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

TL;DR

The work demonstrates that at very high spectral resolution, fluxes modeled by convolution with broadening kernels can diverge from disk-integrated flux, especially for stars with large $v\sin i$, due to center-to-limb and anisotropic effects. It reframes formation temperatures by emphasizing the distinction between intensity and flux contribution functions, showing that proper treatment of rotation and macroturbulence can shift and broaden the atmospheric regions contributing to line formation. The authors quantify the errors introduced by convolution across representative stellar samples and caution that formation temperatures are a simplified, model-dependent summary rather than exact physical depths. They deliver FormationTemperatures.jl to facilitate accurate, disk-integrated calculations and highlight the need for more sophisticated treatments (e.g., 3D NLTE) in regimes where the convolution assumption breaks down. Overall, the paper elevates awareness of the limitations of standard broadening models and the implications for interpreting high-resolution stellar spectra and RV variability.

Abstract

The modeling of stellar spectra is pervasive in astronomy. Conventionally, the shapes of absorption lines are modeled by convolving thermal profiles (computed given some model stellar atmosphere and line list) with broadening kernels intended to account for the effects of rotation and other nonthermal sources of broadening (i.e., macroturbulence). Here, we show that the assumptions that permit this convolution can break down at high spectral resolution and produce appreciable errors in the modeled flux. We then consider the effects of rotation, microturbulence, and macroturbulence on the intensity and flux contribution functions, which astronomers use to map individual spectral segments to quasi-physical formation ``locations'' in the stellar atmosphere. We show that proper consideration of 1) the distinction between intensity and flux and 2) the inclusion of rotation and macroturbulence in the contribution function can dramatically change the modeled formation temperatures. To complement this analysis, we provide a package -- FormationTemperatures.jl -- which quickly computes model line contribution functions and formation parameters given bulk stellar properties as input. In closing, we emphasize the assumptions inherent to this analysis, consider in which regimes the convolution expression for flux should be avoided, and caution how the concept of a singular ``formation temperature'' can oversimplify some realities of radiative transfer.

Figures (9)

• Figure 1: Example broadening kernels from Gray2008 and Hirano2011 at arbitrary velocity offsets. The following values were adopted to generate these kernels: $v\sin i = 2.1\ \ {\rm km\ s}^{-1}$, $\zeta_{\rm RT} = 1.4\ \ {\rm km\ s}^{-1}$, and quadratic limb darkening coefficients $u_1 = 0.4$ and $u_2 = 0.26$. The broadening kernels have all been normalized between 0 and 1 to better illustrate their shape differences.
• Figure 2: The convolution method for broadening model stellar spectra can create appreciable errors in line shape, especially for stars with high $v \sin i$. Top: Flux computed via convolution (blue curves) and explicit disk-integration (black curves) for various combinations of $v \sin i$ and $\zeta_{\rm RT}$. Bottom: Percent error in flux (expressed as a percentage of the disk-integrated continuum flux).
• Figure 3: The flux error introduced by the convolution approximation decreases at lower spectral resolution. However, the error remains appreciable at the spectral resolving powers characteristic of modern EPRV instruments.
• Figure 4: Example intensity contribution functions (multiplied by the differential $d t_\nu$) for four lines of sight ranging from disk center (left-most panel) to near the limb (right-most panel). Three coordinates from the model stellar atmosphere are shown for the vertical axis: physical depth, optical depth at 5000 Å, and temperature. At disk center, the continuum and lines form deeper in the atmosphere compared to lines of sight nearer the limb. The white dotted vertical lines indicate the wavelength at which slices through the intensity contribution functions are plotted in Figure \ref{['fig:cont_slice']}.
• Figure 5: Example intensity (left axis, colored curves) and flux (right axis, black curve) contribution functions (multiplied by the differential $d t_\nu$) for a single wavelength element denoted by the vertical dotted lines in Figure \ref{['fig:cfuncs_intensity']}. Three coordinates from the model stellar atmosphere are shown for the horizontal axis, like for Figure \ref{['fig:cfuncs_intensity']}. Though intensity and flux have different dimensions (and consequently the scaling between the left and right axes is arbitrary), they are here overplotted to emphasize the differences in their shape over the height of model stellar atmosphere. Specifically, the intensity contribution shifts higher in the atmosphere with decreasing $\mu$, and the flux contribution covers a comparatively wide swath of atmosphere.