New assignments of the CaH $A^{2}\Pi-X^{2}\Sigma^{+}$ transitions in the sunspot umbral spectrum and effective temperature estimation using molecular lines

Technical Summary: New Assignments of CaH $A^2\Pi - X^2\Sigma^+$ Transitions and Effective Temperature Estimation in Sunspot Umbral Spectra

Problem Statement
Calcium monohydrate (CaH) serves as a critical spectroscopic tracer for cool stellar and solar environments, particularly for spectral classification and temperature diagnostics. However, high-excitation transitions of CaH remain poorly characterized. Laboratory spectroscopy often struggles to reproduce transitions involving high-energy states or low transition probabilities due to limitations in temperature and particle density. Consequently, existing molecular line lists may lack the completeness required for accurate radiative transfer modeling in cool astronomical objects, such as sunspot umbrae and exoplanetary atmospheres. Previous studies of sunspot spectra (e.g., Wallace et al. 1999) had assigned CaH lines only up to specific rotational quantum numbers and vibrational bands, leaving higher-energy transitions unassigned.

Methodology
The authors utilized high-resolution sunspot umbral spectra obtained from the McMath–Pierce telescope (Kitt Peak) using a Brault Fourier-transform spectrometer (Wallace et al. 1999), covering the wavenumber range of 14,000–15,000 cm$^{-1}$.

1. Spectral Assignment: Using the PGOPHER simulation software, the authors assigned CaH $A^2\Pi - X^2\Sigma^+$ electronic transitions. They incorporated molecular constants from laboratory studies (Shayesteh et al. 2013) and Einstein A coefficients from quantum chemical calculations (Shayesteh et al. 2017). The simulation accounted for the Herman-Wallis effect, though $J$-dependent terms were not explicitly modeled due to data limitations.
2. Spectral Simulation: Synthetic spectra were generated using the SPECTRUM code. The models included CaH and Titanium Monoxide (TiO) as primary opacity sources, with electronic transitions from $v=0$ to $v=3$ (CaH) and $v=0$ to $v=4$ (TiO). The simulations assumed a solar metallicity model atmosphere (Kurucz grid) with effective temperatures ($T_{\text{eff}}$) ranging from 3,500 K to 4,500 K.
3. Line List Comparison: Two line lists were compared:
   • Line List #1: Included newly assigned transitions with rotational quantum numbers ($J$) up to 99.5 across all vibrational levels ($v=0-3$), including the first identification of the (3-3) band.
   • Line List #2: Restricted to previously assigned transitions (Wallace et al. 1999) with lower $J_{\text{max}}$ values.
4. Temperature Estimation: The effective umbral temperature was determined by minimizing the sum of squared residuals between the observed and simulated spectra. The study also compared these spectroscopic results with brightness temperatures derived from empirical relations linking umbral size to intensity (Mathew et al. 2007; Kiess et al. 2014).

Key Contributions

• New Line Assignments: The study reports the assignment of 224 spectral lines. Notably, 75 lines in the (3-3) vibrational band are identified for the first time.
• Extended Rotational Quantum Numbers: The identification of the (0-0), (1-1), and (2-2) bands was extended to significantly higher rotational quantum numbers ($J_{\text{max}} \approx 49.5$) compared to previous sunspot studies, approaching limits previously only seen in laboratory settings for lower bands.
• Improved Spectral Reproduction: By incorporating these new high-energy transitions into spectral simulations that account for overlapping TiO features, the authors achieved a modest but measurable improvement in reproducing the observed umbral spectrum compared to models using older line lists.

Results

• Effective Temperature: Minimizing residuals between observed and simulated spectra yielded an effective umbral temperature of approximately 4000 K. This value represents the best fit for the molecular features (CaH and TiO) within the explored range.
• Line List Sensitivity: Simulations using the extended Line List #1 provided a better fit to the observed black sunspot spectrum than Line List #2, particularly for newly assigned (3-3) lines.
• Comparison with Empirical Data: The spectroscopically derived temperature (~4000 K) aligns with empirical brightness temperature estimates derived from the "core" and "minimum" intensity of the umbra (approx. 3800–4050 K) but is lower than estimates derived from "mean" intensity or previous studies focusing on atomic lines (which suggested ~4700–5100 K).
• Atomic vs. Molecular Discrepancy: The study highlights that while higher-temperature models (>4500 K) may better reproduce certain atomic lines (e.g., Ti I), they fail to reproduce the depth of the strong molecular absorption bands. Conversely, the 4000 K model accurately fits molecular bands but underestimates atomic line depths, suggesting a difficulty in reconciling atomic and molecular diagnostics within a single 1D-LTE framework.

Significance
The paper demonstrates that solar observations serve as a "natural laboratory" for characterizing high-energy molecular transitions that are challenging to reproduce in terrestrial laboratories. The newly assigned CaH lines and the resulting effective temperature of ~4000 K suggest that molecular features in sunspot spectra primarily form in the cool cores of the umbra. This finding reinforces the utility of molecular lines as sensitive "thermometers" for cool astronomical objects. Furthermore, the refined line lists and temperature diagnostics are presented as essential for improving the accuracy of radiative transfer models used in the study of cool stellar atmospheres and the characterization of exoplanetary atmospheres, where species like CaH and TiO are expected to be present.