Is there a tilt in the fundamental (hyper)plane? | Astronomy & Astrophysics (A&A)

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Table 1.

FP and HP equations with the corresponding best-fitting parameters and figure of merits.


	(Hyper-)plane equation	Best-fit parameters				Observed scatter	$Δ \bar{R^{2}}$ $\Delta\overline{{R^2}}$	ΔAIC	ΔBIC

		α	β	ϵ	γ	Δ
FP	log(R_e) = αlog(σ)+βlog(I_e)+γ	1.2 ± 0.1	−0.75 ± 0.04		−0.2 ± 0.1	0.088	0	0	0
M^⋆-HP	log(R_e) = αlog(σ_e8)+βlog(I_e)+ϵlog(M^⋆)+γ	0.3 ± 0.1	−0.63 ± 0.03	0.42 ± 0.06	−3.0 ± 0.6	0.021	+0.081	−67.7	−66.6
f_e^⋆-HP	log(R_e) = αlog(σ)+βlog(I_e)+ϵlog(f_e^⋆)+γ	2.0 ± 0.3	−1.1 ± 0.1	0.8 ± 0.2	−0.68 ± 0.09	0.036	+0.060	−28.6	−27.6

Notes. $Δ \bar{R^{2}}$ $\Delta\overline{{R^2}}$ , ΔAIC, and ΔBIC are the differences relative to the values computed for the FP, which are $\bar{R^{2}} = 0.908$ $\overline{{R^2}}=0.908$ , AIC = −67.3, and BIC = −63.4. A positive value of $Δ \bar{R^{2}}$ $\mathbf{\Delta}\overline{{R^2}}$ and a negative value of ΔAIC and ΔBIC are considered as a strong indication of a better model.

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