In an earlier paper we modeled the far-infrared emission from a star-forming galaxy using the photoionisation code CLOUDY and presented metallicity sensitive diagnostics based on far-infrared fine structure line ratios. Here, we focus on the applicability of the [OIII]88/[NII]122 microns line ratio as a gas phase metallicity indicator in high redshift submillimetre luminous galaxies. The [OIII]88/[NII]122 microns ratio is strongly dependent on the ionization parameter (which is related to the total number of ionizing photons) as well as the gas electron density. We demonstrate how the ratio of 88/$122 continuum flux measurements can provide a reasonable estimate of the ionization parameter while the availability of the [NII]205 microns line can constrain the electron density. Using the [OIII]88/[NII]122 microns line ratios from a sample of nearby normal and star-forming galaxies we measure their gas phase metallicities and find that their mass metallicity relation is consistent with the one derived using optical emission lines. Using new, previously unpublished, Herschel spectroscopic observations of key far-infrared fine structure lines of the z~3 galaxy HLSW-01 and additional published measurements of far-infrared fine structure lines of high-z submillimetre luminous galaxies we derive gas phase metallicities using their [OIII]88/[NII]122 microns line ratio. We find that the metallicities of these z~3 submm luminous galaxies are consistent with solar metallicities and that they appear to follow the mass-metallicity relation expected for z~3 systems.
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