NLTE line formation of Fe in late-type stars II: 1D spectroscopic stellar parameters

Abstract

We investigate departures from local thermodynamic equilibrium (NLTE) in the line formation of neutral and singly ionised iron lines and their impact on spectroscopic stellar parameters. The calculations are performed for an extensive grid of 1D MARCS models of metal-rich and metal-poor late-type dwarfs and giants. We find that iron abundances derived from FeI lines are increasingly underestimated in hotter, lower surface-gravity, and more metal-poor stars, in a simple and well-defined pattern, while LTE is usually a realistic approximation for FeII lines. For the vast majority of dwarfs and giants, the perturbed ionisation balance of FeI and FeII is the main relevant NLTE effect to consider in the determination of spectroscopic parameters, while for extremely metal-poor stars and hot giant stars significant impact is seen also on the excitation balance and on the microturbulence determination from FeI lines.

arXiv comment11 pages, 7 figures, accepted for publication in MNRAS

References in corpus (10)

• A grid of MARCS model atmospheres for late-type stars I. Methods and general properties
• SparsePak: A Formatted Fiber Field Unit for The WIYN Telescope Bench Spectrograph. I. Design, Construction, and Calibration
• Stellar Iron Abundances: non-LTE Effects
• A non-LTE study of neutral and singly-ionized iron line spectra in 1D models of the Sun and selected late-type stars
• The broadening of Fe II lines by neutral hydrogen collisions
• Kinetic equilibrium of iron in the atmospheres of cool stars III. The ionization equilibrium of selected reference stars
• A NLTE line blanketed model of a solar type star
• Effects of line-blocking on the non-LTE Fe I spectral line formation
• The StaggerGrid Project: a Grid of 3D Model Atmospheres for High-Precision Spectroscopy
• Kinetic equilibrium of iron in the atmospheres of cool dwarf stars II. Weak Fe I lines in the solar spectrum

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