On the Grain-Modified Magnetic Diffusivities in Protoplanetary Disks

Abstract

Weakly ionized protoplanetary disks (PPDs) are subject to non-ideal-magnetohydrodynamic (MHD) effects including Ohmic resistivity, the Hall effect and ambipolar diffusion (AD), and the resulting magnetic diffusivities (${\eta }_O,{\eta }_H$ and ${\eta }_A$) largely control the disk gas dynamics. The presence of grains not only strongly reduces disk ionization fraction, but also modify the scalings of ${\eta }_H$ and ${\eta }_A$ with magnetic field strength. We derive analytically asymptotic expressions of ${\eta }_H$ and ${\eta }_A$ in both strong and weak field limits and show that towards strong field, ${\eta }_H$ can change sign (at a threshold field strength $B_{\mathrm{th}}$), mimicking a flip of field polarity, and AD is substantially reduced. Applying to PPDs, we find that when small $\sim 0.1$ ($0.01$)$\mu$m grains are sufficiently abundant [mass ratio $\sim 0.01$ ($10^{-4}$)], ${\eta }_H$ can change sign up to $\sim 2-3$ scale heights above midplane at modest field strength (plasma $\beta \sim 100$) over a wide range of disk radii. Reduction of AD is also substantial towards the AD dominated outer disk and may activate the magneto-rotational instability. We further perform local non-ideal MHD simulations of the inner disk (within 10 AU) and show that with sufficiently abundant small grains, magnetic field amplification due to the Hall-shear instability saturates at very low level near the threshold field strength $B_{\mathrm{th}}$. Together with previous studies, we conclude by discussing the grain-abundance-dependent phenomenology of PPD gas dynamics.

12 pages, 6 figures. submitted to ApJ

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