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We present new HCN and HCO$^+$ ($J$=3-2) images of the nearby star-forming galaxies (SFGs) NGC 3351, NGC 3627, and NGC 4321. The observations, obtained with the Morita ALMA Compact Array, have a spatial resolution of $\sim$290-440 pc and resolve the inner $R_\textrm{gal} \lesssim$ 0.6-1 kpc of the targets, as well as the southern bar end of NGC 3627. We complement this data set with publicly available images of lower excitation lines of HCN, HCO$^+$, and CO and analyse the behaviour of a representative set of line ratios: HCN(3-2)/HCN(1-0), HCN(3-2)/HCO$^+$(3-2), HCN(1-0)/CO(2-1), and HCN(3-2)/CO(2-1). Most of these ratios peak at the galaxy centres and decrease outwards. We compare the HCN and HCO$^+$ observations with a grid of one-phase, non-local thermodynamic equilibrium (non-LTE) radiative transfer models and find them compatible with models that predict subthermally excited and optically thick lines. We study the systematic variations of the line ratios across the targets as a function of the stellar surface density ($\Sigma_\textrm{star}$), the intensity-weighted CO(2-1) ($\langle I_\text{CO}\rangle$), and the star formation rate surface density ($\Sigma_\text{SFR}$). We find no apparent correlation with $\Sigma_\text{SFR}$, but positive correlations with the other two parameters, which are stronger in the case of $\langle I_\text{CO}\rangle$. The HCN/CO-$\langle I_\text{CO}\rangle$ relations show $\lesssim$0.3 dex galaxy-to-galaxy offsets, with HCN(3-2)/CO(2-1)-$\langle I_\text{CO}\rangle$ being $\sim$2 times steeper than HCN(1-0)/CO(2-1). In contrast, the HCN(3-2)/HCN(1-0)-$\langle I_\text{CO}\rangle$ relation exhibits a tighter alignment between galaxies. We conclude that the overall behaviour of the line ratios cannot be ascribed to variations in a single excitation parameter (e.g. density or temperature)., Comment: Accepted for publication in A&A. 14 pages, 8 figures

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We present high resolution [N ii] 205 μm (3P 1-3P0) spectra obtained with Herschel-HIFI towards a small sample of far-infrared bright star forming regions in the Galactic plane: W 31C (G10.6-0.4), W 49N (G43.2-0.1), W 51 (G49.5-0.4), and G34.3+0.1. All sources display an emission line profile associated directly with the H ii regions themselves. For the first time we also detect absorption of the [N ii] 205 μm line by extended low-density foreground material towards W 31C and W 49N over a wide range of velocities. We attribute this absorption to the warm ionised medium (WIM) and find N(N+) ≈ 1.5 × 10 17 cm-2 towards both sources. This is in agreement with recent Herschel-HIFI observations of [C ii] 158 μm, also observed in absorption in the same sight-lines, if ≈ 7-10% of all C+ ions exist in the WIM on average. Using an abundance ratio of [N]/[H] = 6.76 × 10-5 in the gas phase we find that the mean electron and proton volume densities are ~0.1-0.3 cm-3 assuming a WIM volume filling fraction of 0.1-0.4 with a corresponding line-of-sight filling fraction of 0.46-0.74. A low density and a high WIM filling fraction are also supported by RADEX modelling of the [N ii] 205 μm absorption and emission together with visible emission lines attributed mainly to the WIM. The detection of the 205 μm line in absorption emphasises the importance of a high spectral resolution, and also offers a new tool for investigation of the WIM., C.M.P. and J.H.B. acknowledge generous support from the Swedish National Space Board. J.R.G. thanks the Spanish MINECO for funding support under grants CSD2009-00038, AYA2009-07304 and AYA2012-32032.