A&A proofs: manuscript no. paper\_disc\_corona\_LXvsLUV

Fig. 8. 1-, 2- and 3-sigma contours of the emcee regressions in the slope-normalization \( ( { \widehat { \beta } } - { \widehat { \alpha } } ) \) plane of the \( L _ { X } - L _ { U V } \) for both data and models, normalizing all \( L _ { X } \) and \( L _ { I I V } \) to the corresponding median val-ues of XMM-XXL. Dark green contours are related to the cleanest XMM-XXL sample (Appendix B) and the light green ones to the RM-QSO sources. The contour of the models are color coded for \( \mu \) \( = \)\( [ 0 , 0 . 2 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 8 , 1 ] \), as shown in the legend. For simplicity, we report for each \( \mu \) only results obtained with a single \( f _ { m a x } \), correspond-ing to the highest \( r ^ { 2 } \)-score, and fixed \( \alpha _ { 0 } = 0 . 0 2 \). Models that reproduce the observed slope \( _ { - } \) are also the ones that show weaker coronae (lower normalization \( \widehat { \alpha } \)).

single \( f _ { m a x } \), corresponding to the highest \( r ^ { 2 } \)-score (e.g., Fig. 6) for each \( \mu \), and a fixed \( \alpha _ { 0 } = 0 . 0 2 \), for simplicity.

Fig. 8 shows that models reproducing the observed slope, namely the ones with higher \( \mu \) (as in middle panel of Fig. 7), are also the ones that show weaker coronae (lower normalization \( \bar { \alpha } \) b) and overly extended \( L _ { 2 k e V } \)-emission (i.e. higher \( r _ { p e a k } \) and \( r _ { 9 0 } \), top panel of Fig. 7).

# 5.5. The 3D plane: \( L _ { X } \) vs \( L r r \) vs m

As shown by LR17, the \( L _ { X } - L _ { U V } \) relation for AGN is rather a three-dimensional problem, with the mass (or its proxy given by the full-width half-maximum of broad emission lines) playing a significant role as well. The observed \( L _ { X } - L _ { U V } - m \) plane from XMM-XXL can be fit by:

\[ \begin{array} { c } { { \log L _ { 2 k e V } - 2 5 = ( - 0 . 9 1 \pm 0 . 1 3 ) + ( 0 . 3 9 \pm 0 . 0 3 ) ( \log L _ { 3 0 0 0 \hat { \mathbf { A } } } - 2 5 ) } } \\ { { + \left( 0 . 2 3 \pm 0 . 0 4 \right) ( \log m - 7 ) } } \\ { { ( 1 5 ) } } \end{array} \]

and the mock \( L _ { X } - L _ { U V } - m \) from models with \( \mu \)= 0, 0.5 and 1, respectively:

\[ \begin{array} { c } { { \log L _ { 2 k e V } - 2 5 = ( - 3 . 4 9 \pm 0 . 1 5 ) + ( 1 . 0 8 \pm 0 . 0 3 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { - \, ( 0 . 2 7 \pm 0 . 0 3 ) ( \log m - 7 ) } } \\ { { { } } } \\ { { \log L _ { 2 k e V } - 2 5 = ( - 2 . 4 1 \pm 0 . 1 5 ) + ( 0 . 7 3 \pm 0 . 0 1 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { + \, ( 0 . 0 1 3 \pm 0 . 0 0 4 ) ( \log m - 7 ) } } \\ { { { } } } \\ { { \log L _ { 2 k e V } - 2 5 = ( - 2 . 2 8 \pm 0 . 0 8 ) + ( 0 . 5 7 \pm 0 . 0 2 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { { } } } \\ { { + \, ( 0 . 1 4 \pm 0 . 0 2 ) ( \log m - 7 ) } } \end{array} \]

Fig. 9. Same as Fig. 8, with the addition of empty contours for \( \mu \)=0.4, 0.5 and 0.6 (color coded in the legend) obtained with maximally spinning black holes (i.e. with \( \epsilon _ { 0 } = 0 . 3 \) and \( r _ { 0 } \, = \, 1 . 2 4 r _ { g } \). The dashed lines connect them to the non-spinning analogous realizations. Dark-red density spots represent the location of the center of dierent contours of the standard \( \mu \) = 0.5 case, in which the only dierence is the adoption of \( \eta \) (downward scattering component) varying among 0.4, 0.5 and 0.6, going from higher to lower \( \widehat { \alpha } \), respectively.

The comparison in the 3D plane states that the exact dependency is not obtained by any of the models, with \( \mu \) = 1 being the clos-est in qualitatively retrieving the coecients for \( L _ { 3 0 0 0 \mathring { \mathrm { A } } } \) and m. We note that the mass is taken from the observations, thus this mismatch states that the luminosities in the model do not depend on the mass in the correct way.

# 5.6. The impact of the accretion efficiency

Throughout this work we adopted an eciency \( \epsilon _ { 0 } = 0 . 0 5 7 \), typ-ical of non-rotating black holes (e.g. Shapiro 2005), for sim-plicity. Nonetheless, a high spin seems to be preferred to model the blurred relativistic iron line, detected both in the local Uni-verse (Nandra et al. 2007; Reynolds 2013) and up to \( z \sim \) 4 (e.g. Baronchelli et al. 2018). Moreover, flux-limited samples are known to be biased in preferentially detecting high-spinning black holes (Brenneman et al. 2011; Vasudevan et al. 2016), sim-ply because they are brighter than their non-rotating analogous (see Reynolds 2019).

Then, we tested the model using maximally-spinning black holes, with radiative eciency 0.3 and ISCO down to \( r _ { 0 } = \)\( 1 . 2 4 r _ { g } \)(Thorne 1974). This has a major impact on the normal-ization axis of the \( L _ { X } - L _ { U V } . \). Everything else in the source being equal, in a spinning black hole matter can be accreted down to smaller distances with respect to their non-rotating companions, thus the accretion power in the system is much higher. As a mat-ter of fact, changing the radiative eciency has an impact on the numerical equation that regulates f(r): for the same m and \( \dot { m } \) and \( r > \) 3 the values of f is higher, and the transition radius between \( P _ { r a d } - \) and \( P _ { g a s } \)-dominated regions moves at lower radii. This self-consistently aects the disk equations via the \( ( 1 - \tilde { f } ) \) factor (see Appendix A), hence the surface temperature is de-creased at higher radii, where most of the disk emission at 3000Åcomes from. Then, the modeled \( \dot { m } \) value needed to match the observed \( L _ { 3 0 0 0 \mathring \mathrm { A } } \) is higher (see Section 4.2) and, consequently, \( L _ { 2 k e V } \propto f Q _ { + } \) is higher.