Context. According to traditional gas-phase chemical models, O$_{2}$ should be abundant in molecular clouds, but until recently, attempts to detect interstellar O$_{2}$ line emission with ground-... Show moreContext. According to traditional gas-phase chemical models, O$_{2}$ should be abundant in molecular clouds, but until recently, attempts to detect interstellar O$_{2}$ line emission with ground- and space-based observatories have failed. Aims: Following the multi-line detections of O$_{2}$ with low abundances in the Orion and {$ρ$} Oph A molecular clouds with Herschel, it is important to investigate other environments, and we here quantify the O$_{2}$ abundance near a solar-mass protostar. Methods: Observations of molecular oxygen, O$_{2}$, at 487 GHz toward a deeply embedded low-mass Class 0 protostar, NGC 1333-IRAS 4A, are presented, using the Heterodyne Instrument for the Far Infrared (HIFI) on the Herschel Space Observatory. Complementary data of the chemically related NO and CO molecules are obtained as well. The high spectral resolution data are analysed using radiative transfer models to infer column densities and abundances, and are tested directly against full gas-grain chemical models. Results: The deep HIFI spectrum fails to show O$_{2}$ at the velocity of the dense protostellar envelope, implying one of the lowest abundance upper limits of O$_{2}$/H$_{2}$ at {le}6 { imes} 10$^{-9}$ (3{$σ$}). The O$_{2}$/CO abundance ratio is less than 0.005. However, a tentative (4.5{$σ$}) detection of O$_{2}$ is seen at the velocity of the surrounding NGC 1333 molecular cloud, shifted by 1 km s$^{-1}$ relative to the protostar. For the protostellar envelope, pure gas-phase models and gas-grain chemical models require a long pre-collapse phase (~{}0.7-1 { imes} 10$^{6}$ years), during which atomic and molecular oxygen are frozen out onto dust grains and fully converted to H$_{2}$O, to avoid overproduction of O$_{2}$ in the dense envelope. The same model also reproduces the limits on the chemically related NO molecule if hydrogenation of NO on the grains to more complex molecules such as NH$_{2}$OH, found in recent laboratory experiments, is included. The tentative detection of O$_{2}$ in the surrounding cloud is consistent with a low-density PDR model with small changes in reaction rates. Conclusions: The low O$_{2}$ abundance in the collapsing envelope around a low-mass protostar suggests that the gas and ice entering protoplanetary disks is very poor in O$_{2}$. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.Appendices are available in electronic form at http://www.aanda.orgReduced spectra (FITS files) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/558/A58Show less
