Recent observations show that chromospheric Hα activity in late M and L dwarfs is much lower than in the earlier M types. This is particularly surprising given that the late M and L dwarfs are comparatively very rapid rotators: in the early M dwarfs, rapid rotation is associated with high activity levels. One possibility is that the drop-off in activity in the late M and L dwarfs is a result of very high electrical resistivities in their dense, cool, and predominantly neutral atmospheres.We calculate the magnetic field diffusivity in the atmospheres of objects with Teff in the range 3000-1500 K (mid M to late L) using the atmospheric structure models of Allard and Hauschildt. We find that the combination of very low ionization fraction and high density in these atmospheres results in very large resistivities and thus efficient field diffusion. While both ambipolar diffusion and Ohmic decay of currents due to ion-electron collisions occur, the primary diffusion effects are due to current decay through collisions of charged particles with neutrals. Moreover, the latter resistivity is a strong function of both effective temperature and optical depth, increasing rapidly as either Teff or optical depth decreases. This has two implications: (1) Any magnetic field present is increasingly decoupled from atmospheric fluid motions as one moves from mid M to L. In the late M and L dwarfs, atmospheric motions cannot lead to equilibrium field configurations very different from potential ones. That is, the magnitude of magnetic stresses generated by atmospheric motions is very small in these objects. We quantify this effect by a simple Reynolds number calculation. (2) Even if magnetic stresses are easily produced by fluid motions in the hot interior (where the coupling between field and matter is good), their propagation up through the atmosphere will be increasingly hampered by the growing atmospheric resistivity as one moves from mid M to late L. Thus both the generation and propagation of magnetic stresses are increasingly damped with decreasing Teff in these cool dwarfs. As a result, the magnetic free energy available for the support of a chromosphere, and activity becomes smaller and smaller with later type. This can account for the observed drop in Hα activity from mid M to L, assuming that activity in these dwarfs is magnetically driven. To check the latter assumption, we estimate the emergent acoustic fluxes in these objects through a Lighthill-Proudman calculation. While the acoustic fluxes also decrease with decreasing Teff, they appear inadequate to explain the observed Hα fluxes in mid M to L dwarfs. In the absence of acoustic heating, magnetic heating indeed seems the most viable way of generating activity. We emphasize that our calculations are equilibrium ones and do not address time-dependent phenomena. We also do not examine the highest atmospheric layers, where correction is not expected, but which we show are rarefied enough to permit charged particles to remain coupled to the field. Finally, while our calculations do not address flares in late M and L dwarfs, we speculate that the latter could be created by buoyant flux tubes that are generated in the interior and rise rapidly through the atmosphere, dissipating their associated currents in the upper atmospheric layers.
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