The Limits of Line Broadening: Modeling Stellar Spectra and Formation Temperatures at High Resolution

Technical Summary: The Limits of Line Broadening: Modeling Stellar Spectra and Formation Temperatures at High Resolution

Problem Statement
The modeling of stellar spectra is fundamental to various astronomical subdomains, including galactic archeology, stellar population synthesis, and radial velocity (RV) exoplanet science. Conventionally, the shapes of absorption lines in model spectra are generated by convolving thermal profiles (derived from model atmospheres and line lists) with broadening kernels intended to account for non-thermal effects such as stellar rotation, microturbulence, and macroturbulence.

However, this paper argues that the assumptions permitting this convolution approach can break down at the high spectral resolving powers ($R \gtrsim 10^5$) achieved by modern Extremely Precise Radial Velocity (EPRV) spectrographs. Specifically, the reliance on analytical convolution approximations—often assuming solid-body rotation, achromatic limb darkening, and neglecting center-to-limb variations—can introduce appreciable errors in modeled flux. Furthermore, the paper identifies inconsistencies in the literature regarding the definition of "formation temperatures" (specifically $T_{1/2}$), noting that previous works have often conflated intensity contribution functions with flux contribution functions, leading to physically inaccurate interpretations of where spectral features form in the stellar atmosphere.

Methodology
The author employs a rigorous radiative transfer framework to compare two methods of generating broadened stellar spectra:

1. Convolution Approximation: The standard approach where the emergent flux is calculated via a simplified integral (using the second exponential integral, $E_2$) and subsequently convolved with broadening kernels (e.g., the T. Hirano et al. 2011 kernel) that incorporate rotation and radial-tangential macroturbulence.
2. Explicit Disk Integration: A more computationally expensive method where the flux is calculated by explicitly integrating the local intensities over the visible stellar hemisphere. This approach incorporates local variations in rotational velocity and macroturbulent broadening at each point on the disk, accounting for center-to-limb variations without relying on the assumption that intensity profiles are invariant across the disk.

The study utilizes MARCS model solar atmospheres and the Korg.jl spectral synthesis code. The analysis focuses on specific Fe I lines (e.g., at 6301.5 Å and 6302.5 Å) and varies parameters including projected rotational velocity ($v \sin i$), radial-tangential macroturbulence ($\zeta_{RT}$), and spectral resolving power.

To address the formation temperature issue, the paper defines and compares:

• Intensity Contribution Functions: The integrand of the emergent intensity equation, dependent on the viewing angle $\mu$.
• Flux Contribution Functions: The integrand of the emergent flux equation, which effectively represents a weighted disk average of intensity contributions.

Key Contributions

1. Quantification of Convolution Errors: The paper demonstrates that the convolution approximation introduces flux errors that scale with rotational velocity ($v \sin i$). For stars with high $v \sin i$, these errors can reach several percent of the continuum flux, a magnitude significant for EPRV science. While instrumental broadening (lower spectral resolution) can "blur out" these errors, they remain appreciable at the resolving powers of modern instruments.
2. Distinction Between Intensity and Flux: The author clarifies that using the intensity contribution function (evaluated at disk center, $\mu=1$) to calculate formation temperatures for unresolved stars is physically incorrect. The flux contribution function, which accounts for the entire visible disk, yields systematically cooler formation temperatures. The difference is not a simple offset but alters the shape of the formation temperature spectrum, particularly in line cores.
3. Impact of Broadening on Formation Temperatures: The study shows that while the error introduced by using convolution vs. integration for formation temperatures is relatively small (tens of Kelvin) compared to the error from using intensity vs. flux (hundreds of Kelvin), proper treatment of rotation and macroturbulence is still necessary for precision.
4. Software Release: The author provides FormationTemperatures.jl, a Julia package wrapping Korg.jl, to facilitate the computation of model line contribution functions and formation parameters that properly account for these effects.

Results

• Flux Errors: Figure 2 and Figure 3 show that flux errors increase with $v \sin i$. At $v \sin i \approx 10-16$ km s$^{-1}$, errors can exceed 3-4% of the continuum. These errors persist at resolving powers $R > 10^4$, which is the regime of modern EPRV instruments.
• Formation Temperature Discrepancies: Figure 6 illustrates that formation temperatures derived from the disk-center intensity contribution function are significantly hotter than those derived from the flux contribution function. The difference is not uniform; it varies across the spectral line profile.
• Contribution Function Diversity: Figure 9 demonstrates that spectral elements sharing the same formation temperature ($T_{1/2}$) do not necessarily originate from the same physical regions of the atmosphere. Their contribution functions can have markedly different distributions over height and velocity, suggesting that $T_{1/2}$ is a summary statistic that can obscure underlying radiative transfer complexities.
• Parameter Space: Figure 8 indicates that flux errors are most prevalent in stars with higher effective temperatures ($T_{\text{eff}} \gtrsim 6000$ K) and higher rotational velocities, corresponding to the Kraft break where magnetic braking becomes inefficient.

Significance and Claims
The paper claims that the assumptions inherent in standard convolution-based spectral synthesis are insufficient for the precision required by current and future EPRV surveys. The author emphasizes that:

• The concept of a singular "formation temperature" is a simplification that can oversimplify the realities of radiative transfer, as light emerges from an extended region of the atmosphere rather than a single layer.
• Researchers must distinguish between intensity and flux when modeling unresolved stars to avoid artificially hot formation temperatures.
• While the convolution method is computationally efficient, it should be avoided in regimes where high precision is required and where $v \sin i$ is significant, as the resulting flux errors can bias the interpretation of stellar variability and line shapes.

The work serves as a cautionary analysis for the community, urging a reconsideration of modeling assumptions in high-resolution spectroscopy and providing tools to implement more physically rigorous calculations.