Granulation signatures as seen by Kepler short-cadence data - I. A decoupling between granulation and oscillation timescales for dwarfs

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Volume 707 (March 2026)

A&A, 707 (2026) A220

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Table C.2

Compilation of the general priors for the complete background model terms describing stellar activity, the oscillation excess, and white noise.

Fit parameters	Prior distribution	Prior setup
Activity component priors
$M_{act} = \frac{a_{2}^{2} / b_{2}}{1 + {(\frac{v}{b_{2}})}^{2}}$ $Mathematical equation: $\mathcal{M}_{\text {act }}=\frac{a_{2}^{2}/b_{2}}{1+\left(\frac{v}{b_{2}}\right)^{2}}$$
a₂	beta	$\begin{matrix} a, b = data-driven \\ loc = 0.0, scale = 3 \times a_{2}^{est} \\ mode = a_{2}^{est} = \sqrt{2 P_{peak} b_{2}^{est}} \\ P_{peak} = max (Power [v < 0.15 v_{max}]) \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=\text {data-driven} \\ \text {loc}=0.0,\ \text {scale}=3 \times a_2^{\text {est}} \\ \text { mode}=a_2^{\text {est}}=\sqrt{2 P_{\text {peak}} b_2^{\text {est}}} \\ P_{\text {peak}}=\max \left(\text { Power}\left[v<0.15 v_{\text {max}}\right]\right) \end{gathered}$$
b₂	beta	$\begin{matrix} a, b = data-driven‖ \\ l o c = 1 \times 10^{- 6}, scale‖ = (1 / 8) b_{g r a n} (v_{max}) - 1 \times 10^{- 6} \\ mode‖ = b_{2}^{e s t} = arg max‖ (Power [v < 0.15 v_{max}]) \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=\text { data-driven } \\ \mathrm{loc}=1 \times 10^{-6},\ \text { scale }=(1 / 8) b_{\mathrm{gran}}\left(v_{\max }\right)-1 \times 10^{-6} \\ \text { mode }=b_2^{\mathrm{est}}=\text { arg max }\left(\operatorname{Power}\left[v<0.15 v_{\max }\right]\right) \end{gathered}$$
Gaussian excess specific priors
$M_{osc} = P_{osc} exp \frac{- {(v - v_{max})}^{2}}{2 σ^{2}}$ $Mathematical equation: $\mathcal{M}_{\text {osc}}=P_{\text {osc}} \exp \frac{-\left(v-v_{\text {max}}\right)^{2}}{2 \sigma^{2}}$$
σ_osc	beta	$\begin{matrix} a, b = data-driven‖ \\ loc = 1 e - 4, scale‖ = 3 \times mode‖ - 1 e - 4 \\ mode‖ = Δ v \cdot max (1, 4 {(v_{max} / 3090)}^{0.2}) / (2 \sqrt{2 \ln 2}) \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=\text { data-driven } \\ \operatorname{loc}=1 \mathrm{e}-4, \text { scale }=3 \times \text { mode }-1 \mathrm{e}-4 \\ \text { mode }=\Delta v \cdot \max \left(1,4\left(v_{\max } / 3090\right)^{0.2}\right) /(2 \sqrt{2 \ln 2}) \end{gathered}$$
P_osc	beta	$\begin{matrix} a, b = data-driven‖ \\ loc‖ = 1 e - 4, scale‖ = 15 \times mode‖ - 1 e - 4 \\ mode‖ = α \cdot median‖ (Power [v_{max} \pm 2 σ_{o s c}])) \\ α = 0.4 (v_{max} < 1000 μ H z), 0.1 (v_{max} > 1000 μ H z) \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=\text { data-driven } \\ \text { loc }=1 \mathrm{e}-4, \text { scale }=15 \times \text { mode }-1 \mathrm{e}-4 \\ \text { mode } \left.=\alpha \cdot \text { median }\left(\operatorname{Power}\left[v_{\max } \pm 2 \sigma_{\mathrm{osc}}\right]\right)\right) \\ \alpha=0.4\left(v_{\max }<1000 \mu \mathrm{~Hz}\right), 0.1\left(v_{\max }>1000 \mu \mathrm{~Hz}\right) \end{gathered}$$
ν_max	beta	$\begin{matrix} a, b = 11, 11 \\ l o c = 0.75 v_{max}, s c a l e = 1.25 v_{max} - 0.75 v_{max} \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=11, 11 \\ \mathrm{loc}=0.75\ v_{\max}, \mathrm{scale}=1.25\ v_{\max}-0.75\ v_{\max} \end{gathered}$$
Peakbogging specific priors
H	beta	$\begin{matrix} a, b = 1, 5 \\ loc, scale = 0, 1 \end{matrix}$ $Mathematical equation: $\begin{gathered} a, b=1,5 \\ \text {loc, scale}=0,1 \end{gathered}$$
β	beta	$\begin{matrix} a, b = 1, 10 \\ loc, scale = 1, 19 \end{matrix}$ $Mathematical equation: $\begin{gathered} a, b=1,10 \\ \text {loc, scale}=1,19 \end{gathered}$$
White noise prior
W	beta	$\begin{matrix} a, b = 1.5, 4 \\ loc, scale = 0.5 W_{est‖}, 2.5 W_{est} \\ W_{est} = median (Power [- 100 :]) \end{matrix}$ $Mathematical equation: $\begin{gathered} \mathrm{a}, \mathrm{~b}=1.5,4 \\ \text { loc, scale}=0.5 W_{\text {est }}, 2.5 W_{\text {est}} \\ \left.W_{\text {est}} = \text{median}(\text {Power}[\hbox{-}100:]\right) \end{gathered}$$

¹ Oscillation excess width prior set as in the SYD pipeline (Huber et al. 2009) and its python implementation pySYD (Chontos et al. 2021), using an approximation of Δ ν (Stello et al. 2009a; Hekker et al. 2009; Huber et al. 2011).

Notes. The functional forms of the stellar activity (standard Harvey profile, Harvey 1985) and Gaussian oscillation excess components are provided for clarity of the parameter definitions. When the prior uses ν_max as input, it is the observed value obtained as specified in Sect. 2.

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