Irregular dust grains are subject to radiative torques when irradiated by interstellar starlight. It is shown how these radiative torques may be calculated using the discrete dipole approximation. Calculations are carried out for one irregular grain geometry and three different grain sizes. It is shown that radiative torques can play an important dynamical role in spin-up of interstellar dust grains, resulting in rotation rates that may exceed even those expected from H_2 formation on the grain surface. Because the radiative torque on an interstellar grain is determined by the overall grain geometry rather than merely the condition of the grain surface, the resulting superthermal rotation is expected to be quite long-lived. By itself, long-lived superthermal rotation would permit grain alignment by normal paramagnetic dissipation on the " Davis-Greenstein" timescale τ_(DG). However, radiative torques arising from anisotropy of the starlight background can act directly to alter the grain alignment on times short compared to τ_(DG). Radiative torques must therefore play a central role in the process of interstellar grain alignment. The radiative torques depend strongly on the grain size, measured by a_(eff), the radius of a sphere of equal volume. In diffuse clouds, radiative torques dominate the torques due to H_2 formation for a_(eff) = 0.2 μm grains, but are relatively unimportant for a_(eff) ≤ 0.05 μm grains. We argue that this may provide a natural explanation for the observation that a_(eff) approx > 0.1 μm grains in diffuse clouds are aligned, while there is very little alignment of a_(eff) approx < 0.05 μm grains. We show that radiative torques are ineffective at producing superthermal rotation within quiescent dark clouds, but can be very effective in star-forming regions such as the M17 molecular cloud.
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