[C II] emission and star formation in late-type galaxies

We study the relationship between gas cooling via the [C II] ($\rm \lambda = 158~\mu m$) line emission and dust cooling via the far-IR continuum emission on the global scale of a galaxy in normal (i.e. non-AGN dominated and non-starburst) late-type systems. It is known that the luminosity ratio of total gas and dust cooling, ${L_{\mathrm{C~II}}/L_{\mathrm{FIR}}\rm}$, shows a non-linear behaviour with the equivalent width of the $\rm H \alpha$ ($\rm \lambda = 6563~\AA$) line emission, the ratio decreasing in galaxies of lower massive star-formation activity. This result holds despite the fact that known individual Galactic and extragalactic sources of the [C II] line emission show different [C II] line-to-far-IR continuum emission ratios. This non-linear behaviour is reproduced by a simple quantitative theoretical model of gas and dust heating from different stellar populations, assuming that the photoelectric effect on dust, induced by far-UV photons, is the dominant mechanism of gas heating in the general diffuse interstellar medium of the galaxies under investigation. According to the model, the global $L_{\mathrm{C~II}}/L_{\mathrm{FIR}}\rm$ provides a direct measure of the fractional amount of non-ionizing UV light in the interstellar radiation field and not of the efficiency of the photoelectric heating. The theory also defines a method to constrain the stellar initial mass function from measurements of $L_{\mathrm{C~II}}\rm$ and $L_{\mathrm{FIR}}\rm$. A sample of 20 Virgo cluster galaxies observed in the [C II] line with the Long Wavelength Spectrometer on board the Infrared Space Observatory is used to illustrate the model. The limited statistics and the necessary assumptions behind the determination of the global [C II] luminosities from the spatially limited data do not allow us to establish definitive conclusions but data-sets available in the future will allow tests of both the reliability of the assumptions behind our model and the statistical significance of our results.