Appendix: Acronyms

Following a suggestion of the referee, we present here the definition of the many acronyms used in this paper.

$T_{\rm eff}$: effective temperature of the atmosphere to be illuminated

$T_{\rm h}$: heating star's effective temperature

$\gamma$: line of sight angle (see Fig.1)

$\mu$: $\cos\gamma$

$\theta$: incidence angle (see Fig.1)

$\nu$: $\cos\theta$

$\lambda$: wavelength

$I_{\lambda}(\mu)$: beam intensity at the wavelength $\lambda$,direction $\mu$

$R_{\lambda}(\mu)$: limb-darkening law

$x_{\lambda}$: linear limb-darkening coefficient

$y_{\lambda}$: non-linear limb-darkening coefficient

$r_{\rm h}$: the apparent radius of the illuminating star (see Fig.1)

g: local surface gravity acceleration

$\alpha$: the mixing length parameter

$\beta$: the gravity brightening exponent

Q: the quality factor of the limb-darkening law adjustment

$F(\lambda)$: the monochromatic flux

$F_{{\rm rel}, \nu}$: bolometric incident flux relative to the one emitted by the illuminated star at the illuminated point of its surface

$F^{\prime}$: total emergent flux obtained by integrating the intensities calculated from the adjusted limb-darkening coefficients

$F_{\rm m}$: total flux obtained from the model intensities.