Arc-minute-scale studies of the interstellar gas towards HESS$\,$J1804$-$216: Still an unidentified TeV $\gamma$-ray source

The Galactic TeV $\gamma$-ray source HESS$\,$J1804$-$216 is currently an unidentified source. In an attempt to unveil its origin, we present here the most detailed study of interstellar gas using data from the Mopra Southern Galactic Plane CO Survey, 7 and 12$\,$mm wavelength Mopra surveys and Southern Galactic Plane Survey of HI. Several components of atomic and molecular gas are found to overlap HESS$\,$J1804$-$216 at various velocities along the line of sight. The CS(1-0) emission clumps confirm the presence of dense gas. Both correlation and anti-correlation between the gas and TeV $\gamma$-ray emission have been identified in various gas tracers, enabling several origin scenarios for the TeV $\gamma$-ray emission from HESS$\,$J1804$-$216. For a hadronic scenario, SNR$\,$G8.7$-$0.1 and the progenitor SNR of PSR$\,$J1803$-$2137 require cosmic ray (CR) enhancement factors of $\mathord{\sim} 50$ times the solar neighbour CR flux value to produce the TeV $\gamma$-ray emission. Assuming an isotropic diffusion model, CRs from both these SNRs require a slow diffusion coefficient, as found for other TeV SNRs associated with adjacent ISM gas. The morphology of gas located at 3.8$\,$kpc (the dispersion measure distance to PSR$\,$J1803$-$2137) tends to anti-correlate with features of the TeV emission from HESS$\,$J1804$-$216, making the leptonic scenario possible. Both pure hadronic and pure leptonic scenarios thus remain plausible.