{3}}, \end{aligned}$$(53)$$\begin{aligned} \mu = 2 \xi  1. \end{aligned}$$(54)

After initialisation, the pool of packets is processed with each being propagated through the sphere following the principles outlined above. This includes drawing the random interaction \(\tau \)values (code line 143) and calculating distances to boundary crossing (code line 145). Packet interaction occurs when the randomly drawn optical depth is reached before the boundary of the simulation (code line 149). Whenever a packet interacts, Eq. (51) is used to decide whether the packet is absorbed (destroyed in this case) or scattered. Following scattering, a new direction (code line 172) is drawn with Eq. (52) and the propagation continues.

Each packet is followed until it either is absorbed or escapes through the surface of the sphere, and the entire MCRT simulation ends when all packets are processed in this manner. Finally, the escape probability is calculated by dividing the number of escaped packets by the total number of packets which have been initialised.
6.6 MCRT: timedependent applications
In addition to tracking the current time for each packet the basic propagation scheme as outlined in Sects. 6.1–6.4 has to be further extended to account for the subdivision of the MCRT simulation into a series of time steps. Whenever the internal clock of a MC packet has progressed to the end of the current simulation time step, \(t^{n+1}\), the packet’s propagation is interrupted. The instantaneous state of the packet, i.e. its position, current frequency, energy, propagation direction and any further properties, is stored and the next packet of the active population is treated. At the end of the propagation process, all packets stored are transferred to the next simulation cycle and the packets continue their propagation at \(t = t^{n+1}\) with the saved properties.
7 Thermal and line emission in MCRT
The treatment of absorption and pure scattering processes as outlined above are relatively standard and the principles used are very well established. In contrast, the manner in which emission is handled in MCRT schemes is relatively varied and much of the sophistication and ongoing developments in the MCRT field relate to the manner in which this is done.
In this section we aim to review some of the approaches to treating emissions within a computational domain. To be clear, this is distinct from questions of how MC quanta might be injected at some computational boundary: in Sect. 5.3 we already reviewed how e.g. a population of packets might be injected to represent a seed blackbody radiation field as might be appropriate as an initial condition in a timedependent simulation. Likewise, we described how packets might be injected at some boundary surface to represent a radiation source external to the simulation domain, for example a photospheric boundary condition. Here, instead, our focus is on how emissivity within the computational domain can be taken into account.
7.1 Known emissivity
The most obvious case to handle is any for which the emissivity is externally known (or can be easily estimated) without prior knowledge of the RT process within the domain. One such example might be a nonequilibrium plasma that is predominantly heated and ionized by nonradiative processes.
In this case, the emissivity can simply be sampled using standard sampling techniques (Sect. 4.2) to create a population of packets with properties that represent the emission process (i.e. photon frequency, weights/energies, propagation directions etc.). This population is simply injected alongside any packets due to external radiation field boundary conditions, and their subsequent propagation followed in the same manner.^{21}
7.2 Radiative equilibrium (RE)
For several of the applications to which MCRT has been applied, the emissivity is not known a prior. Indeed, for many astronomical RT problems (e.g. stellar/disk atmospheres, winds, SNe etc.), RE is a good approximation and the emissivity is effectively determined by the radiation field itself (i.e. nearequilibrium is achieved between absorption and emission of radiation). In such cases, the emissivity usually cannot be anticipated independently of a radiation transport simulation, which poses a challenge for consistent modelling. In the following sections we review methods that can be applied to problems with this requirement.
7.3 Radiative equilibrium in MCRT by iteration
One approach for RE problems is to use an iteration scheme to determine the conditions of the medium (temperature, ionization state, level populations etc.) on which the emissivity depends. Here, an iterative sequence of RT simulations would be performed: in each iteration the current best estimate of the conditions in the medium would be adopted to calculate the emissivity, and the outcome of the RT calculations^{22} used to make an improved estimate for those conditions in the next cycle. This approach can be applied to schemes based on photon packets and/or energy packets and it has been used for modelling at least some part of the emissivity (e.g. the part associated with radiative cooling by Long and Knigge 2002) and works well provided that the complexity of the problem is not too severe.
However, as is well known from the history of modelling stellar atmospheres (cf. Hubeny and Mihalas 2014), the nonlocal character of RT problems can lead to significant convergence problems for this type of iteration scheme, especially when considering regions associated with high optical depth. In particular, in its pure form, this scheme suffers from the issue that energy conservation is only achieved asymptotically (i.e. once/if a converged equilibrium solution is found). As a result, during the iteration process, over or underestimated emissivities (due e.g. to unconverged temperatures) will lead to spurious sources and sinks of radiation that might slow or inhibit convergence.
7.4 “Onthefly” radiative equilibrium in MCRT via indivisible energy packets
As described/developed in the works of Lucy (Abbott and Lucy 1985; Lucy 1999a, 2005), it is actually very simple to rigorously enforce the conservation of energy required by RE via an indivisible^{23} energy packet MCRT scheme. The principle is straight forward: RE implies that at all points there is (local) balance between the rates of absorption and emission of energy from/to the radiation field. Since, in an energy packet discretizaion the MC quanta already represent (local) bundles of radiative energy, the condition of RE is trivially enforced by insisting that the MC quanta are never destroyed, or otherwise degraded in weight, in the course of the simulation. Thus, all interactions between MC packets and the medium—even those representing pure absorption processes—become effective scatterings controlled by rules devised for the MCRT simulation. The rules for how packets should be altered in these effective scattering events depend on the physical process(es) being simulated (Lucy 2002, 2003, 2005): commonly, considerations of statistical equilibrium and/or thermal equilibrium (TE) will form the basis for formulating these rules. In the following sections we will elaborate these principles more generally, but for concreteness we first discuss one of the simplest specific examples.
7.4.1 Example: effective resonant scattering in a twolevel atom
This particular example is almost trivial but, as will be elaborated below, the basic idea of combining RE (indivisible packets) with statistical equilibrium (traffic flow rules to process packet interactions) can be extended to much more sophisticated cases. Before discussing more general cases, however, we pause to comment on some of the key features that this simple example already highlights.
7.4.2 Fluorescence and thermal emissivity via redistribution parameters
Early MCRT implementations, such as Abbott and Lucy (1985), applied the twolevel effective resonance line treatment of opacity, essentially as outlined above. The twolevel approximation is relatively welljustified for many of the strong metal lines in the ultraviolet (UV, such as C iv 1550 Å or N v 1242 Å) that were relevant to studies of stellar winds (Abbott and Lucy 1985) and also later studies of accretion disk winds (Long and Knigge 2002; Kusterer et al. 2014). However, the twolevel atom approximation has limited utility and is not realistic for problems in which flux redistribution via fluorescence and/or thermal reprocessing of radiation is important.
Various methods, with differing degrees of sophistication, can be employed to simulate flux redistribution in indivisible packet MCRT. One approach is to assume that the radiation–matter interactions can be modelled as a combination of resonance scattering and some form of complete flux redistribution across the spectrum. In this approach, a redistribution parameter, \(\varLambda \), is introduced and used to determine the outcome of each packet interaction by drawing a random number (\(\xi \)) and comparing: if \(\xi > \varLambda \) then the MC packet is assumed to undergo coherent scattering (i.e. a new direction is assigned but the CMF frequency is conserved as it would be in electron scattering or resonance line scattering); otherwise an incoherent (effective) scattering is executed in which a new random direction of propagation is assigned along with a new frequency. When the incoherent case is selected, the new frequency must be drawn from some suitable normalised emissivity distribution. One simple possibility is the local thermodynamic equilibrium (LTE) thermal emissivity (\(\chi _{\nu } B_{\nu }\)), but alternative choices could be made. The redistribution parameter (\(\varLambda \)) can be set globally or made a function of the interaction process (e.g. for linescattering problems it might be estimated by comparing the relative importance of collisional and radiative deexcitation, similar to the considerations by Long and Knigge 2002). The effectiveness of this approach naturally depends on the problem under consideration. However, at least for some applications studied with MCRT it has been shown that this scheme is effective. In particular, as demonstrated by Baron et al. (1996), Pinto and Eastman (2000a, b) and Kasen et al. (2006), flux redistribution in Type Ia Supernova (SN Ia) modelling can be quite effectively approximated via a simple (thermal) redistribution parameter, achieving good agreement with more detailed treatments without too much sensitivity to the particular value of \(\varLambda \) adopted (see also Magee et al. 2018). We note that, in the limit \(\varLambda \rightarrow 1\), it may appear that this type of approach seems very similar to that outlined in Sect. 7.1: selecting postinteraction properties of the MC packets depends on knowing the material state sufficiently well to estimate an emissivity distribution from which to draw e.g. photon frequencies. However, the notable difference is that here the absolute normalisation of the emissivity is not used: i.e. although the emissivity distribution is used to select most properties, the packet energies remain fixed by the indivisible packet principle. As a consequence, strict RE is still enforced in the radiation/matter interactions.
7.4.3 Fluorescence and redistribution: macro atom method
Approaches similar to those outlined above (i.e. that treat interactions as either coherent or fully redistributive) are easy to implement, fast to run and, with appropriate parameter choices, can capture many of the essential features of scattering and redistribution. However, not all physical processes are readily captured this way: for example fluorescence (and cascades) between energy levels in an atom or ion certainly leads to a coupling of emission in different parts of the spectrum, but it cannot be well described via a single “redistribution emissivity” that can be sampled for all interactions. In general, we must acknowledge that every distinct radiation–matter interaction can lead to its own distinct set of outcomes. For example, consider a threelevel atom in a problem for which statistical and RE are assumed (cf. Fig. 6): if a MCRT packet is absorbed in the transition from the lowest level (1) to the highest (3), it is expected that the range of outcomes following that event should at least include a combination of reemission in the \(3 \rightarrow 1\) transition plus the cascade \(3 \rightarrow 2\) and \(2 \rightarrow 1\). It is therefore desirable to construct sets of rules for processing packet interactions in MCRT simulations that can accurately describe this physics.
One way to handle the threelevel atom example would simply be to split the original interacting packet, whether a photon or energy packet, in proportion to the number of emitted photons or corresponding energy flow for each of the outgoing channels and continue the simulation. For the threelevel case, this is quite feasible but the drawback of such an approach in general is that, for interactions with very many possible outcomes (e.g. atomic models with large numbers of levels) this can lead to a proliferation of packets that is computationally too expensive to follow. Moreover, it is nontrivial to generalise that approach to handle e.g. the inverse: our threelevel atom ought to also be able to absorb \(1 \rightarrow 2\) and \(2 \rightarrow 3\), and then emit \(3 \rightarrow 1\) (inverse fluorescence). How can this process be captured in such a redistribution scheme?
Nevertheless, the Lucy (1999b) downbranching scheme is still not complete and does not address all the issues raised even by our simple threelevel atom example (e.g. while it will reproduce flux redistribution between the \(1\rightarrow 3\) and \(2 \rightarrow 3\) transitions—because they share an upper level—it does not connect the \(1 \rightarrow 2\) transition to the cascade). A more complete solution that can incorporate all transitions in multilevel atoms was later devised by Lucy (2002, 2003) via what he called the macro atom method. This approach provides a generalised approach to formulating rules to process interactions of MC energy packets in accordance with the requirements of radiative and statistical equilibrium. In essence, we can view all of the possible excited levels of the matter as energy pools. Energy flows into/out of each pool via the set of transitions into that energy level and the equilibrium condition (namely that the energy associated with each such pool is stationary) is satisfied by imposing a traffic flow set of rules to process interactions for each possible energy level. The extra sophistication compared to the downbranching scheme is that we include the fact that physical processes represent not only energy flow to and from the radiation field, but also between the various energy pools associated with the different available levels of the atoms/ions/molecules in the medium. Expressions for the general macro atom transition probabilities and their interpretation are derived by Lucy (2002). We will not repeat the general case here but, in the example below, illustrate its application to our example threelevel atom in order to clarify the principles.
7.4.4 Example and discussion: macro atom scheme for a threelevel atom
 (A)
Whenever an active radiation packet is absorbed by any of the three transitions we view this as a discrete realisation of the corresponding \(\dot{A}\) term in the macro atom equation. We say that this process has activated a macro atom in the corresponding energy level.
 (B)
We then inspect the sink terms (i.e. RHS terms) for the activated level of the macro atom and use a random number to select an outcome with probabilities proportional to the energy flows implied by the system of macro atom equations. Thus, for example, if the macro atom is activated to level 2, with probability \(\dot{E}_{21} / D_2\) we select emission in the \(2\rightarrow 1\) transition, and with probabilities of \(\varepsilon _2 R_{23} / D_2\) and \(\varepsilon _1 R_{21}/ D_2\) we select internal macro atom transitions \(2\rightarrow 3\) and \(2\rightarrow 1\), respectively (\(D_2 = \dot{E}_{21} + \varepsilon _2 R_{23} + \varepsilon _1 R_{21}\) is selected to normalise the probabilities correctly).
 (C)
 (i)
If the selection corresponds to an emission \(\dot{E}\) term, the macro atom deactivates and the radiation packet is returned to the main MC simulation with new properties (photon frequency, direction etc.) set in accordance with the properties of the corresponding emission process. The total energy carried by the packet (in the CMF) remains equal to that when the packet was absorbed (in accordance with the requirements of RE).
 (ii)
Alternatively, if an internal transition term is selected, the macro atom remains active but is switched from its current state to a new state in accordance with the selected term [e.g. selecting the \(\varepsilon _2 R_{23} / D_2\) term results in a transition from macro atom state 2 to state 3, conceptually representing the “sink” on the RHS of Eq. (67) into the matching “source” term on the LHS of Eq. (68)]. The algorithm then returns to step B and processes the activated macro atom again. This continues until deactivation occurs.
 (i)
First, we note that all rates \(R_{ij}\) are directly proportional to the level population \(n_i\), which would imply that, like the normal line emissivity, Eq. (58), determining the terms in the macro atom equations depends on already knowing the level populations. However, because of the normalisation process in step (B), this leading dependence cancels out from the transition probabilities. Of course, additional effects (e.g. corrections for stimulated emission to absorption rates or introduction of Sobolev escape probabilities; see Sect. 8.2) can still lead to dependencies on the populations. Nevertheless, cancelling of the leadingorder effect means that the macro atom transition rates can be relatively well determined even in the absence of a converged set of level populations. This property can be rather powerful when treating complex systems for which exact calculations of excited state level populations (and therefore a direct evaluation of absolute emissivities) is challenging: as shown by Lucy (2002, fig. 5), even for a complicated ion such as Fe ii the macro atom scheme produces fairly accurate excited state effective emissivities without any iteration to determine level populations.^{25}
Second, we note that the first of the set of macro atom traffic flow equations [Eq. (66)] involves no activation (\(\dot{A}\)) terms and no deactivation (\(\dot{E}\)) terms: it is a balance only between internal transition rates. This makes sense because it follows from the equation of statistical equilibrium for the lowest energy state: there are no channels for absorption of energy directly to that state nor emission of energy directly from it. Moreover, we note that the choice \(\varepsilon _1 = 0\)^{26} trivially satisfies Eq. (66) and also eliminates the corresponding internal transition terms from Eqs. (67) and (68). Making use of this definition will therefore (slightly) simplify the macro atom algorithm by effectively removing the need to explicitly consider the ground state.
Third, it can be seen that both of the simpler treatments introduced earlier for handling atomic line interactions are special cases of the full macro atom. Specifically, the effective resonance scattering approach used in several early studies (example in Sect. 7.4.1) is a twolevel macro atom with \(\varepsilon _1 = 0\). The downbranching scheme by Lucy (1999b) outlined in Sect. 7.4.3 is the macro atom scheme with all internal transition terms suppressed (formally, this can be derived from the general macro atom algorithm by assuming (i) downwards transition rate coefficients dominate and (ii) for all transitions between upper level u and lower level l, \(\varepsilon _u \gg \varepsilon _l\)).
Repeated cycling through steps (B) and (C)(ii) in the algorithm above can make it computationally inefficient, particularly when the scheme is extended to also include coupling to the thermal pool. This can be addressed in several ways, however. As noted by Lucy (2002), the macro atom algorithm can be viewed as recursive application of the set of transition/deactivation probabilities and recently, Ergon et al. (2018) have presented a Markovchain approach to the macroatom machinery. This method effectively solves the problem without the need to follow internal macroatom state transitions which can be a substantial advantage in terms of computational efficiency.
7.4.5 The thermal energy pool
7.5 Indivisible energy packets beyond radiative equilibrium
As before, the three terms on the RHS of Eq. (70) are the sink terms for the kpacket pool and so they are sampled to determine the manner in which the energy flow out of the kpacket pool behaves. \(C_{\mathrm{R}}\) and \(C_{\mathrm{C}}\) can be simulated just as before: packets are fed back to the radiation field (\(C_{\mathrm{R}}\)) or to the excitation energy of macro atom pools via collisional excitation (\(C_{\mathrm{C}}\)). The additional term, \(C_{\mathrm{E}}\), can be treated as a true external sink term: i.e. with probability \(C_{\mathrm{E}}/(C_{\mathrm{R}}+C_{\mathrm{C}}+C_{\mathrm{E}})\) energy packets that flow into the thermal pool are terminated. Alternatively, this term could be treated via a reduction in packet energies: i.e. one could opt to sample the RHS of Eq. (70) only considering \(C_{\mathrm{R}}\) and \(C_{\mathrm{C}}\) but reduce the energy of all packets processed through this channel by a factor of \((C_{\mathrm{R}}+C_{\mathrm{E}})/(C_{\mathrm{R}}+C_{\mathrm{E}}+C_{\mathrm{C}})\). We note that, in the limit where the external sources and sink term (\(H_{\mathrm{E}}\) and \(C_{\mathrm{E}}\) above) become dominant, an indivisible energy packet simulation performed with this machinery will essentially reproduce the elementary scheme explained in Sect. 7.1.
The example above illustrates how the indivisible packet scheme can be altered to take account of specific departures from RE, and this is done in many of the existing implementations of this method, particularly to account for adiabatic cooling (e.g. Long and Knigge 2002; Kasen et al. 2006; Kromer 2009; Vogl et al. 2019). In principle a similar logic could be employed to deal with departures from statistical equilibrium (affecting the macro atom transition rules) or TE (further affecting the kpacket transition rules). Specifically, if the inflow and outflow rates are not in balance such that there is a net rateofchange of the energy reservoir, then terms representing the ongoing accumulation (or loss) of energy from the pool could be built into the formulation (i.e. retain terms including the derivatives of the level populations and/or the kinetic temperature). Provided that values for those derivatives are known they could also then be included in the packet flow. To our knowledge, however, extensions of the macro atom/kpacket schemes that consider such terms have not yet been implemented.
8 MCRT: application in outflows and explosions
A prominent field in astrophysics, in which MCRT methods are very popular and successful, is the study of fast mass outflows. For example, MCRT schemes can be used to calculate massloss rates and the structure of hotstar winds (see e.g. Abbott and Lucy 1985; Lucy and Abbott 1993; Schaerer and Schmutz 1994; Schmutz 1997; Vink et al. 1999, 2000; Sim 2004; Müller and Vink 2008; Muijres et al. 2012b; Lucy 2012b; Noebauer and Sim 2015; Vink 2018), to determine synthetic light curves and spectra for SNe (see e.g. Mazzali and Lucy 1993; Lucy 1999a, 2005; Mazzali 2000; Kasen et al. 2006; Sim 2007; Kromer and Sim 2009; Wollaeger et al. 2013; Wollaeger and van Rossum 2014; Kerzendorf and Sim 2014; Bulla et al. 2015; Magee et al. 2018) or to treat RT in winds emanating from accretion discs of cataclysmic variables (Knigge et al. 1995; Long and Knigge 2002; Noebauer et al. 2010; Kusterer et al. 2014; Matthews et al. 2015) or active galactic nuclei (Sim 2005; Sim et al. 2010, 2012; Higginbottom et al. 2013; Matthews et al. 2016, 2017; Tomaru et al. 2018). In applications such as these, our current implicit assumption, namely that RT occurs in static media or in environments with material velocities low enough to be safely ignored, can no longer be maintained. Instead, special relativistic effects play an important role and have to be taken into account. In the following, we outline some important aspects of performing MCRT in moving media. While many of the described concepts are generic, the treatment of line interactions using the Sobolev approximation (see Sect. 8.2) is specific to MCRT in expanding media, such as SN ejecta or winds.
8.1 The mixedframe approach
As introduced in Sect. 2, there are two fundamental frames of reference for RT, namely the LF and the local rest frame (CMF). Until this point, we have largely ignored the distinction between these frames since RT was assumed to occur in static media or in lowvelocity environments. When the material velocities become large, however, this simplification is no longer justified. In these situations, MCRT schemes often rely on a socalled mixedframe approach (see for example Lucy 2005). This exploits the fact that the handling of different tasks involved in MCRT simulations is easier in one or other of the two frames. Specifically, the spatial and temporal mesh is usually defined in the lab frame, making it most convenient for measuring distances and thus for tracking packets and simulating their propagation. Radiation–matter interactions, on the other hand, are more easily described in the local rest frame of the material. Here, the material functions take their simplest form. Consequently, MCRT schemes adopting the mixedframe approach propagate packets in the LF but treat all interactions in the CMF.
Finally, we note that one also has to decide in which frame the MC packets are launched during the initialisation. Often, the CMF is the natural choice for this process, e.g. when representing a thermal radiation field. In such cases, packet properties are drawn in the CMF and then transformed into the LF using the rules given here and in Sect. 2 before starting the propagation.
8.2 Line interactions in outflows
As described above, treating frequencydependent opacities in the presence of large material velocities is challenging. However, in the case of boundbound processes, the situation can be significantly simplified with the socalled Sobolev approximation (Sobolev 1960). Indeed, RT through fast expanding mass outflows is the classical example for the use of the Sobolev approximation. We refrain from a detailed description of Sobolev theory since it is a widely used technique in astrophysical RT problems (see e.g. Castor 2007 for a general overview of the approximation and Rybicki and Hummer 1978, 1983; Hummer and Rybicki 1985; Jeffery 1993, 1995 for various extensions of the original formulation) but highlight some key aspects and describe how a Sobolev line interaction scheme can be easily incorporated into MCRT simulations for fast outflows. An illustrative overview of this approximation can be found in Lamers and Cassinelli (1999).
Many MCRT applications in outflows adopt the Sobolev approximation and follow a line interaction scheme similar to the one just outlined. Examples include the studies by Abbott and Lucy (1985), Lucy and Abbott (1993), Vink et al. (1999), Sim (2004), Noebauer and Sim (2015) dealing with hot star winds, or the works by Long and Knigge (2002) performing MCRT in disc winds and Mazzali and Lucy (1993), Mazzali (2000), Kasen et al. (2006), Sim (2007), Kromer and Sim (2009), Kerzendorf and Sim (2014) who use MCRT in SN ejecta. There are several studies that treat line interactions without relying on the Sobolev approximation such as Knigge et al. (1995) and Kusterer et al. (2014). Here, the conceptual and computational effort is, however, significantly higher.
8.3 MCRT and expansion work
In RE, packet energy is conserved in the CMF during interactions, which partly motivates the introduction of the mixedframe approach for MCRT in moving media. We emphasize, however, that packet energy conservation does not necessarily hold in the LF. In fact, depending on the flow of radiation relative to the moving ambient material, photons may either lose or gain energy in interactions. This is a crucial process in astrophysical applications involving strong mass outflows, for example hotstar winds. Here, photons collectively lose energy in interactions by performing expansion work, ultimately driving and maintaining the outflow (cf. Lamers and Cassinelli 1999; Puls et al. 2008). In the following, we briefly demonstrate that MCRT schemes adopting the mixedframe approach readily capture this work term (indeed this was one of the original motivating factors for the approach; Abbott and Lucy 1985).
9 Extracting information from MCRT simulations
With the algorithms outlined above, the flight paths of MC quanta can be determined and tracked during MCRT simulations. In general, the individual trajectories are not of primary interest. Rather, meaningful information that effectively represents the radiation field needs to be extracted from them. In some cases, only radiation escaping from the simulation box may be of interest to construct synthetic spectra, light curves or images. For other applications, the most important outcome may be a characterisation of the radiation field internal to the system. In this part of the review, we present a number of common approaches that can be used to extract physical information from MCRT simulations. We preface this by a brief discussion of MC noise, which is a fundamental, often undesired property of MCRT simulations that motivates the design of the extraction techniques described below.
9.1 MC noise
MCRT simulations are probabilistic by nature. Consequently, results obtained with these approaches will generally be subject to stochastic fluctuations. This fundamental and inherent property of MC calculations is often referred to as Monte Carlo noise or simply noise. Here, we briefly present the basic behaviour of this noise component and discuss the implications for devising techniques to extract or reconstruct physical information from a MC simulation. More details about this subject may be found in the standard literature, e.g. in Kalos and Whitlock (2008).
9.2 Direct counting of packets
9.3 Volumebased estimators
Lucy (1999a) introduced a technique to reconstruct properties of the internal radiation field that is less vulnerable to noise than direct counting approaches since information along the entire packet trajectory is used instead of only a momentary snapshot of the packet distribution. These techniques have been refined by Lucy (1999b, 2003, 2005) and are often referred to as volumebased estimators.^{32} The effective use of such estimators has been a key consideration for many MCRT studies relying on Lucy’s approach (e.g. Sim 2004, 2007; Kasen et al. 2006; Kromer and Sim 2009; Harries 2011, 2015; Noebauer et al. 2012; Kerzendorf and Sim 2014).
The volumebased estimator approach rests on the idea that instead of considering packets at certain discrete instances, timeaveraged estimates of radiation field properties can be constructed by incorporating information from the full packet propagation path. The fundamental notion is that the packet flight histories form an ensemble of trajectory elements that statistically represent the radiation field. To better illustrate this principle, we follow Lucy (1999a) and repeat the formulation of a volumebased estimator for the radiation field energy density.
9.3.1 Example: formulation of volumebased estimator for the radiation energy density
The advantage of the volumebased estimator scheme compared to simple direct counting measurements is twofold. First, a single packet can contribute to the estimators in multiple cells, provided that its trajectory intersects these cells during the time step. Second, the same packet can in principle contribute repeatedly to the estimator in a specific cell, if it is scattered in the cell or backscattered from a different cell. Both features of the volumebased estimator scheme drastically increase statistics and thus reduce the amount of MC noise in the reconstructed quantity. Also, this technique reduces the risk of obtaining undetermined results. In the direct counting approach, at least one packet must reside in the cell at the instant considered to obtain a nonzero result. This condition is mitigated to the much less restrictive requirement that at least one packet has resided in the cell at any point during the time step.
9.3.2 Constructing volumebased estimators: radiation field quantities
9.3.3 Constructing volumebased estimators: extracting physical rates
For many problems, simulation of the radiation field serves not only to predict synthetic observables but also to determine thermodynamic conditions of the astrophysical plasma: e.g. often the radiation field is crucial for determining the ionization/excitation state and heating (e.g. Mazzali and Lucy 1993; Bjorkman and Wood 2001; Long and Knigge 2002; Ercolano et al. 2003, 2005, 2008). In such cases, we therefore wish to extract information on the relevant rates of physical processes in the simulations. Following the principle outlined in Sect. 9.2, this could be achieved simply by counting the rate at which individual packet events corresponding to the process in question occur during the simulation. However, such an approach relies on a sufficient number of such interactions happening to achieve acceptable statistics and an accurate result. This becomes very challenging in optically thin regions since very few packets or even none interact.
Again, the volumebased estimator approach offers a significant improvement since it takes a broader view and includes the information encoded in the entire packet propagation paths instead of only considering a series of isolated snapshots. In particular, a volumebased estimator can be formulated for any quantity that depends on the radiation field by applying constructions similar to those outlined in Sect. 9.3. The general principle will be that the rate of energy extracted from the radiation field by some process can be described in terms of a sum over packet trajectories weighted by the appropriate absorption coefficient. These energy flow rates can then be recast in other forms (e.g. transitions rates), as required.
9.3.4 Example: photoionization rate estimators
9.3.5 Volumebased estimators for energy and momentum flow
This idea also readily generalises to provide volumebased estimators for momentum transfer (see e.g. Noebauer et al. 2012; Roth and Kasen 2015), which are instrumental for MCbased Radiation Hydrodynamics (RH) calculations (see Sect. 11). For continuum driving, the form of these estimators is very similar to the F estimator, Eq. (100).
Lucy (1999b) used similar considerations for the formulation of estimators in applications which are dominated by atomic line interactions that are treated in the Sobolev limit, such as stellar winds or the ejecta of thermonuclear SNe. Here, the formulation is slightly more complicated and the form of the energy/momentum flow rate estimators is different from those outlined above: they are formed as summations over all packets that have come into Sobolev resonance within a grid cell (see also Sim 2004; Noebauer and Sim 2015). The principal advantage compared to direct counting still applies since all resonances contribute, regardless of whether the packet actually undergoes interaction. Given the potential importance of forests of weak lines to heating/driving of outflows, as is for example the case in hot star winds where many weak iron lines drive the outflow (Vink et al. 1999), this is a critical advantage.
9.4 Biasing
In many MCRT applications, only a subset of the packet population is of interest. For example, when creating a synthetic image, only packets that escape towards the virtual observer are relevant. It is therefore desirable to selectively invest more effort into propagating packets that are crucial for the determination of the quantity or process of interest instead of treating packets that do not contribute. This selective increase in statistics can be achieved with the help of socalled biasing techniques. The underlying basic principle is known as importance sampling in the field of MC integration.
9.4.1 Biased emission
Biased emission is a simple but powerful illustration of a biasing scheme. This approach helps in problems where we wish to accurately describe the emission from sources with very different emissivities. These can be external sources, such as stars irradiating some environment, or simply the internal emissivity of the ambient material occupying grid cells of a computational mesh. Biased emission is frequently used in dust RT, for example by YusefZadeh et al. (1984) and Juvela (2005). A detailed account of the technique is given by Baes et al. (2016).
9.4.2 Forced scattering
9.4.3 Peeloff
Constructing properties of the emerging radiation field by simply examining the properties of escaping packets often yields unsatisfactory results, particularly in multidimensional simulations: typically only a small fraction of the packet population escapes towards the observer meaning that the reconstruction will suffer from strong noise. Here, the socalled peeloff technique (sometimes also referred to as next event estimate) helps (e.g. YusefZadeh et al. 1984; Wood and Reynolds 1999; Baes et al. 2011; Steinacker et al. 2013; Lee et al. 2017). In the context of MCRT in fast mass outflows, this method is sometimes referred to as viewpoint technique or virtual packet scheme (Woods 1991; Knigge et al. 1995; Long and Knigge 2002; Kerzendorf and Sim 2014; Bulla et al. 2015).
Since every interaction any packet performs contributes to the reconstruction, the improvement in statistics in peeloff methods is substantial. However, the ray tracing exercise of the peeloff technique adds significantly to the overall computational effort of the MCRT calculation, sometimes even dominating the computational costs.
We note that variants of methods similar to peeloff have been used in specific applications. Lucy (1991, 1999b) introduced a ray tracing technique for variance reduction specifically designed for applications in which a photosphere approximation can be adopted and in which the medium is freely expanding, e.g. SN ejecta. During the MCRT simulation the source function is reconstructed from the packet interaction histories and then used in a formal integration step to calculate the emergent radiation field along cast rays. By relying on this technique, virtually noisefree spectra can be determined.^{36} Also, as described by Bulla et al. (2015), the peeloff technique can be applied not only to interaction events but instead to all MC packet trajectory elements. Here, packets can contribute to the synthetic observation even when no interactions occur: the synthetic observables are obtained by a sum over contributions from all packet trajectories weighted similar to Eq. (112) but including an additional multiplicative term that gives the probability that an interaction event could have happened along the trajectory element.
9.4.4 Further biasing techniques
In addition to the schemes outlined so far, a variety of other biasing techniques have been developed and are actively used. Among them are, for example, techniques called path length stretching (Baes et al. 2016), continuous absorption (known also as packet splitting or survival biasing Carter and Cashwell 1975; Steinacker et al. 2013; Lee et al. 2017) or polychromatism (Jonsson 2006; Steinacker et al. 2013). We refer the reader to the literature for example to the review by Steinacker et al. (2013) and the book by Dupree and Fraley (2002) for detailed accounts.
9.4.5 Limitations—Russian Roulette and composite biasing
Naturally, biasing techniques are not a universal remedy and are also afflicted by drawbacks. Here, we highlight some of the more severe limitations and discuss techniques that have been proposed and developed to alleviate them.
When applying biasing techniques that can act multiple times on the same packet, also small packet weights can become a hindrance. Packets with very small weights only contribute insignificantly to the reconstructed property but roughly the same computational effort has to be invested to follow their propagation as for important packets. Based on this costbenefit argument, it is advisable to terminate the propagation once the weight and thus importance of a packet has decreased beyond some predefined threshold. In this context, the socalled Russian Roulette method provides a stochastic framework to remove lowweight packets from the simulation, while still retaining energy conservation in a statistical sense (see e.g. Carter and Cashwell 1975; Dupree and Fraley 2002). In its simplest form, a termination probability \(p_T\) is defined. Whenever a packet enters the roulette, the termination probability is sampled and the packet propagation is terminated if the sampling outcome is positive. Otherwise, the packet survives and its weight increases to \(w / p_T\). This way, the weights of the terminated packets are distributed probabilistically onto the surviving ones and energy/weight conservation is ensured statistically. A detailed description of the Russian Roulette technique, and more sophisticated realisations, is given by Dupree and Fraley (2002).
10 Implicit and diffusion Monte Carlo techniques
Conventional MCRT methods, built upon the techniques outlined so far, inherently rely on explicitly tracking packet flight paths. Although this has a range of compelling benefits, not least the conceptual ease with which it can be developed, it has limitations particularly in regard to efficiency for many applications. For example, MCRT calculations become prohibitively slow when applied in optically thick media since the number of physical and numerical events that has to be explicitly tracked increase drastically. Another challenge is posed in Thermal Radiative Transfer (TRT) applications where successive absorptions and reemissions occur frequently. Achieving a stable and accurate solution of the evolution of the ambient medium and of the radiation field typically requires a drastic reduction of the size of the physical time step. In the following, we outline a number of developments and techniques that have been proposed and are actively used to alleviate these shortcomings.
10.1 Implicit Monte Carlo
Standard explicit MC techniques face challenges when dealing with TRT problems since these involve a rapid succession of absorption and emission processes. In this situation sufficiently short time steps have to be used so that the ambient conditions (temperature etc.) can properly react to absorption–emission imbalances. Otherwise, the radiation source term may deplete the internal energy reservoir of the ambient material between successive temperature updates and lead to unphysical conditions (e.g. negative temperatures).
These difficulties are addressed by the socalled Implicit Monte Carlo (IMC) method, introduced in the seminal work by Fleck and Cummings (1971). Here, the sequence of absorption and emission events is replaced by an effective scattering prescription and only the net imbalance remains as a true absorption and emission contribution. Despite the name, the IMC method does not constitute a truly implicit solution approach, comparable to techniques encountered in the field of solving differential equations. Instead, a semiimplicit recasting of the discretized RT equation is performed. This procedure leads to the main advantage of the IMC approach, namely the introduction of unconditional stability. In the following, we briefly outline the formulation of the IMC technique and discuss some important properties of this approach. For an indepth discussion of the method, we refer to the original work by Fleck and Cummings (1971) and to the recent detailed review by Wollaber (2016) on the subject.
This unconditional stability constitutes the main advantage of IMC and a substantial improvement over conventional MCRT approaches. This beneficial property has led to widespread adoption of the IMC scheme. In the astrophysics community, IMC schemes are predominantly applied in the field of RT in SN ejecta. Abdikamalov et al. (2012) have incorporated the method in a MCRT scheme for neutrino transport, Wollaeger et al. (2013), Wollaeger and van Rossum (2014) have developed a MC tool for RT in SNe based on IMC and recently Roth and Kasen (2015) have included IMC into the MCRT code Sedona (Kasen et al. 2006) and demonstrated its utility in onedimensional radiation hydrodynamical calculations.
The stability benefit of IMC does, however, come at a cost and some of the less desirable features of this technique should not go unmentioned. In general, the construction of the governing IMC equations introduces a time discretization error which is formally of \(\mathcal {O}(\varDelta t)\). As a consequence, the scheme becomes increasingly inaccurate as the time step becomes larger. Moreover, Wollaber (2016) cite the socalled maximum principle violation which can occur within IMC calculations as its main weakness. Here, temperatures within a computational domain can nonphysically exceed the imposed boundary temperatures in the absence of internal sources. Larsen and Mercer (1987) formulate a time step constraint under which these violations may be avoided. However, these conditions are very restrictive and limit the applicability of IMC. More information about the maximum principle violation, and about efforts to alleviate it within the IMC framework as well as other drawbacks, such as accurately reproducing the diffusion limit, the introduction of damped oscillations or teleportation errors, are summarized by Wollaber (2016).
Finally, we note that the linearisation, semiimplicit recasting and discretisation proposed by Fleck and Cummings (1971) and reviewed here constitutes only one possibility to improve numerical stability. The recent review by Wollaber (2016) provides a comprehensive overview of a number of alternative approaches. In particular, we draw attention to the family of techniques, mainly shaped by Brooks and collaborators (e.g. Brooks 1989; Brooks et al. 2005), denoted Symbolic Implicit Monte Carlo (SIMC), which leave the thermal emission term formally unknown by introducing unknown symbolic packet weights. This technique may be denoted as a truly implicit MC method in the same sense as applied in the field of solving differential equations (see Wollaber 2016).
10.2 Efficient Monte Carlo techniques in optically thick media
While conventional MC techniques are well suited for problems with a moderate or low optical depth, their efficiency decreases dramatically in optically thick applications. In a pure scattering environment, packets are frequently deflected by collisions and their propagation effectively becomes a random walk. Explicitly following and treating the multitude of interactions as is required in conventional MC approaches becomes very inefficient and computationally expensive. The situation is similar in problems with high absorption opacities. At first glance the short packet trajectories due to rapid truncation by frequent absorption events seem to argue for a efficient application of MC techniques in this regime. However, in equilibrium/steadystate problems this would need to be countered by very large numbers of quanta to describe the propagation while in explicit timedependent MC treatments, small time steps are required to ensure numerical stability (see discussion in Sect. 10.1). As detailed above, the IMC approach offers a solution to the timestep problem since it ensures unconditional stability. However, the IMC approach suffers equally in efficiency in the optically thick regime since the Fleck factor is very small in such situations and the vast majority of interactions proceed as effective scatterings.
A number of authors have developed techniques that improve the efficiency of MC calculations in optically thick regimes. These acceleration techniques replace the conventional MC transport process by a diffusion treatment that efficiently transports MC quanta through regions of high optical depth. The appropriate probabilities for these transport processes are found by a stochastic interpretation of the diffusion equation that constitutes the correct physical limit for RT processes in the presence of high opacities. Typically, these MC diffusion techniques are interfaced with a conventional, often IMC transport approach to yield a hybrid scheme that efficiently solves RT in problems with varying optical thickness. In the following, we briefly outline two popular flavours of these diffusion techniques, which predominantly differ in how the diffusion regions, in which the normal transport simulation is switched off, are treated. These are the socalled random walk or Modified Random Walk (MRW) techniques originally developed by Fleck and Canfield (1984) and the Discrete Diffusion MC (DDMC) methods (see e.g. Densmore et al. 2007, and references therein).
10.2.1 Modified random walk
The Random Walk (RW, or MRW as coined by Min et al. 2009) was developed by Fleck and Canfield (1984) as an extension to their IMC method (see Sect. 10.1) to improve the computational efficiency in applications with regions of high optical depth. The main idea underlying this approach is the introduction of spherical diffusion regions whenever the optical depth is high. Instead of following the multitude of effective scatterings in these regions with IMC, the conventional packet transport process is switched off and replaced by a diffusion procedure. Here, packets are able to traverse the diffusion regions in onestep processes. The probabilities governing this propagation mode are derived by Fleck and Canfield (1984) by examining the statistical properties of the random walk process and the solution to the diffusion equation. While the original MRW scheme has been derived for IMC applications, it naturally applies to explicit MC approaches as well after setting the Fleck factor to 1.
Recently, the MRW approach has been applied in astrophysical RT problems by Min et al. (2009) and Robitaille (2010). There, the scheme is incorporated into MC approaches to dust RT and specifically helps to transport packets through optically thick parts of dusty discs. However, it seems very challenging to adapt this scheme to applications in which complex opacities, particularly Sobolevtype line opacities, have to be taken into accounted.
10.2.2 Discrete diffusion Monte Carlo
In the MRW scheme, only spherical subregions of grid cells are designated diffusion zones. As outlined above, constraints imposed on the size of the sphere lead to efficiency problems when packets are located close to grid cell boundaries. This drawback is eliminated in other MC diffusion approaches. In socalled Discrete Diffusion Monte Carlo (DDMC) techniques, entire grid cells are treated as diffusion regions. Within, DDMC packets are generated that can traverse these cells efficiently in onestep processes. The propagation rules for this procedure are again extracted from a probabilistic interpretation of the discretized diffusion equation. In analogy to the MRW method, DDMC schemes are commonly used in hybrid approaches in combination with IMC transport techniques to ensure an efficient applicability to problems with regions of different optical thickness (see e.g. Gentile 2001; Densmore et al. 2007).
DDMC techniques have their origin in neutron transport problems (see overview by Densmore et al. 2007) but a popular variant designed for photon RT has been presented by Densmore et al. (2007). Another flavour of the diffusion technique has been developed by Gentile (2001) and is often referred to as Implicit Monte Carlo Diffusion (IMD).^{41} The main difference with respect to the DDMC approach by Densmore et al. (2007) lies in the treatment of how time is tracked by the DDMC packets. While both DDMC and IMD have originally been presented for grey problems, multigroup extensions appropriate for frequencydependent applications have already been devised, in particular by Densmore et al. (2012) and Cleveland et al. (2010) respectively.
Of the DDMC schemes, the variant of Densmore et al. (2007, 2012) seems to currently have experienced the most attention in the astrophysical community. Abdikamalov et al. (2012) have developed a hybrid DDMCIMC approach for neutrino transport in corecollapse SNe and Wollaeger et al. (2013) and Wollaeger and van Rossum (2014) have introduced a MC method for RT in SN ejecta, constructed around a DDMCIMC core. Consequently, we only focus on the DDMC scheme of Densmore et al. (2007) in the following, where we briefly highlight the guiding principles of discrete diffusion techniques. We refer the reader to Gentile (2001), Cleveland et al. (2010) and Cleveland and Gentile (2015) for details about the closely related IMD approach.
An equation similar to Eq. (139) is found for DDMC interface cells, which are at the edge of the diffusion regions, after imposing appropriate boundary conditions. Instead of relying on the Marshak boundary condition, Densmore et al. (2007) propose a condition inspired by the asymptotic diffusionlimit. This ensures an accurate behaviour of the DDMC scheme in situations in which the incoming transport packet population has a very anisotropic angular distribution too (see Densmore et al. 2007). The resulting spacediscretized diffusion equation has the same structure as for interior cells apart from an additional source term that describes the influx of radiation from the transport region (or from outside of the computational domain if the interface is at the domain edge). This source term can be converted into a probability which is sampled every time a MC packet from the transport region or from the domain boundary condition impinges onto the diffusion region to decide whether the packet is converted into a DDMC particle or reflected back. The complementary process of DDMC packets leaking out of the diffusion region is handled by placing them isotropically onto the interface. Such packets then continue propagating according to the conventional MC transport scheme.
11 MCRT and dynamics
In Sect. 9 we reviewed how estimators can be constructed to determine the rate of transfer of energy and momentum from the radiation field to the ambient medium. This transfer can become dynamically important and drastically affect the evolution of a system. In the astrophysical realm, prominent examples for such circumstances include radiatively driven mass outflows from hot stars (see review by Puls et al. 2008) or accretion discs (e.g. Proga et al. 1998, 2000; Proga and Kallman 2004), the star formation process (see review by McKee and Ostriker 2007, and references therein) or the shock outbreak phase in SNe (see e.g. overview in Mihalas and Mihalas 1984). In situations such as these, a decoupled treatment of hydrodynamics and RT is no longer accurate but a coupled RH solution approach has to be followed.
Historically, RH studies have been typically performed with deterministic solution techniques. But particularly in the field of linedriven winds from hot stars, there is a substantial literature based on MC studies by Abbott and Lucy (1985) and, among others, Lucy and Abbott (1993), Vink et al. (1999, 2000) and Müller and Vink (2008). The main motivation for relying on MC schemes certainly lies in their ease of treating the Sobolevtype line opacities encountered in these winds. Specifically, a MC calculation is used to determine the momentum deposition in the outflow material according to which a steadystate wind structure is calculated. In addition, fully dynamic RH approaches which rely on MC methods have been developed and applied. For example, Nayakshin et al. (2009) and Acreman et al. (2010) coupled Smoothed Particle Hydrodynamics (SPH) approaches with MCRT calculations. Haworth and Harries (2012) investigated triggered star formation with a RH approach in which the gas temperature is adjusted by a MCbased photoionization calculation. Harries (2015) and Harries et al. (2017) continued the development of MCbased RH methods for star formation problems. Noebauer et al. (2012) and Roth and Kasen (2015) introduced MCbased RH techniques with a generalpurpose scope, with a particular focus on IMC techniques in the latter. Implicit MC diffusion methods were coupled with hydrodynamics calculations by Cleveland and Gentile (2015). This limited list of examples illustrates that the possibility of using MCRT techniques in fully dynamic applications is actively researched and developed. In the following, we briefly sketch how energy and momentum transfer terms may be reconstructed from MCRT calculations and included in fluid dynamics calculations.
11.1 Reconstructing energy and momentum transfer terms
As an alternative to the CMFbased reconstruction approaches detailed above, the radiation force components can also be determined in the LF. A corresponding reconstruction procedure within the volumebased estimator approach was outlined by Noebauer et al. (2012).
11.2 Coupling to fluid dynamics
11.3 Example application
As originally suggested by Ensman (1994), solving the structure of radiative shocks has become a standard test problem for RH solution techniques. In these shocks, a radiative precursor emerging from the shocked domain penetrates the upstream material preheating and compressing it (for a detailed overview of these phenomena, we refer the reader to Zel’dovich and Raizer 1969). Depending on the strength of the preheating, sub and supercritical shocks are distinguished. The temperature in the precursor region remains below that of the shocked material in the subcritical case but reaches it in supercritical shocks. Thanks to the seminal works by Lowrie and Rauenzahn (2007) and Lowrie and Edwards (2008), analytic steadystate solutions are available for these shocks.
As a test of the methods, Noebauer et al. (2012) and Roth and Kasen (2015) have used operatorsplitting techniques to successfully calculate the structure of radiative shocks with MCbased RH approaches. Here, we discuss the success of these tests—further details about the physical and numerical setup of these simulations are given in Appendix A.4.
11.4 Challenges and limitations
The stiff source term problem is not unique to the MC RH problem but a general challenge when dealing with source terms in hydrodynamical calculations (cf. LeVeque 2002). A common approach to address this problem is to rely on implicit solution techniques. In this context, the IMC techniques outlined in Sect. 10.1 seem very promising. In fact, Roth and Kasen (2015) coupled an IMC RT scheme with a fluid dynamical calculation and successfully applied it to test problems in which the radiative time scales are smaller than the fluidflow time scales. Nevertheless, as stressed in Sect. 10.1, IMC methods are not truly implicit in the traditional sense and also suffer from other potential downsides, e.g. maximum principle violation (cf. Wollaber 2016).
A completely different approach to the stiff source term problem was suggested by Miniati and Colella (2007). An unsplit Godunov scheme was developed, consisting of a modified predictor and a semiimplicit corrector step which incorporates the effects of the source term. This method was adapted to RH by Sekora and Stone (2010) and Jiang et al. (2012). In principle, the hybrid Godunov approach could also be utilised in MCbased RH calculations, but a successful application of this scheme in conjunction with MCRT methods has yet to be demonstrated.
Notwithstanding the challenges, MCbased techniques constitute a valuable alternative approach to RH. Such methods offer the possibility to benefit from the same advantages that MC techniques already bring to pure RT calculations, namely a straightforward generalization to multidimensional geometries and the ease with which complex interaction processes are incorporated.
12 Example astrophysical application
We conclude this article by presenting a concrete example from our own experience of how MCRT methods can be used to solve RT problems in astrophysics. In this first version of our Living Review, we will focus on a discussion of calculating synthetic spectra for SNe Ia. This example, makes use of many techniques outlined in this review, particularly, the indivisible energy packet scheme (cf. Sect. 5.2), a variant of the macroatom scheme (cf. Sect. 7), volumebased estimators (cf. Sect. 9.3) and the peelingoff technique for variance reduction (cf. Sect. 9.4). Throughout the discussion we make use of the open source code Tardis (Kerzendorf and Sim 2014; Kerzendorf et al. 2018), which is readily available^{48} for inspection (or use) by the interested reader.
In future verstions of this Review we will plan to gradually extend our discussion of examples. In particular, we aim to summarise closely related work on the modelling of fast outflows for other classes of astrophysical sources such as hot stars and accretion disk winds (see references in Sect. 3). Such applications also make use of many of the techniques outlined in this review and are generally quite closely related to the methods used in the SN Ia example discussed here. The most important difference, arguably, is that the SN problem often requires only an homologous velocity law, which leads to a number of simplifications (see Sect. 8.2). In contrast, more general stellar/disk wind applications require that more complicated velocity fields are considered.
12.1 Type Ia supernovae
SNe Ia are transient events that have been instrumental in establishing our currently accepted cosmological standard model and are still widely used in precision cosmology (see e.g. Goobar and Leibundgut 2011). In particular, Riess et al. (1998) and Perlmutter et al. (1999) pioneered the use of SNe Ia as standardisable distance indicators to map out the recent expansion history of our Universe, finding an accelerated expansion. Apart from their relevance in cosmological studies, SNe Ia play an important role in many other branches of astrophysics as well, for example in galactic chemical evolution (e.g. Kobayashi et al. 1998; Seitenzahl and Townsley 2017). Notwithstanding the importance of SNe Ia, a full understanding of the exact nature of these transients still remains elusive and a range of proposed models remain under study (see e.g. Hillebrandt and Niemeyer 2000; Hillebrandt et al. 2013; Röpke 2017; Röpke and Sim 2018). One important strategy to study SNe Ia is to model their observed spectra with the aim of inferring the ejecta composition and structure as a means to understand the explosion itself. Tardis, which we use for this demonstration, is a tool aimed at this problem in which highly parameterized and flexible RT simulations are used to interpret observations.
MCRT methods are wellsuited for calculating synthetic observables in SNe Ia. Due to the absence of hydrogen and helium and the dominance of heavy elements in the ejecta of SNe Ia, RT is mainly driven by boundbound interactions. As a consequence, SN Ia spectra show no true continuum but rather a pseudocontinuum, generated by the flux redistribution achieved in a multitude of nonresonant line interactions. This property in combination with the fact that many models predict anisotropies in the overall morphology and chemical structure of SN Ia ejecta make MCRT an attractive choice for treating RT. Popular numerical approaches relying on MCRT for SN Ia studies include Artis (Kromer and Sim 2009), Sedona (Kasen et al. 2006), SuperNu (Wollaeger et al. 2013; Wollaeger and van Rossum 2014), Tardis (Kerzendorf and Sim 2014; Kerzendorf et al. 2018; Vogl et al. 2019), the scheme developed by Mazzali and Lucy (1993) and Mazzali (2000) and Sumo (Jerkstrand et al. 2011, 2012).
12.2 Model type Ia supernova
Since Tardis was specifically designed as a highly parameterized MCRT approach for spectral synthesis in SNe Ia, it adopts a number of simplifications. For a detailed overview we refer to the original publication by Kerzendorf and Sim (2014) and the publicly available documentation.^{49} Here, we only highlight some of the key aspects of the MCRT machinery of Tardis.
Similar to the approach by Mazzali and Lucy (1993), Tardis adopts the elementary SN model of Jeffery and Branch (1990). Here, the SN ejecta are approximated as spherically symmetric and divided into two domains, the continuumforming region and the atmosphere. A photosphere separates both regions. It is assumed that thermalization processes are only relevant below the photosphere and that interactions in the atmosphere are either electron scatterings in the Thomson limit or line interactions. Tardis follows the spectral synthesis process in the atmosphere with a timeindependent, frequencydependent MCRT approach. Packets are launched from the photosphere at the inner computational boundary from a thermal distribution according to the photospheric temperature and followed as they propagate through the envelope until escaping through either boundary. An important aspect of the Tardis approach is the determination of a selfconsistent plasma state and photospheric temperature, which is achieved using volumebased estimator techniques akin to those outlined in Sect. 9.3 in an iterative process. Only after a converged plasma state has been found, the final synthetic spectrum is calculated. Tardis includes electron scattering and boundbound interactions relying on the Sobolevapproximation (see Sect. 8.2). Fluorescence can be treated either using the downbranching scheme by Lucy (1999b) or a simplified version of the macro atom scheme by Lucy (2002, 2003, see Sect. 7). To reduce the MC noise in the synthetic spectra, a variant of the peeloff technique can be used, referred to as virtual packet scheme (see Sect. 9.4). Different assumptions about the ioniziation and excitation state can be adopted but for the Tardis simulations presented below, a modified nebular approximation (see Mazzali and Lucy 1993) was used together with a diluteBoltzmann excitation treatment. Finally, Tardis relies on a discrete representation of the ejecta state in terms of density and velocity on a spherical grid. For each grid cell, the mass density, the velocity at the cell interfaces and the chemical composition have to be specified. Internally, perfect homology is assumed, for example when progressing through the Sobolev line interaction scheme (see Sect. 8.2).
12.3 Spectral synthesis with MCRT
13 Summary and conclusions
In this work, we provide an overview of some of the MCRT techniques used in astrophysics. We have presented a variety of evidence that this approach has evolved into a competitive and very successful method to solve radiative transfer problems. With its probabilistic approach, MCRT offers a number of compelling advantages that make this technique ideal for a variety of astrophysical applications. Whenever irregular multidimensional geometries are encountered or complex interaction processes, particularly scatterings, have to be accounted for, MCRT methods are typically a good choice for addressing radiative transfer problems. For this reason, the MCRT framework finds widespread application in astrophysics, from modelling massoutflows from stars and accretion discs, to simulating radiative transfer through dusty environments or studying ionization on cosmological scales. Recently, MCRT schemes have even been included in fully dynamic radiation hydrodynamics calculations.
Relying on MCRT approaches, however, always comes at the cost of introducing statistical fluctuations into the solution process. Nevertheless, a variety of variance reduction techniques have been developed over the years to keep this noise component under control—many of these methods have been reviewed in this work. Also, conventional MCRT approaches are illsuited for the application to optically thick environments and to problems with short cooling time scales. Extensions and modifications, particularly MC diffusion schemes and the IMC approach, have been developed to alleviate these deficiencies and have already found their application in astrophysical MCRT calculations.
Finally, we want to emphasize an important aspect of MCRT methods, the value of which should not be underrated: the MCRT approach of performing a simulation of radiative transfer by following the propagation of packets is very intuitive since it closely resembles the microphysical processes realised in nature. Furthermore, the fundamental MCRT concepts are quite simple and basic computer programs can be developed quickly with only a handful of instructions. The directness of the physics and simplicity of the algorithms also mean that it is typically fairly easy to develop codes by gradually upgrading the physics: incorporating new physical processes rarely requires any fundamental overhaul. All this, together with the fact that many stateoftheart MCRT simulation codes for astrophysical applications are open source and freely available, makes the entrance barrier quite low for the adoption of MCRT. As the continuous increase in the availability of computational resources seems to hold and since MC calculations can easily be distributed over multiple computation units, it seems more than likely that the success MCRT will continue.
Footnotes
 1.
Note, however, that this situation changes when the data structures holding for example atomic data or the computational grid become too large to fit into the memory of a single computing node. Then these data structures have to be split and communicating MC particles between threads becomes the performance bottleneck (see, e.g. Harries 2015, for more details on possible parallelization schemes for such situations).
 2.
We neglect general relativistic effects in this article.
 3.
Due to the local definition of the CMF, it is not an inertial frame (see e.g. detailed discussion of this in Mihalas and Mihalas 1984).
 4.
 5.
The resulting numbers can be mapped onto the unit interval [0, 1[ by dividing by M.
 6.
For the linear congruential methods as defined by Eq. (16), the period can at most reach M.
 7.
For the sake of brevity we will omit the attribute “pseudo”.
 8.
This is a common procedure to sample discrete probabilities (see e.g. Carter and Cashwell 1975).
 9.
We assume that \(\min \rho _{X}(x) = 0\) and \(\max \rho _{X}(x) = 1\). Otherwise, the draws for \(\xi _2\) have to be scaled and shifted appropriately.
 10.
Unpublished Lawrence Radiation Laboratory internal report, cf. Fleck and Cummings (1971).
 11.
For the moment, we neglect polarization.
 12.
Throughout this review we generally assume monochromatic packets for simplicity. Some of the techniques presented here can also be generalized to polychromatic packets (see, e.g. Steinacker et al. 2013, for more information on polychromatism).
 13.
We note that packet energies and weights are somewhat interchangeable concepts. Thus, we will make use of both terminologies in this review.
 14.
The difference between this case and the isotropic initialization of the initial radiation field is that the procedure is based on the flux in the former and on the energy density in the latter case.
 15.
Note that in the indivisible energy packet scheme proposed by Lucy (1999a) for RE applications, packets are immediately reemitted.
 16.
 17.
However, as we shall discuss later (see Sect. 9), various techniques are available to alleviate the issue of MC noise in determining rates for rare physical processes.
 18.
In such cases, the issue of how to handle the computational cost of packets with weights attenuated to the point where they may become negligible can be handled using strategies such as Russian Roulette (see Sect. 9.4).
 19.
Additional complications can arise from the frequency dependence of the opacity in fast flows (see Sect. 8).
 20.
In this illustration, we deviate from the indivisible energy packet scheme introduced by Lucy (1999a) and do not immediately reemit absorbed packets.
 21.
In a timedependent simulation, packets representing ongoing emissivity can be gradually injected during the course of a numerical time step (i.e. the time at which they are injected to the simulation is also a property to be sampled).
 22.
See Sect. 9 for further details of how such information can be optimally extracted from a MCRT simulation.
 23.
One might argue that the key property here is that the packets are indestructible rather than indivisible, but we retain the more usual name for this approach for consistency.
 24.
We make the specific rearrangement such that all terms are positive: this is to facilitate interpretation of the resulting equation in terms of energy flow probabilities.
 25.
Of course, the macro atom scheme can also be coupled to an iterative solution for the level populations to provide accurate level populations upon convergence. Depending on the problem, it may be anticipated that the use of the macro atom scheme in such an approach can aid convergence since it gives a relatively good estimate of the true emissivity even before convergence of the level populations has been achieved.
 26.
This is, of course, the standard definition for the zero of (excitation) energy—namely that the energy of the lowest lying level (ground state) is defined to be zero.
 27.
For more complicated cases involving incoherent scattering and/or departures from equilibrium, the principles discussed in Chapter 7 can all be applied to the packet energy and frequency in the CMF.
 28.
See Rybicki and Hummer (1978) for an extension to nonmonotonous flows.
 29.
It was obtained by a formal integration of the RT problem according to the scheme outlined by Jeffery and Branch (1990).
 30.
The law of large numbers states that this convergence proceeds almost surely (cf. Kalos and Whitlock 2008).
 31.
The applicability of this theorem is not a necessity. Qualitatively equivalent estimates can be derived when only weaker statements can be made about the random processes (Kalos and Whitlock 2008).
 32.
Och et al. (1998) presented reconstruction schemes which use control surfaces.
 33.
For a timedependent calculation, the appropriate \(\varDelta t\) will be the duration of the current time step. In a timeindependent/steadystate calculation, it will be the implied length of the time interval being simulated.
 34.
Note that the increase in uncertainty in the inner regions is simply a consequence of the numerical setup. Since the grid has been chosen equidistant in r and since the packets carry all the same energy \(\varepsilon \), fewer packets are spawned in inner regions.
 35.
\(\gamma \) gives the number of photoionization events per second per unit volume per photoionization target atom/ion.
 36.
Note, however, that the resulting spectrum will still vary once a different RNG seed is chosen since the source function used in the formal integration is determined within a MCRT simulation.
 37.
Indeed, Fleck and Cummings (1971) introduce the IMC approach both for grey and nongrey applications.
 38.
Fleck and Cummings (1971) point out that different timeaveraging prescriptions can in principle be chosen for the various quantities.
 39.
See e.g. Hubeny and Mihalas (2014) for definition and discussion of the Rosseland mean opacity.
 40.
Fleck and Canfield (1984) argue that the Rosseland mean free path tends to be much smaller than the Planck mean free path, which describes the typical distance between collisions.
 41.
Densmore et al. (2007) still classifies the IMD approach as a member of the class of DDMC techniques.
 42.
A crucial difference in the IMD scheme is that the timederivative in the diffusion equation is discretized by finite differences as well (cf. Gentile 2001).
 43.
I.e. assume \(\eta _{\nu } = \chi _{\mathrm{a}} B_{\nu }\).
 44.
Note, however, that the applicability of the Sobolev approximation (1960) to line opacity is assumed in these radiation force estimators.
 45.
A Python implementation for this task can be found at https://github.com/unoebauer/publicastrotools.
 46.
To \(\mathcal {O}(v/c)\).
 47.
The Courant condition essentially limits the duration of a simulation time step relative to the gridcell crossing time for the characteristic fluid waves.
 48.
The code can be obtained from https://github.com/tardissn/tardis.
 49.
 50.
This subluminous SN Ia belongs to a peculiar subclass of these transients, which is named after the prototypical event, SN 1991bg. SN 2005bl is wellstudied and a spherically symmetric approximation to its ejecta structure has been previously estimated by Hachinger et al. (2009) using the abundance tomography method developed by Stehle et al. (2005).
 51.
This limitation has only bookkeeping reasons. There are no conceptual obstacles to record and diagnose the entire interaction histories of all packets.
 52.
Note that this test is performed without a specific frequency association.
 53.
 54.
Ensman (1994) found it necessary to include this slight gradient to avoid numerical problems in their calculations.
 55.
We use a Python implementation of the solution strategy which is available at https://github.com/unoebauer/publicastrotools.
Notes
Acknowledgements
We dedicate this work to the memory of Leon B. Lucy who, with his many invaluable contributions, had a profound influence on the development of Monte Carlo radiative transfer techniques in astrophysics. One of us (SAS) had the privilege of working with LBL during his time at Imperial College London and wishes to acknowledge the large number of insightful discussions with LBL. We both (UMN/SAS) wish to thank Knox S. Long who has, for over a decade, been a collaborator and sounding board for many projects. UMN first came into contact with Monte Carlo radiative transfer when working with KSL, an experience that instilled a fascination for the subject that ultimately lead to the development of this work. Nick Higginbottom, Wolfgang Kerzendorf, Christian Knigge, James Matthews, Jorick Vink and Christian Vogl are thanked for many fruitful discussions on topics included in this review. We are also very grateful to Rémi Kazeroni and Jérôme Guilet for their help with the original publications in French that were used for the historical sketch of the Monte Carlo approach. Finally, we would like to express our sincere thanks to Markus Kromer and Wolfgang Hillebrandt for their thoughtful comments and suggestions in the prepration of this review, and for many interesting and productive collaborations over the years.
Supplementary material
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