[Paper Review] Formation depths of Fraunhofer lines

É. A. Gurtovenko, V. A. Sheminova|arXiv (Cornell University)|May 5, 2015

Optical Polarization and Ellipsometry1 references3 citations

TL;DR

This paper establishes that the formation depth of Fraunhofer lines—specifically the line depression—must be determined using a depression contribution function (F_D), not the standard emission contribution function. By applying Unsold’s weighting function to the line depression, the authors demonstrate that formation depths of weak lines correlate strongly with excitation potential, resolving inconsistencies in earlier methods that failed to distinguish between emission and depression formation regions.

ABSTRACT

We have summed up our investigations performed in 1970--1993. The main task of this paper is clearly to show processes of formation of spectral lines as well as their distinction by validity and by location. For 503 photospheric lines of various chemical elements in the wavelength range 300--1000 nm we list in Table the average formation depths of the line depression and the line emission for the line centre and on the half-width of the line, the average formation depths of the continuum emission as well as the effective widths of the layer of the line depression formation. Dependence of average depths of line depression formation on excitation potential, equivalent widths, and central line depth are demonstrated by iron lines.

Motivation & Objective

• To resolve the long-standing ambiguity in determining the true formation depth of Fraunhofer lines in the solar photosphere.
• To address the failure of standard emission contribution functions (F_E) to correctly predict formation depths for weak lines with varying excitation potentials.
• To validate the use of the depression contribution function (F_D) derived from Unsold’s weighting function as the correct approach for line depression formation depth.
• To provide a comprehensive dataset of average formation depths for 503 photospheric lines across 300–1000 nm, categorized by element, wavelength, and line strength.
• To clarify the distinction between formation regions of line emission and line depression, which had been conflated in prior studies.

Proposed method

• Derives the emergent line depression intensity (D_l(0)) using the integral of the depression contribution function F_D over continuum optical depth τ_c.
• Introduces F_D as the product of the depression weighting function g′ and the line-to-continuum opacity ratio η, with g′ based on the Planck function and exponential extinction.
• Uses the expression D_l(0) = ∫ F_D dτ_c to compute the line depression, where F_D = g′η exp(−τ_l), ensuring physical consistency with radiative transfer.
• Applies the method to 503 photospheric lines from various elements (Fe, Ni, Zn, Sr, Zr, Ba, Ce, Nd, Sm, Dy, Pb), calculating formation depths from F_D profiles.
• Compares results using F_D with those from standard emission CF (F_E) and finds F_D correctly reproduces the known trend of deeper formation for higher-excitation potential lines.
• Validated the method by showing that F_D-based depths exhibit clear dependence on excitation potential, unlike F_E-based depths which were uniform across lines of different potentials.

Experimental results

Research questions

• RQ1 Why do standard emission contribution functions (F_E) fail to reproduce the known trend that weak lines of higher excitation potential form in deeper layers of the solar photosphere?
• RQ2 What is the correct contribution function to use for determining the formation depth of the line depression (i.e., the absorption feature) rather than the line emission?
• RQ3 How does the formation depth of the line depression depend on excitation potential, equivalent width, and central depth for iron lines?
• RQ4 Can the depression contribution function (F_D) derived from Unsold’s weighting function accurately reproduce the observed physical behavior of weak Fraunhofer lines?
• RQ5 What is the distinction between the formation regions of line emission and line depression, and why is it critical for accurate atmospheric parameter determination?

Key findings

• The use of the standard emission contribution function (F_E) leads to identical formation depths for all weak lines of similar wavelength, regardless of excitation potential, contradicting established physical expectations.
• The depression contribution function (F_D), derived from Unsold’s weighting function and applied to the line depression, correctly reproduces the known trend: higher excitation potential lines form at greater depths.
• For iron lines, the average formation depth of the line depression increases with excitation potential, with a clear and measurable dependence confirmed across the 300–1000 nm range.
• The average formation depth of the line depression is strongly correlated with central line depth and equivalent width, with deeper lines forming in higher optical depth (deeper) layers.
• The effective width of the layer responsible for line depression formation is found to be narrow, typically less than 100 km, indicating that the line formation is confined to a thin atmospheric layer.
• The method successfully distinguishes between formation regions of line emission and line depression, resolving a long-standing ambiguity in solar spectral line analysis.