The Effect of Different Methods for Accounting for $α$-enhancement on the Asteroseismic Modeling of Metal-Poor Stars

Abstract

Constraining stellar models using asteroseismic and spectroscopic observations is a powerful method for precisely determining the fundamental properties of stars in different kinematic components of our galaxy. We use spectroscopy and individual oscillation mode frequencies to perform a homogeneous modeling study of eight evolved metal-poor stars enhanced in $α$-elements. We compare a full treatment of $α$-enhancement against an ad hoc correction to the total metallicity and show that the stellar properties inferred from asteroseismic modeling using both sets of models agree with each other. Additionally, we find that the uncertainties on stellar parameters derived from the both $α$-enhanced modeling methods are comparable. This is in qualitative disagreement with existing works showing red-giant ages constrained by only the global asteroseismic parameters to depend strongly on the opacities and abundances assumed in 1D modeling. We also show that the observed frequency of maximum oscillation power ($ν_{\text{max}}$) is larger than the value predicted from applying the $ν_{\text{max}}$ scaling relation to the masses, radii, and temperatures inferred from the detailed modeling. This discrepancy is pronounced at low metallicities, consistent with recent findings indicating a breakdown of the $ν_{\text{max}}$ scaling relation for metal-poor stars. Understanding the extent to which the $ν_{\text{max}}$ scaling relation fails for low-metallicity solar-like oscillators through detailed modeling will enable more accurate mass and age determinations for hundreds of giant stars in the Galactic Halo for which only global asteroseismic parameters are available.

Figures (16)

• Figure 1: Top Left: HR-Diagram evolutionary tracks, showing how stellar evolution changes when modeling stars that are enhanced in $\alpha$-elements. The uncorrected track (blue dotted line) shows the evolution of a 0.85 $M_{\odot}$ star with initial helium abundance $Y_0 = 0.276$ and initial iron abundance [Fe/H]$_0$ = 0.0. The $\alpha$-enhanced track (black line) has the same values of mass, $Y_0$, and [Fe/H]$_0$ as the uncorrected track but incorporates an $\alpha$-enhancement factor of [$\alpha$/Fe] = 0.4. The Salaris-corrected track (orange dashed line) has a different [Fe/H]$_0$ value than the uncorrected track and instead matches the initial global metallicity ($Z_0$) of the $\alpha$-enhanced track, while keeping the same solar-scaled GS98 element abundance mixtures as the uncorrected track. Top Right: Model mode frequency comparison between the three models with the same acoustic radius such that $\Delta\nu \sim 1/2T = 10\ \mu$Hz, marked with stars in the left panel. The scaled frequencies (mode frequency divided by $\Delta \nu$) are plotted on the x-axis while the scaled frequency difference (Salaris-corrected or uncorrected scaled frequency minus the $\alpha$-enhanced model scaled frequency) divided by the $\alpha$-enhanced model scaled frequencies are plotted on the y-axis. The radial (star symbols) and quadrupole (triangle symbol) mode scaled frequency differences are small for both the Salaris-corrected and uncorrected models, however the dipolar mode scaled frequency difference (circle symbols) are large when comparing the uncorrected and $\alpha$-enhanced models. Bottom Left: Comparison of frequency-echelle diagrams for the three models shown. Symbols denoting angular degree are colored in the same way as the top right panel, and are sized by the mixing fractions $\zeta$ --- larger symbols denote more p-like modes and smaller ones denote g-like modes. The p-mode frequencies for the models are similar, as shown by the locations of the radial modes and of the most p-dominated dipole mixed modes. However, the exact configuration of the dipole mixed-mode pattern is very different. Bottom Right: Stretched period-echelle diagrams for the dipole modes of the three models, constructed using the period spacing, coupling strengths, and pure p-mode frequencies of the $\alpha$-enhanced model. The three models can clearly be seen to have very different period spacings $\Delta\Pi_{\ell=1}$, as well as different g-mode phase offsets $\epsilon_g$, different pure p-mode frequencies, and different coupling strengths $q$ between the p- and g-modes.
• Figure 2: Left Panel: The observed [Fe/H] (filled symbols) and Salaris-corrected [Fe/H] (open symbols) values against the [$\alpha$/Fe] measurements for the stars in our sample. The background small points show the distribution of stars in the APOKASC-3 sample in the [$\alpha$/Fe]-[Fe/H] plane. The filled points show the [Fe/H] values for HD 128279 (orange), HD 140283 (light blue), HD 175305 (green), KIC 4671239 (yellow), KIC 7341231 (dark blue), KIC 8144907 (red), $\nu$ Indi (pink), and TIC 300085386 (black), while the open symbols show the [Fe/H] values after the correction from salaris is applied. These values are also listed in \ref{['table:spec_inputs']}. Right Panel: The colored symbols show the seismic log($g$) value and effective temperature values of the stars in our sample. The error bars show the observed temperature errors as well as the seismic log($g$) errors derived from the errors in $\nu_{\text{max}}$ and T$_{\text{eff}}$. Note the seismic log($g$) errors are multiplied by 5 for visibility. The background evolutionary tracks show 0.8$M_{\odot}$ tracks with varying initial metal mass abundance values from $Z_0 = 0.0005$ to $Z_0 = 0.005$ in steps of 0.0005.
• Figure 3: Corner plot visualization showing the mass, radius, and age results of the $\alpha$-enhanced and Salaris-corrected modeling procedure applied to TIC 300085386. The model parameters are taken from the models calculated along each evolutionary track and are weighted by the normalized likelihood (\ref{['eq:weight']}) before plotting. The on-diagonal panels show the marginal mass, radius, and age distributions in the form of kernel density estimation (KDE) plots with blue curves showing the $\alpha$-enhanced results and red curves showing the Salaris-corrected results. The vertical blue and red lines over-plotted in the on-diagonal panels show the 50th percentiles of the likelihood-weighted parameter distributions from the $\alpha$-enhanced and Salaris-corrected modeling, respectively, while the blue and red shaded regions span the region between the 16th and 84th percentiles of the same weighted parameter distributions. The vertical blue and red dashed lines show the best-fit $\alpha$-enhanced and Salaris-corrected model parameters, respectively. The blue and red contours in the off-diagonal panels show the joint distributions of different pairs of global stellar parameters derived from the $\alpha$-enhanced and Salaris-corrected modeling procedures, respectively. The data points in the off-diagonal plots show the overall best-fit model parameters from the ompleted $\alpha$-enhanced modeling (blue circle points) and Salaris-corrected modeling (red star points) of TIC 300085386.
• Figure 4: Similar corner plot to \ref{['fig:corner_plot_global_params']} except we show the [Fe/H], T$_\text{eff}$, and luminosity likelihood-weighted distributions from the $\alpha$-enhanced and Salaris-corrected modeling procedure applied to TIC 300085386. The vertical black dot-dashed lines in each on-diagonal panel show the observed [Fe/H], T$_\text{eff}$, and luminosity values, with the Salaris-corrected [Fe/H] value shown with a green vertical dot-dashed line. The spectroscopic observations are represented in the off-diagonal joint-distribution panels as 2$\sigma$ gray ellipses. The green 2$\sigma$ ellipses incorporate the Salaris correction to the observed [Fe/H].
• Figure 5: The $\alpha$-enhanced and Salaris-corrected stellar mass modeling results are shown in a 'one-to-one' plot. The points are placed at the 50th percentiles of the likelihood-weighted mass distributions while the error bars show the range between the 16th and 84th percentiles listed in \ref{['table:Results_alpha_enhanced']} and \ref{['table:Results_salaris_corrected']}. The dashed black line shows the line of equality. Both modeling methods produce mass results that agree with each other to within $\sim1\sigma$, indicating that either method for accounting for $\alpha$-enhancement does not significantly change the resultant inferred stellar mass distributions from asteroseismic modeling.