An adiabatic approximation for grain alignment theory

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The alignment of interstellar dust grains is described by the joint distribution function for certain ``internal'' and ``external'' variables, where the former describe the orientation of a grain's axes with respect to its angular momentum, J, and the latter describe the orientation of J relative to the interstellar magnetic field. I show how the large disparity between the dynamical timescales of the internal and external variables--- which is typically 2--3 orders of magnitude--- can be exploited to greatly simplify calculations of the required distribution. The method is based on an ``adiabatic approximation'' which closely resembles the Born-Oppenheimer approximation in quantum mechanics. The adiabatic approximation prescribes an analytic distribution function for the ``fast'' dynamical variables and a simplified Fokker-Planck equation for the ``slow'' variables which can be solved straightforwardly using various techniques. These solutions are accurate to order epsilon, where epsilon is the ratio of the fast and slow dynamical timescales. As a simple illustration of the method, I derive an analytic solution for the joint distribution established when Barnett relaxation acts in concert with gas damping. The statistics of the analytic solution agree with the results of laborious numerical calculations which do not exploit the adiabatic approximation.