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Table C.1

Compilation of the granulation background model specific priors used for the three prescriptions in Table 1.

Model specific fit parameters Model J Model H Model T
Correlated-inference setup (Sect. 3.1)
σa –Samples a freely as below– lognormal μ=0σ=0.1Mathematical equation: $\begin{aligned} \mu=0 \\ \sigma=0.1 \end{aligned}$ lognormal μ=0σ=0.1Mathematical equation: $\begin{aligned} \mu=0 \\ \sigma=0.1\end{aligned}$
σb  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered} \text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$
σd  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$  beta‖a,b=6,6 loc, scale‖=0.6,0.8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=6,6 \\ \text { loc, scale }=0.6,0.8 \end{gathered}$
Free variable inference
a  lognormal‖μ=f(ac=3.555,ae=−1.006,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=f\left(a_c=3.555, a_e=-1.006, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(ac=3.530,ae=−0.609,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=f\left(a_c=3.530, a_e=-0.609, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(ac=3.530,ae=−0.609,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=f\left(a_c=3.530, a_e=-0.609, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$
b  lognormal‖μ=f(bc=−0.499,be=0.970,vmax)σ=0.1Mathematical equation: $\begin{gathered}\text { lognormal } \\ \mu=f\left(b_c=-0.499, b_e=0.970, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$ μ=f(bc=−0.499,be=0.970,vmax) lognormal‖σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal }\\ {\mu=f\left(b_c=-0.499, b_e=0.970, v_{\max }\right)} \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(bc=−0.499,be=0.970,vmax)σ=0.1Mathematical equation: $\begin{gathered}\text { lognormal } \\ \mu=f\left(b_c=-0.499, b_e=0.970, v_{\max }\right) \\ \sigma=0.1\end{gathered}$
d  lognormal‖μ=f(dc=−0.020,de=0.992,vmax)σ=0.1Mathematical equation: $\begin{gathered}\text { lognormal } \\ \mu=f\left(d_c=-0.020, d_e=0.992, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(dc=−0.020,de=0.992,vmax)σ=0.1Mathematical equation: $\begin{gathered}\text { lognormal } \\ \mu=f\left(d_c=-0.020, d_e=0.992, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(dc=−0.020,de=0.992,vmax)σ=0.1Mathematical equation: $\begin{gathered}\text { lognormal } \\ \mu=f\left(d_c=-0.020, d_e=0.992, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$
Universal parameter priors
c –  lognormal‖μ=f(cc=3.477,ce=−0.609,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=f\left(c_c=3.477, c_e=-0.609, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$  lognormal‖μ=f(cc=3.477,ce=−0.609,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=f\left(c_c=3.477, c_e=-0.609, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$
e – – lognormal  lognormal‖μ=12f(cc=3.477,ce=−0.609,vmax)σ=0.1Mathematical equation: $\begin{gathered} \text { lognormal } \\ \mu=\frac{1}{2} f\left(c_c=3.477, c_e=-0.609, v_{\max }\right) \\ \sigma=0.1 \end{gathered}$
f – –  beta‖a,b= data-driven‖loc=0.9vmax scale‖=min(4vmax,0.9vNyq)−locmode=1.1vmaxMathematical equation: $\begin{gathered}\text { beta } \\ a, b=\text { data-driven } \\ \operatorname{loc}=0.9 v_{\max } \\ \text { scale }=\min \left(4 v_{\max }, 0.9 v_{\mathrm{Nyq}}\right)-\mathrm{loc} \\ \operatorname{mode}=1.1 v_{\max }\end{gathered}$
l  beta‖a,b=1.8,3.0 loc, scale‖=1,8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=1.8,3.0 \\ \text { loc, scale }=1,8\end{gathered}$  beta‖a,b=1.8,3.0 loc, scale‖=1,8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=1.8,3.0 \\ \text { loc, scale }=1,8\end{gathered}$  beta‖a,b=1.8,3.0 loc, scale‖=1,8Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=1.8,3.0 \\ \text { loc, scale }=1,8\end{gathered}$
k  beta‖a,b=2.0,5.0 loc, scale‖=1,9Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=2.0,5.0 \\ \text { loc, scale }=1,9\end{gathered}$ beta  beta‖a,b=2.0,5.0 loc, scale‖=1,9Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=2.0,5.0 \\ \text { loc, scale }=1,9\end{gathered}$  beta‖a,b=2.0,5.0 loc, scale‖=1,9Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=2.0,5.0 \\ \text { loc, scale }=1,9\end{gathered}$
m – – beta  beta‖a,b=2.0,5.0 loc, scale‖=1,10Mathematical equation: $\begin{gathered}\text { beta } \\ a, b=2.0,5.0 \\ \text { loc, scale }=1,10\end{gathered}$

1 Amplitude prior for model J follows the prescription in Larsen et al. (2025).

2 Shape parameters a(m), b(m) set according to bounds such that mode lies just above vmax.

Notes. The log-space coefficients for the power law f (constant, exponent, vmax)=constant+log10(vmax)exponent are specified directly when used and originate from Kallinger et al. (2014), except for the amplitude a of model J which is from Larsen et al. (2025). When the prior uses vmax as input, it is the observed value obtained as specified in Sect. 2.

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