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Our objective is to investigate the physical properties of the ionised gas of the low-metallicity dwarf galaxy, IC 10, at various spatial scales: from individual HII regions to the entire galaxy scale and examine whether diagnostics for integrated measurements introduce bias in the results. We modeled the ionised gas combining the mid- and far-infrared fine-structure cooling lines observed with Spitzer/IRS and Herschel/PACS, with the photoionisation code Cloudy. The free parameters of the models are the age of the stellar cluster, the density and the ionisation parameter of the ionised gas as well as the depth of the cloud. The latter is used to investigate the leakage of the ionising photons from the analysed regions of IC 10. We investigate HII regions in the main star-forming body, on scales of ~25 pc, three in the main star-forming region in the center of the galaxy and two on the first arc. We then consider larger sizes on the scale of ~200 pc. We find that most clumps have nearly identical properties, density ~10$^{2.} $ - 10$^{2.6}$ cm$^{-3}$, ionisation parameter between 10$^{-2.2}$ and 10$^{-1.6}$ and age of the stellar cluster ~5.5 Myr. All of them are matter-bounded regions, allowing ionising photons to leak. The relatively uniform physical properties of the clumps suggest a common origin for their star formation activity, which could be related to the feedback from stellar winds or supernovae of a previous generation of stars. The properties derived for ~200 pc size "zones" have similar properties as the HII regions they encompass, but with the larger regions tending to be more radiation-bounded. Finally, we investigate the fraction of [CII] 157.7 {\mu}m, [SiII] 34.8 {\mu}m, and [FeII] 25.9 {\mu}m, emission arising from the ionised gas phase and we find that most of the emission originates from the neutral gas, not from the ionised gas., Comment: accepted for publication in A&A

Stars form out of the densest parts of molecular clouds. Far-IR emission can be used to estimate the Star Formation Rate (SFR) and high dipole moment molecules, typically HCN, trace the dense gas. A strong correlation exists between HCN and Far-IR emission, with the ratio being nearly constant, over a large range of physical scales. A few recent observations have found HCN to be weak with respect to the Far-IR and CO in subsolar metallicity (low-Z) objects. We present observations of the Local Group galaxies M33, IC10, and NGC6822 with the IRAM 30meter and NRO 45m telescopes, greatly improving the sample of low-Z galaxies observed. HCN, HCO$^+$, CS, C$_2$H, and HNC have been detected. Compared to solar metallicity galaxies, the Nitrogen-bearing species are weak (HCN, HNC) or not detected (CN, HNCO, N$_2$H$^+$) relative to Far-IR or CO emission. HCO$^+$ and C$_2$H emission is normal with respect to CO and Far-IR. While $^{13}$CO is the usual factor 10 weaker than $^{12}$CO, C$^{18}$O emission was not detected down to very low levels. Including earlier data, we find that the HCN/HCO$^+$ ratio varies with metallicity (O/H) and attribute this to the sharply decreasing Nitrogen abundance. The dense gas fraction, traced by the HCN/CO and HCO$^+$/CO ratios, follows the SFR but in the low-Z objects the HCO$^+$ is much easier to measure. Combined with larger and smaller scale measurements, the HCO$^+$ line appears to be an excellent tracer of dense gas and varies linearly with the SFR for both low and high metallicities., Comment: 9 pages, 13 figures, A&A in press