Top left: LLmodels photospheric spectrum of the Sun in the region of the Ca ii K line core (black solid line) modified by adding a line core emission of 112 erg cm^-2 s^-1 (red solid and dashed lines). This composite spectrum has then been modified by two ISM absorption features, located at the line centre, calculated with Ca ii column densities of 10¹² and 10¹⁶ cm^-2 (blue and green dashed and solid lines). Bottom left: comparison between a high resolution HIRES spectrum of the young Sun-like star EK Dra and the corresponding synthetic spectrum obtained by adding a line core emission of E = 112 erg cm^-2 s^-1 to the photospheric spectrum. The two spectra return the same S_MW value of 0.440. The synthetic spectrum accounts also for the instrumental spectral resolution and the stellar projected rotational velocity (≈15 km s^-1). Right: same as the left panels, but for the hyper-velocity halo Sun-like star HD 29587. In this case, the emission flux is 15 erg cm^-2 s^-1, corresponding to a S_MW value of 0.144. The synthetic spectrum does not match the line profile in detail, possibly due to non-LTE effects, but it reproduces the empirical S value. In the top panels, the red solid lines are visible just in the very core of the line, as it is otherwise covered by the blue solid line.