Improving the stellar age determination through joint modeling of binarity and asteroseismology - Grid modeling of the seismic red giant binary KIC 9163796

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Volume 696 (April 2025)

A&A, 696 (2025) A42

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Table 3.

Cases considered for modeling KIC 9163796 and the associated constraints.

Case	Observables used in figure of merit	Constraints in addition to Age₁ = Age₂
$\begin{matrix} A 0 \\ A 1 \\ A 2 \\ A 3 \end{matrix}}$ $\left.\begin{array}{c} A0 \\ A1\\ A2\\ A3\\ \end{array}\right\}$	T_eff, 1,T_eff, 2, log g₁, log g₂, q	${\begin{matrix} - \\ Z_{1} = Z_{2}, Y_{1} = Y_{2}, \\ Z_{1, 2} = Z_{spec}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec} \pm σ_{Z_{spec}}, Y_{1} = Y_{2} \end{matrix}$ $\left\{\begin{array}{l} - \\ Z_1\,=\,Z_2,\;Y_1\,=\,Y_2,\\ Z_{1,2}\,=\,\mathrm{Z_{spec}},\;Y_1\,=\,Y_2\\ Z_{1,2}\,=\,\mathrm{Z_{spec}}\,\pm\,\sigma_{\mathrm{Z_{spec}}},\;Y_1\,=\,Y_2 \end{array}\right.$
$\begin{matrix} B 1 \\ B 2 \\ B 3 \end{matrix}}$ $\left.\begin{array}{c} B1 \\ B2\\ B3\\ \end{array}\right\}$	T_eff, 1,T_eff, 2log g₂,q,ν_max, 1	${\begin{matrix} Z_{1} = Z_{2}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec} \pm σ_{Z_{spec}}, Y_{1} = Y_{2} \end{matrix}$ $\left\{\begin{array}{l} Z_1\,=\,Z_2, Y_1\,=\,Y_2 \\ Z_{1,2}\,=\,\mathrm{Z_{spec}},~Y_1\,=\,Y_2 \\ Z_{1,2}\,=\,\mathrm{Z_{spec}}\,\pm\,\sigma_{\mathrm{Z_{spec}}},~Y_1\,=\,Y_2 \end{array}\right.$
$\begin{matrix} C 1 \\ C 2 \\ C 3 \end{matrix}}$ $\left.\begin{array}{c} C1 \\ C2\\ C3\\ \end{array}\right\}$	T_eff, 1,T_eff, 2, log g₂,q, ν_max, 1,Δν₁	${\begin{matrix} Z_{1} = Z_{2}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec} \pm σ_{Z_{spec}}, Y_{1} = Y_{2} \end{matrix}$ $\left\{\begin{array}{l} Z_1\,=\,Z_2,~Y_1\,=\,Y_2 \\ Z_{1,2}\,=\,\mathrm{Z_{spec}},~Y_1\,=\,Y_2 \\ Z_{1,2}\,=\,\mathrm{Z_{spec}}\,\pm\,\sigma_{\mathrm{Z_{spec}}},~Y_1\,=\,Y_2 \end{array}\right.$
$\begin{matrix} D 1 \\ D 2 \\ D 3 \end{matrix}}$ $\left.\begin{array}{c} D1 \\ D2\\ D3\\ \end{array}\right\}$	T_eff, 1,T_eff, 2, log g₂,q,ν_max, 1,Δν₁,Δm_V	${\begin{matrix} Z_{1} = Z_{2}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec}, Y_{1} = Y_{2} \\ Z_{1, 2} = Z_{spec} \pm σ_{Z_{spec}}, Y_{1} = Y_{2} \end{matrix}$ $\left\{\begin{array}{l} Z_1\,=\,Z_2,~Y_1\,=\,Y_2\\ Z_{1,2}\,=\,\mathrm{Z_{spec}},~Y_1\,=\,Y_2\\ Z_{1,2}\,=\,\mathrm{Z_{spec}}\,\pm\,\sigma_{\mathrm{Z_{spec}}},~Y_1\,=\,Y_2 \end{array}\right.$

Notes. The first column names the different optimization cases. The second column indicates those parameters included in the χ_red² optimization. The third column indicates any constraints in the model input in addition to the requirement that the components exhibit the same age, which has been applied to all cases.

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