Where the Mass Sits in a Power-Law Grain Population: Mass and Surface Budgets of the Size Distribution

A power-law grain size distribution contains very different answers depending on whether mass, surface area, or number is counted. This paper evaluates the analytic mass and surface integrals for power-law indices from 3.0 to 4.0 and upper size cutoffs from 0.25 to 1.0 microns. For the classic index of 3.5 with an upper cutoff of 0.25 microns, 0.4281 of the mass lies in grains larger than 0.1 microns, while 0.9688 of the grains by number are smaller than 0.02 microns, and the mass-to-surface ratio corresponds to an effective radius of 0.0471 microns. Flattening the index to 3.0 and extending the cutoff to 1 micron raises the large-grain mass fraction to 0.9045, whereas steepening to 4.0 lowers it to 0.2342 at the standard cutoff. Extinction, emission, chemistry, and dynamics each weight the distribution differently, so quoting a single typical grain size without stating the weighting is not a well-posed description. All values are computed from stated assumptions, not measured.