Stellar mass functions and implications for a variable IMF

Spatially resolved kinematics of nearby galaxies has shown that the ratio of dynamical- to stellar population-based estimates of the mass of a galaxy ($M_*^{\rm JAM}/M_*$) correlates with $\sigma_e$, if $M_*$ is estimated using the same IMF for all galaxies and the stellar M/L ratio within each galaxy is constant. This correlation may indicate that, in fact, the IMF is more dwarf-rich for galaxies with large $\sigma$. We use this correlation to estimate a dynamical or IMF-corrected stellar mass, $M_*^{\rm \alpha_{JAM}}$, from $M_{*}$ and $\sigma_e$ for a sample of $6 \times 10^5$ SDSS galaxies for which spatially resolved kinematics is not available. We also compute the `virial' mass estimate $k(n,R)\,R_e\,\sigma_R^2/G$, where $n$ is the Sersic index, in the SDSS and ATLAS$^{\rm 3D}$ samples. We show that an $n$-dependent correction must be applied to the $k(n,R)$ values provided by Prugniel & Simien (1997). Our analysis also shows that the shape of the velocity dispersion profile in the ATLAS$^{\rm 3D}$ sample varies weakly with $n$: $(\sigma_R/\sigma_e) = (R/R_e)^{-\gamma(n)}$. The resulting stellar mass functions, based on $M_*^{\rm \alpha_{JAM}}$ and the recalibrated virial mass, are in good agreement. If the $M_*^{\rm \alpha_{JAM}}/M_* - \sigma_e$ correlation is indeed due to the IMF, and stellar M/L gradients can be ignored, then our $\phi(M_*^{\rm \alpha_{JAM}})$ is an estimate of the stellar mass function in which $\sigma_e$-dependent variations in the IMF across the population have been accounted for. Using a Fundamental Plane based observational proxy for $\sigma_e$ produces comparable results. By demonstrating that cheaper proxies are sufficiently accurate, our analysis should enable a more reliable census of the mass in stars for large galaxy samples, at a fraction of the cost. Our results are provided in tabular form.