2021ApJ...908..218H

Dust polarization induced by aligned grains is widely used to study magnetic fields in various astrophysical environments. However, the question of to what optical depth grain alignment still exists in a dense molecular cloud (MC) is unclear. In this paper, we derive analytical formulae for the minimum size of aligned grains (a_align) and rotational disruption (a_disr) by RAdiative Torques (RATs) as a function of the local physical parameters within MCs. We first find the analytical approximations for the radiation strength and mean wavelength of the attenuated radiation field in a dense MC with and without embedded stars, and then derive analytical formulae for a_align and a_disr as functions of the visual extinction A_V and gas density. We find that, within a starless core of density n_H∼10⁵cm^–3, grains of size a< 0.25µm can be aligned at A_V ∼ 5 by RATs, whereas micron-sized grains can still be aligned at A_V∼50. The increase in a_align with A_V can explain the presence of polarization holes observed toward starless cores. For MCs with an embedded protostar, the efficiency of both alignment and rotational disruption increases toward the protostar due to the increasing radiation strength. Such a disruption effect results in the decrease of the polarization degree with A_V or emission intensity, reproducing the popular polarization holes observed toward the location of protostars. Finally, we derive the formula for the maximum A_V where grain alignment still exists in a starless core, and we discuss its potential for constraining grain growth.