  We present a sample of 32 stars of spectral types G and K, and luminosity
classes I to V, with moderate activity levels, covering four orders of
magnitude of surface gravity and a representative range of effective
temperature. For each star we obtained high S/N TIGRE-HEROS spectra with a
spectral resolving power of $R\approx20,000$ and have measured the Ca II K
line-widths of interest, $W_0$ and $W_1$. The main physical parameters are
determined by means of iSpec synthesis and Gaia EDR3 parallaxes. Mass estimates
are based on matching to evolution models. Using this stellar sample, that is
highly uniform in terms of spectral quality and assessment, we derive the
best-fit relation between emission line width and gravity $g$, including a
notable dependence on effective temperature $T_{\rm eff}$, of the form $W_1
\propto g^{-0.229} T_{\rm eff}^{+2.41}$. This result confirms the physical
interpretation of the Wilson-Bappu effect as a line saturation and photon
redistribution effect in the chromospheric Ca II column density, under the
assumption of hydrostatic equilibrium at the bottom of the chromosphere. While
the column density (and so $W_1$) increases towards lower gravities, the
observed temperature dependence is then understood as a simple ionization
effect -- in cooler stars, Ca II densities decrease in favor of Ca I.
