Kinematic scaling relations of CALIFA galaxies: A dynamical mass proxy for galaxies across the Hubble sequence

We used ionized gas and stellar kinematics for 667 spatially resolved galaxies publicly available from the Calar Alto Legacy Integral Field Area survey (CALIFA) 3rd Data Release with the aim of studying kinematic scaling relations as the Tully $\&$ Fisher (TF) relation using rotation velocity, $V_{rot}$, the Faber $\&$ Jackson (FJ) relation using velocity dispersion, $\sigma$, and also a combination of $V_{rot}$ and $\sigma$ through the $S_{K}$ parameter defined as $S_{K}^2 = KV_{rot}^2 + \sigma^2$ with constant $K$. Late-type and early-type galaxies reproduce the TF and FJ relations. Some early-type galaxies also follow the TF relation and some late-type galaxies the FJ relation, but always with larger scatter. On the contrary, when we use the $S_{K}$ parameter, all galaxies, regardless of the morphological type, lie on the same scaling relation, showing a tight correlation with the total stellar mass, $M_\star$. Indeed, we find that the scatter in this relation is smaller or equal to that of the TF and FJ relations. We explore different values of the $K$ parameter without significant differences (slope and scatter) in our final results with respect the case $K=0.5$ besides than a small change in the zero point. We calibrate the kinematic $S_{K}^2$ dynamical mass proxy in order to make it consistent with sophisticated published dynamical models within $0.15\ dex$. We show that the $S_{K}$ proxy is able to reproduce the relation between the dynamical mass and the stellar mass in the inner regions of galaxies. Our result may be useful in order to produce fast estimations of the central dynamical mass in galaxies and to study correlations in large galaxy surveys.