The historical arc of the field is one of steady, incremental extension into adjacent problems. The first synthetic supernova spectrum was presented by David Branch in 1980, who adapted the Schuster-Schwarzschild approach from stellar atmospheres and introduced the Sobolev formalism for line transfer. His tools, which eventually became the widely used SYNOW code, allowed detailed comparisons between white dwarf explosion models and observations of Type I supernovae. In the same year, Timothy Axelrod’s doctoral thesis laid the foundation for nebular-phase modeling, establishing the treatment of non-thermal and non-local thermodynamic equilibrium (NLTE) physics that still sets the standard for many codes today. The explosion of SN 1987A then drew new workers into the field, including Claes Fransson in Stockholm and Leon Lucy in London, whose Monte Carlo techniques would prove transformative.

Around 2005, a wave of development swept through the field, much of it inspired by Lucy’s work on Monte Carlo methods. Three-dimensional codes operating under the assumption of local thermodynamic equilibrium (LTE) appeared, including SEDONA and ARTIS, while one-dimensional NLTE codes such as SUMO, NERO, and CMFGEN were developed for higher-fidelity spectral calculations. The first public Schuster-Schwarzschild code, TARDIS, was released in 2014 and remains popular for rapid line identification. When the first kilonova, AT2017gfo, was discovered in 2017 following the gravitational-wave detection GW170817, the supernova toolkit was ready to be adapted. Metzger and colleagues extended SEDONA to kilonovae as early as 2010, and codes such as SuperNu, POSSIS, and a kilonova version of SUMO followed, opening the door to NLTE modeling of neutron star merger ejecta.

The review emphasizes that astrophysics grows by small extension steps into related areas, a process that takes decades. Sometimes growth in computing power drives these steps, but more often they are application-driven. This has a practical consequence: methods derived under computing constraints orders of magnitude stricter than today’s may carry simplifications that are no longer necessary. Studying the history of a code or methodology therefore reveals where modern computing power can be exploited to improve upon legacy approximations, a point the author urges researchers to keep in mind whenever they adopt an existing model.

Why invest such effort in spectral synthesis? The review identifies three main science drivers. First, identifying which elements produce which line features and determining the elemental abundances in the ejecta. For decades this seemed almost insurmountable, because supernovae blend lines through rapid Doppler expansion, cascade energy over six orders of magnitude, and involve non-thermal effects, asymmetries, molecules, and dust. Today, clear diagnostic methods exist for seventeen elements from hydrogen to nickel, and direct full-ejecta spectral tests of explosion models have been performed for all major supernova classes. Second, understanding the origin of the elements across the periodic table: with AT2017gfo, the upper two-thirds of the table opened for direct analysis, with diagnostic potential already established for elements including strontium, yttrium, tellurium, lanthanum, cerium, neodymium, and tungsten. Third, determining the progenitor systems and explosion mechanisms, since spectra diagnose both the hydrostatic life of the progenitor star and the physics of its death, from neutrino-driven core collapse to accretion disk outflows near black holes.

The physical situation these codes must capture is extreme. The ejecta expand homologously, with faster fragments outrunning slower ones, at characteristic velocities of roughly 5,000 kilometers per second for core-collapse supernovae, 8,000 for thermonuclear supernovae, and 50,000 for kilonovae. Kilonovae carry about one hundred times less mass than supernovae but comparable kinetic energy, hence their higher speeds and their rise and decline over just days. Both classes are powered by radioactivity: supernovae chiefly by the decay of nickel-56 and cobalt-56, kilonovae by a vast array of radioactive r-process nuclides whose energy is carried mainly by leptons rather than gamma rays. Because the ejecta are semi-transparent, with line opacity producing lingering radiative transfer effects for years or even decades, much of the optical and infrared spectra of both supernovae and kilonovae is formed by fluorescence.

Temperature emerges as the most fundamental quantity in the modeling. It governs ionization and excitation, sets the regime of spectral formation, and strongly influences opacity. The governing law is energy conservation, expressed either through the internal energy of the gas or, equivalently in NLTE treatments, through the evolution of thermal kinetic energy balanced by heating and cooling. Codes differ in whether they retain the time-derivative terms: STELLA, SuperNu, CMFGEN, and, since 2022, SUMO can solve the time-dependent energy equation with implicit or semi-implicit finite-difference schemes, typically using time steps of about ten percent, which is well matched to both the homologous expansion timescale and radioactive decay timescales. The review shows that this choice of stepping has a solid physical anchoring for both supernova and kilonova applications across all conceivable epochs.

The treatment of line cooling exposes a subtle but consequential divide between LTE and NLTE approaches. In LTE codes, a parameterized thermalization probability is often assigned to line absorptions, and surprisingly, realistic light curves and spectra require this probability to be near unity, effectively mimicking the fluorescence process that the simplified treatment omits. But when radioactivity dominates the heating, such an inflated absorption coefficient can drive temperature estimates a factor of ten too low, likely explaining why LTE codes yield significantly lower temperatures than NLTE ones from quite early phases. In NLTE, cooling is computed from net electron collision rates, which represent the small difference between two large terms and demand careful formulation. A code comparison for a simple Type Ia test model shows that even state-of-the-art tools produce temperature profiles that vary considerably, underscoring how much work remains to achieve robust temperature determinations.

Solving the NLTE rate equations presents its own numerical challenges. With potentially more than one hundred thousand levels per grid cell, the full system is split into blocks, typically one per ion, and solved iteratively. Matrix storage scales with the square of the system size and inversion time with its cube, so the standard LAPACK routine DGESV, based on LU decomposition, remains efficient only up to a few hundred levels. Superlevels, which bundle groups of levels assuming LTE distributions within them, offer dramatic savings: in one full-composition Type Ia model, 10,605 levels were grouped into 2,338 superlevels, saving a factor of sixteen in storage and up to sixty-four in matrix inversion time. Negative populations, false singularities, and oscillating Newton-Raphson steps all require layered fall-back strategies, and co-solving all ions of an element, as done in recent 3D codes, reduces such pathologies.

Radioactive powering completes the triad of central modeling blocks. High-energy decay particles transfer their energy through ionization, excitation, and heating, with the Bethe stopping-power formula, derived in the early 1930s, describing the continuous energy loss of fast particles. Gamma rays Compton-scatter to create primary electrons of 0.1 to 1 megaelectronvolt, which cascade into secondaries; the distribution of these secondaries, measured experimentally by Opal and colleagues, feeds into steady-state Boltzmann solutions of the degradation spectrum. For kilonovae, time-dependent thermalization effects become important within weeks, and modeling the powering has already allowed constraints to be placed on the composition of AT2017gfo. Looking forward, the review charts a roadmap: better physical treatments, accurate atomic data for trans-iron elements, 3D hydrodynamic explosion models, and high-quality observations from the ultraviolet to the mid-infrared. Eighteen of the thirty lightest elements now have good or moderate diagnostic potential, and with kilonova spectroscopy advancing rapidly, the goal of determining the origin of the elements directly at their production sites is coming within reach.

Subject of Research: Computational spectral synthesis modeling of supernova and kilonova spectra to infer ejecta composition and explosion physics

Article Title: Spectral synthesis techniques for supernovae and kilonovae

Article References: Jerkstrand, A. (2025). Spectral synthesis techniques for supernovae and kilonovae. Living Reviews in Computational Astrophysics, 11(1), Article 1. https://doi.org/10.1007/s41115-025-00022-2

Image Credits: AI Generated

DOI: 10.1007/s41115-025-00022-2

Keywords: supernovae, kilonovae, spectral synthesis, radiative transfer, NLTE modeling, r-process nucleosynthesis, Monte Carlo methods, neutron star mergers, radioactive powering, astrophysical transients, computational astrophysics, origin of the elements

Radiation transfer is, in essence, the calculation of how emitted light travels through a medium before reaching an observer. In a simplified one-dimensional model, an atmosphere can be represented as a stack of horizontal layers whose properties vary smoothly with depth. Real atmospheres are less orderly. Stars contain rising hot granules and sinking cooler material, magnetic structures, winds, and large-scale convective cells. Exoplanets have day-night temperature contrasts, circulation, clouds, and externally supplied stellar radiation. Kilonovae consist of rapidly expanding, chemically complex ejecta whose geometry and physical state evolve. A three-dimensional model retains variations in all spatial directions, while a time-dependent treatment can follow changes in the gas rather than assuming a permanent steady state.

The NLTE part addresses a second limitation of standard modeling. Local thermodynamic equilibrium assumes that collisions dominate the internal energy distribution of atoms and molecules, allowing their populations to be estimated with Saha-Boltzmann statistics from the local temperature and density. That assumption often fails near an open surface, where photons escape and the radiation field can control excitation and ionization. In NLTE calculations, the population of each energy level is determined by balancing radiative and collisional transitions. The radiation field affects those populations, while the populations in turn alter the opacity and emissivity that shape the radiation field. Solving the problem therefore requires repeated, coupled calculations rather than a single local evaluation.

At the center of the calculation is the radiative-transfer equation for the specific intensity, the amount of radiation traveling in a particular direction at a particular frequency. In a common time-independent form, the change in intensity along a ray depends on the difference between the intensity and the source function, which is the ratio of emissivity to extinction. The calculation integrates this relationship over optical depth, a measure of how opaque the material is. In three dimensions, the code must trace many rays through a spatial grid, interpolate temperature, density, velocity, opacity, and source-function values onto each photon path, and then combine the directional intensities to obtain the mean radiation field. That mean field enters the rate equations governing atomic and molecular populations.

The review emphasizes that numerical choices can influence the answer as much as the underlying physics. Long-characteristics methods follow rays through an entire model and preserve sharp spectral structures well, but they can be expensive. Short-characteristics methods connect neighboring layers and are easier to parallelize, although interpolation can diffuse intense beams. Higher-order interpolation can improve accuracy but may create artificial overshoots, including unphysical negative opacities or intensities. Monotonic schemes suppress those artifacts but may sacrifice accuracy or complicate convergence. The angle quadrature, which selects and weights the rays used to integrate the radiation field, also matters. Relatively few directions may be sufficient for calculating mean intensities in statistical-equilibrium equations, whereas emergent line profiles and centre-to-limb variations generally require more.

One practical compromise is the so-called 1.5D approach. Each vertical column in a three-dimensional atmospheric simulation is treated as an independent one-dimensional atmosphere, preserving the local temperature, density, and velocity structure but ignoring horizontal radiation exchange between columns. The resulting fluxes can then be averaged across the model. This approach can greatly reduce computational demands and has produced useful results for several stellar problems, especially when the photon mean free path is short. However, it is not universally reliable. In extremely metal-poor stars, for example, the review describes cases in which 1.5D and full 3D NLTE calculations produced abundance differences as large as 0.22 dex for iron lines. The approximation must therefore be tested against the specific diagnostic and physical regime.

The scientific payoff is clearest in stellar spectroscopy. Convection gives spectral lines distinctive asymmetric shapes and Doppler shifts because rising and sinking gas contribute different amounts of light at different velocities. Three-dimensional models reproduce observed line bisectors more successfully than traditional hydrostatic one-dimensional models in several comparisons, while predicted convective shifts can reach hundreds of metres per second. Such effects matter for radial-velocity measurements, chemical-abundance studies, and efforts to separate stellar surface variability from planetary signals. Centre-to-limb observations provide another stringent test. For the solar oxygen line near 7772 angstroms, the review reports that 1D LTE models can overestimate the inferred abundance by 0.6 dex when the limb is analyzed, whereas 3D NLTE calculations offer a way to account for the changing geometry and radiation field.

These corrections extend directly to exoplanet research. During a transit, a planet blocks different portions of its host star, each with its own brightness, velocity, magnetic activity, and spectral-line shape. The resulting Rossiter-McLaughlin signal can reveal the projected alignment between stellar rotation and the planetary orbit, but it can also be distorted by inaccurate models of the stellar surface. Studies summarized in the review find that 3D NLTE treatment improves some diagnostics, including those based on sodium and potassium lines, although other features remain to be explored. Three-dimensional and NLTE methods are also being adapted to irradiated planetary atmospheres, where the host star supplies an external radiation field and can drive photoionization, photodissociation, heating, and atmospheric escape. Clouds and global circulation add further spatial complexity.

Kilonovae present a different but equally demanding challenge. Their spectra arise from rapidly expanding material produced in compact-object mergers, with radioactive decays supplying energy and heavy elements providing dense forests of spectral transitions. Expanding shells are often modeled with spherical symmetry and escape-probability approximations, but their composition, velocity structure, and ionization state can evolve rapidly. The review places such systems within the broader push toward time-dependent, multidimensional NLTE calculations, while noting that direct spatially resolved tests are not currently available for exoplanet or kilonova photospheres. Progress will require reliable atomic and molecular data, including transition probabilities, photoionization cross-sections, collision rates, and information for complex ions and molecules. It will also require algorithms that balance physical fidelity with the immense cost of solving millions of coupled radiation and population equations. The central message is not that every observation needs the most elaborate possible model, but that astronomers must understand when common simplifications introduce systematic errors. As high-resolution spectrographs, transit surveys, and time-domain observatories deliver increasingly precise data, 3D NLTE radiation transfer provides a framework for turning subtle spectral details into more dependable knowledge of some of the universe’s most dynamic environments.

A useful distinction in these calculations is between the radiation field inside a model and the observables ultimately compared with data. A simulation can predict an angle- and frequency-dependent specific intensity at the surface, as well as a flux integrated over directions. The intensity contains information about viewing angle and spatial structure, whereas the integrated flux provides a spectral energy distribution for the object as a whole. This difference is important for phenomena such as stellar surface inhomogeneities, where two observers may receive different line profiles from the same model depending on which regions are visible.

The transfer calculation is also only one part of a larger physical modeling chain. A model must first specify or compute the state of the gas, including quantities such as density, temperature, velocity, and composition. Radiation transfer then uses detailed opacities and emissivities on a finer frequency grid to produce a more realistic spectrum than the coarse radiative description used in many underlying fluid or atmosphere calculations. In NLTE work, the sequence is not strictly one-way: the radiation field changes the energy states of atoms and molecules, and those changed populations modify the opacity and emissivity. Iteration is therefore needed until the matter and radiation descriptions become mutually consistent.

The relevant microscopic data can be a major source of uncertainty. The review stresses the need for atomic and molecular information alongside fluid dynamics, statistical mechanics, and energy transport. Rates for radiative and collisional processes determine how strongly particles respond to the radiation field, while the available transitions establish which frequencies can absorb or emit. These inputs become especially consequential when spectra are used to infer detailed chemical abundances. A numerical solution may be internally converged yet still inherit systematic limitations from incomplete or inaccurate physical data.

Geometry is not an optional refinement in every regime. Multidimensional treatment becomes necessary when the structure encountered by a photon varies substantially across space or changes non-monotonically along its path. This criterion can apply to convective stellar surfaces, strongly irradiated atmospheres, or expanding merger ejecta. The review consequently treats 3D NLTE as a family of coupled problems rather than a single universal algorithm. Different systems demand different compromises among spatial resolution, frequency coverage, angular sampling, time dependence, and the representation of matter-radiation coupling.

These methodological issues connect radiation-transfer modeling to several broader observational programs. In stellar studies, synthetic spectra help constrain chemical evolution, convection, magnetism, and mass loss. In exoplanet work, they support interpretation of transit and atmosphere measurements. For kilonovae, evolving spectra can provide clues to the composition and physical state of rapidly changing ejecta. The review also places related applications in a wider computational landscape that includes interstellar-medium diagnostics, circumstellar polarization, dusty galaxy discs, active-galaxy accretion discs, and supernova modeling. Across these settings, the value of greater realism is measured by whether it changes an inferred physical parameter or resolves a discrepancy with observations, not simply by the number of dimensions in the calculation.

Subject of Research: Three-dimensional non-local thermodynamic equilibrium radiation transfer in astrophysical atmospheres

Article Title: 3D NLTE radiation transfer: theory and applications to stars, exoplanets, and kilonovae

Article References: Bergemann, M., & Hoppe, R. (2026). 3D NLTE radiation transfer: theory and applications to stars, exoplanets, and kilonovae. Living Reviews in Computational Astrophysics, 12(1), Article 7. https://doi.org/10.1007/s41115-026-00029-3

Image Credits: AI Generated

DOI: 10.1007/s41115-026-00029-3

Keywords: radiative transfer, 3D modeling, NLTE, stellar atmospheres, exoplanets, kilonovae, spectroscopy, astrophysical simulations, radiation, transfer, theory, applications