Reference study to characterise plasma and magnetic properties of ultra-cool atmospheres

Radio and X-ray emission from brown dwarfs suggest that an ionised gas and a magnetic field with a sufficient flux density must be present. We perform a reference study for late M-dwarfs, brown dwarfs and giant gas planet to identify which ultra-cool objects are most susceptible to plasma and magnetic processes. Only thermal ionisation is considered. We utilise the {\sc Drift-Phoenix} model grid where the local atmospheric structure is determined by the global parameters T$_{\rm eff}$, $\log(g)$ and [M/H]. Our results show that it is not unreasonable to expect H$_{\alpha}$ or radio emission to origin from Brown Dwarf atmospheres as in particular the rarefied upper parts of the atmospheres can be magnetically coupled despite having low degrees of thermal gas ionisation. Such ultra-cool atmospheres could therefore drive auroral emission without the need for a companion's wind or an outgassing moon. The minimum threshold for the magnetic flux density required for electrons and ions to be magnetised is well above typical values of the global magnetic field of a brown dwarf and a giant gas planet. Na$^{+}$, K$^{+}$ and Ca$^{+}$ are the dominating electron donors in low-density atmospheres (low log(g), solar metallicity) independent of T$_{\rm eff}$. Mg$^{+}$ and Fe$^{+}$ dominate the thermal ionisation in the inner parts of M-dwarf atmospheres. Molecules remain unimportant for thermal ionisation. Chemical processes (e.g. cloud formation) affecting the most abundant electron donors, Mg and Fe, will have a direct impact on the state of ionisation in ultra-cool atmospheres.