4.2 Stellar atmosphere models and opacity

Stellar Opacity and Atmosphere Models

Stellar opacity governs how radiation moves through a star. It determines whether energy escapes freely or gets trapped, shaping everything from a star's temperature profile to the spectrum we observe from Earth. Atmosphere models then use opacity as a key input to predict emergent spectra and extract stellar parameters from observations.

Concept of Stellar Opacity

Opacity measures a material's resistance to the flow of radiation. In a stellar context, it tells you how far a photon can travel before being absorbed or scattered. High opacity means photons get stuck; low opacity means they stream through.

The opacity coefficient $\kappa$ quantifies absorption and scattering per unit mass, with units of $\text{cm}^2/\text{g}$. You can think of it as an effective cross-section per gram of stellar material.

Closely related is optical depth $\tau$, which tracks how transparent a column of material is along a path $s$:

$d\tau = -\kappa \rho \, ds$

where $\rho$ is the mass density. A region with $\tau \gg 1$ is optically thick (opaque), while $\tau \ll 1$ means optically thin (transparent). The photosphere, the surface you "see" when you observe a star, sits roughly where $\tau \approx 1$.

Why does opacity matter so much?

• It controls the temperature gradient through the atmosphere, directly shaping the emergent spectrum.
• It determines whether energy transport in a given layer is radiative or convective. Where opacity is high enough that radiation can't carry the flux efficiently, convection takes over (as described by the Schwarzschild criterion).
• It sets evolutionary timescales: higher interior opacity traps energy longer, affecting how quickly a star evolves through different phases.

Sources of Atmospheric Opacity

Opacity doesn't come from a single process. Several physical mechanisms contribute, and which ones dominate depends on temperature, density, and composition.

Bound-bound transitions produce discrete absorption lines when an electron jumps between bound energy levels in an atom or ion. The Hydrogen Balmer series is the classic example, but thousands of metal lines (Fe, Ca, Mg, etc.) collectively contribute significant line blanketing, redistributing flux from UV wavelengths to longer ones.

Bound-free transitions (photoionization) occur when a photon has enough energy to eject a bound electron entirely. These produce continuous opacity with sharp edges at threshold wavelengths. The negative hydrogen ion ($\text{H}^-$) is the single most important continuous opacity source in solar-type stars. It forms when a neutral hydrogen atom captures a free electron; its low binding energy (~0.75 eV) means it absorbs efficiently across the visible and near-infrared.

Free-free transitions (Bremsstrahlung) happen when a free electron gains or loses energy while passing near an ion. This process produces continuous opacity and becomes increasingly significant in hot, highly ionized plasmas.

Electron scattering dominates in the hottest stellar atmospheres (O and early B stars), where most atoms are fully ionized and bound-state processes are unavailable. Thomson scattering is the low-energy limit, where the photon energy is much less than the electron rest mass energy. At higher photon energies, Compton scattering becomes relevant, and the photon transfers some energy to the electron.

Molecular opacity is crucial for cool stars (spectral types K and M). Molecules like TiO, $\text{H}_2\text{O}$, and CO have dense forests of rotational-vibrational bands that can dominate the spectrum entirely, making cool-star atmospheres far more complex to model.

Rayleigh scattering contributes in the UV and blue-visible regions of cool stellar atmospheres, where photon frequencies are well below electronic resonance frequencies of atoms and molecules.

Role of Atmosphere Models

Atmosphere models are the bridge between stellar interior theory and what you actually measure with a telescope. Their central task is solving the radiative transfer equation, which describes how radiation intensity changes as it propagates through material that absorbs, emits, and scatters:

$\cos\theta \, \frac{dI_\nu}{d\tau_\nu} = I_\nu - S_\nu$

where $I_\nu$ is the specific intensity, $\tau_\nu$ is the monochromatic optical depth, $\theta$ is the angle from the surface normal, and $S_\nu$ is the source function. Under LTE, the source function equals the Planck function $B_\nu(T)$.

A model atmosphere typically solves this equation at many frequencies across many depth points, enforcing constraints like radiative equilibrium (total flux is conserved) and hydrostatic equilibrium (pressure supports against gravity). The output is a predicted emergent spectrum.

From this, atmosphere models let you:

• Determine fundamental stellar parameters such as effective temperature $T_\text{eff}$, surface gravity $\log g$, and metallicity $[\text{Fe/H}]$ by fitting observed spectra to synthetic ones.
• Perform chemical abundance analysis by matching observed line strengths to model predictions, yielding element-by-element compositions.
• Construct stellar population models for galaxies and clusters, where libraries of synthetic spectra are combined to interpret integrated light.
• Provide boundary conditions for stellar evolution calculations and help interpret asteroseismological data.

Widely used model grids include ATLAS (Kurucz), which computes plane-parallel LTE models, MARCS, which is particularly strong for cool stars with molecular opacity, and PHOENIX, which handles non-LTE effects and extended atmospheres.

Limitations of Atmosphere Models

No model captures the full complexity of a real stellar atmosphere. Understanding where models break down is just as important as knowing how to use them.

LTE assumption. Most classical models assume Local Thermodynamic Equilibrium, meaning the source function equals the Planck function at the local temperature. This works well deep in the atmosphere where collisions dominate, but breaks down in the low-density outer layers where radiation fields decouple from local conditions. Non-LTE (NLTE) effects are particularly important for strong lines, UV wavelengths, and low-metallicity stars, but NLTE calculations require extensive and often incomplete atomic data.

Plane-parallel geometry. Standard models treat the atmosphere as a series of infinite parallel layers. This is reasonable for main-sequence and dwarf stars, where the atmosphere is thin compared to the stellar radius. For giants and supergiants with extended atmospheres, spherical geometry becomes necessary.

Static, 1D structure. Classical models assume a time-independent, one-dimensional atmosphere. Real atmospheres have:

• Convection that produces granulation patterns (the Sun's surface shows ~1000 km granules)
• Stellar winds that carry mass away, especially in hot massive stars
• Pulsations that periodically change the atmospheric structure

Full 3D hydrodynamical models (e.g., Stagger-grid, CO5BOLD) now exist and show that 1D models can introduce systematic errors in derived abundances, sometimes at the 0.1–0.3 dex level.

Other missing physics. Magnetic fields are usually excluded despite their importance for active stars and starspots. Chromospheres and coronae are omitted, limiting the modeling of emission lines and high-energy phenomena. In the coolest stars, dust formation and its feedback on opacity remain difficult to treat self-consistently. The common practice of parameterizing unresolved velocity fields as microturbulence (a single free parameter $\xi_t$) is a known oversimplification of what are really complex, scale-dependent motions.