NLTE Radiative Transfer Models

• NLTE Radiative Transfer Models are computational frameworks that solve coupled, time-dependent equations for radiation and atomic level populations in explosive astrophysical environments.
• They integrate detailed atomic models with radiative transfer, energy, and statistical equilibrium equations to reproduce evolving spectral features and light curves in supernovae.
• The approach rigorously handles line blanketing and ionization freeze-out, offering high-fidelity constraints on progenitor properties and explosion energetics.

Non-local thermodynamic equilibrium (NLTE) radiative transfer models are computational frameworks designed to predict the emergent spectra, spectral line profiles, and multi-band light curves of astrophysical plasmas where the assumption of local thermodynamic equilibrium (LTE) breaks down. In supernovae, rapid dynamical expansion and strong radiative fields cause atomic level populations and radiation fields to become strongly coupled and time-dependent, making a full NLTE treatment essential for quantitative modeling. Contemporary NLTE models, as exemplified by the time-dependent, full-ejecta approach applied to supernova SN1987A, solve the coupled radiative transfer equation, energy equation, and statistical equilibrium equations governing atomic level populations, explicitly retaining time-dependent terms and consistently treating radiative line blanketing.

1. Time-dependent NLTE Radiative Transfer Formalism

NLTE radiative transfer models for supernovae solve the full time-dependent evolution of both the matter and radiation fields as the ejecta expand and cool. Unlike steady-state or LTE models where ionization and excitation equilibrium is assumed at each epoch, these models retain time derivatives in all coupled equations:

• Statistical equilibrium equations for the atomic level populations $n_i$ of each ion/atom,
• The energy (first law of thermodynamics) equation governing the internal energy and heating/cooling by radiation and radioactivity,
• Zeroth and first (moment) equations of the radiative transfer equation with full time dependence and frequency redistribution.

For the i-th atomic level, the Lagrangian (comoving frame) statistical equilibrium including time-dependence is

$$\rho \frac{D}{Dt}\left(\frac{n_i}{\rho}\right) = \sum_j \left( n_j R_{ji} - n_i R_{ij} \right)$$

where $R_{ij}$ and $R_{ji}$ are the total (radiative plus collisional) transition rates, $\rho$ is the mass density, and $D/Dt$ is the Lagrangian derivative. This formulation captures the dynamical non-equilibrium evolution of level populations and allows for explicit treatment of phenomena such as ionization freeze-out, which is crucial when the recombination timescale is comparable to or longer than the expansion timescale.

The energy equation is written as

$$\rho \frac{De}{Dt} - \frac{P}{\rho}\frac{D\rho}{Dt} = 4\pi \int \left(\chi_\nu J_\nu - \eta_\nu \right) d\nu + \frac{De_{decay}}{Dt}$$

where $e$ is the internal energy per unit mass, $P$ is pressure, $\chi_\nu$ and $\eta_\nu$ are opacity and emissivity at frequency $\nu$, $J_\nu$ is mean intensity, and $e_{decay}$ accounts for radioactive energy deposition.

The radiative transfer is solved using moment equations (in spherical symmetry, for homologously expanding ejecta) such as

$$\frac{1}{c r^3} \frac{D(r^3 J_\nu)}{Dt} + \frac{1}{r^2} \frac{\partial (r^2 H_\nu)}{\partial r} - \frac{\nu V}{r c}\frac{\partial J_\nu}{\partial \nu} = \eta_\nu - \chi_\nu J_\nu$$

where $J_\nu$ is the zeroth moment (mean intensity), $H_\nu$ the first moment (flux), $V$ the local velocity, and $r$ is the radius.

2. Atomic Kinetics and Ionization Freeze-Out

The explicit treatment of time-dependent statistical equilibrium in NLTE models allows for rigorous modeling of ionization freeze-out. In the outer ejecta, where the expansion timescale rapidly exceeds recombination and cooling timescales, recombination of hydrogen, helium, and metals cannot keep pace with dilution and expansion:

• Populations remain more highly ionized than any steady-state equilibrium solution would predict,
• This non-equilibrium ionization sustains strong hydrogen Balmer and helium lines even at lower temperatures,
• Synthetic spectra reproduce observed features (e.g., the He I lines in SN1987A) using realistic, unmixed progenitor compositions without invoking artificially enhanced helium abundances.

The integration of such time-dependent effects is especially important in the early post-explosion ($<1\,\mathrm{d}$) and photospheric epochs ($\lesssim 3$ weeks) when line strengths and profiles rapidly evolve due to recombination, cooling, and radiative transfer.

3. Line Blanketing, Atomic Models, and Opacity Treatment

Accurate NLTE radiative transfer modeling of supernovae requires sophisticated handling of line blanketing — the collective absorption and reemission of radiation by a vast number of overlapping spectral lines, especially from iron-group and CNO elements. The approach involves:

• Large, detailed atomic models including hundreds of levels and thousands of bound-bound transitions for key species (e.g., iron, nickel, cobalt),
• Direct inclusion of all relevant line opacities and emissivities in the statistical equilibrium, radiative transfer, and energy equations,
• Self-consistent calculation of the local radiation field $J_\nu$ and photoionization rates that evolve as ionization stages shift (e.g., Fe V–VII to Fe II–IV) during cooling.

The radiative impact of blanketing is directly encoded through wavelength-dependent opacities:

• In the earliest phases ($<$1 d), the spectral energy distribution peaks in the far-UV ($300-2000$ Å) and is heavily suppressed by line blanketing from highly ionized metals and CNO,
• As recombination proceeds and Fe transitions dominate in the UV and optical, blanketing shifts and modulates both the emergent flux and the energy balance in the ejecta.

4. Direct Comparison with Observational Data

Application of NLTE radiative transfer models to SN1987A demonstrates both the accuracy and physical completeness of the approach:

• Input hydrodynamical ejecta models specify density, velocity, and composition as a function of radius; no photospheric or "core" boundary conditions are imposed,
• Synthetic spectra and broad-band (BVRI) light curves match observed optical fluxes to within 10–20% throughout 0.27 to 20.8 days after explosion,
• The synthetic H α line profile (width, shape, absorption/emission balance) closely follows observational data,
• UV flux is somewhat overpredicted (by tens of percent), but the trend of rapid fading due to strong blanketing is correctly captured,
• Historical difficulties with the strength of He I and H I lines disappear when time-dependent and full NLTE effects are included, validating a blue supergiant progenitor composition and obviating the need for ad hoc composition adjustment.

5. Sensitivities, Limitations, and Physical Implications

The self-consistent, time-dependent NLTE approach enables critical astrophysical inference:

• Progenitor properties (e.g., radius, envelope mass, metallicity) and explosion energetics (velocity profiles) are tightly constrained by spectral and light curve fits,
• The technique provides sensitivity to ejecta structure, thermalization depths, and mixing of radioactive isotopes,
• Realistic modeling of line blanketing and ionization freeze-out ensures reliable forward inference without parameter degeneracies common in LTE or steady-state models.

Limitations include:

• Computational cost, due to the full time-dependent solution and the need for detailed atomic data,
• The restriction to 1D spherical symmetry in this formulation, though the physical robustness allows extension to multi-dimensional or hydrodynamical input with sufficient computational resources,
• The greater mismatch in the faintest UV regions, attributable in part to atomic data uncertainties and possible 3D effects or asymmetric mixing.

6. Summary Table of Key Formal Components

Equation/Component	Description	Role in NLTE Modeling
$$\rho \frac{D}{Dt}(\frac{n_i}{\rho}) = \sum_j (n_j R_{ji} - n_i R_{ij})$$	Time-dependent level populations	Captures recombination/ionization lags; crucial for freeze-out
$$\rho \frac{De}{Dt} - \frac{P}{\rho}\frac{D\rho}{Dt} = 4\pi \int (\chi_\nu J_\nu - \eta_\nu)d\nu +(De_{decay}/Dt)$$	Energy equation	Tracks thermal state evolution, including radioactive input
$$(1/cr^3)\frac{D(r^3 J_\nu)}{Dt} + (1/r^2)[\partial (r^2 H_\nu)/\partial r]- (\nu V/(rc))(\partial J_\nu/\partial\nu) = \eta_\nu - \chi_\nu J_\nu$$	Zeroth-moment transfer	Radiation field and spectral energy density evolution
Large atomic models + full line blanketing	Atomic data and opacities	Condition emergent flux and capable of reproducing observed SED

7. Broader Impact and Future Directions

The NLTE, time-dependent radiative transfer approach presented for supernovae provides a paradigm for modeling rapidly evolving, dynamically complex astrophysical plasmas. The high-fidelity agreement with observables — in both line profiles and broad-band fluxes — validates the underlying hydrodynamic explosion models and constrains both progenitor properties and explosion mechanisms. The full-ejecta, first-principles approach adopted here forms the quantitative foundation for:

• Deriving the pre-supernova evolution histories of massive stars,
• Disentangling the roles of expansion, recombination, and radioactive heating in shaping observed supernova spectra and light curves,
• Extending models to other core-collapse and thermonuclear supernova types, facilitating systematic population studies.

Future development may include generalization to multi-dimensional or magnetohydrodynamic ejecta, refinement of atomic data for improved UV predictions, and incorporation of additional physics such as polarization or asphericity, enabled by the same robust NLTE, time-dependent formalism.