The $L - \sigma$ relation for massive bursts of star formation

The validity of the emission line luminosity vs. ionised gas velocity dispersion ($L - \sigma$) correlation for HII galaxies (HIIGx), and its potential as an accurate distance estimator are assessed. For a sample of 128 local ($0.02\lesssim z\lesssim 0.2$) compact HIIGx with high equivalent widths of their Balmer emission lines we obtained ionised gas velocity dispersion from high S/N high-dispersion spectroscopy (Subaru-HDS and ESO VLT-UVES) and integrated H$\beta$ fluxes from low dispersion wide aperture spectrophotometry. We find that the $L - \sigma$ relation is strong and stable against restrictions in the sample (mostly based on the emission line profiles). The `gaussianity' of the profile is important for reducing the rms uncertainty of the distance indicator, but at the expense of substantially reducing the sample. By fitting other physical parameters into the correlation we are able to significantly decrease the scatter without reducing the sample. The size of the starforming region is an important second parameter, while adding the emission line equivalent width or the continuum colour and metallicity, produces the solution with the smallest rms scatter=$\delta \log L(\mathrm{H}\beta) = 0.233$. The derived coefficients in the best $L - \sigma$ relation are very close to what is expected from virialized ionising clusters, while the derived sum of the stellar and ionised gas masses are similar to the dynamical mass estimated using the HST corrected Petrosian radius. These results are compatible with gravity being the main mechanism causing the broadening of the emission lines in these very young and massive clusters. The derived masses range from about 2 $\times10^6$ $\mathrm{M}_{\odot}$ to $10^9$ $\mathrm{M}_{\odot}$ and their `corrected' Petrosian radius, from a few tens to a few hundred parsecs.