A new method (a variant of tumble finishing) for polishing and achieving local planarization on precision spherical, plastic capsules is described. Such capsules have niche applications, such as ablators used in high-peak-power laser targets for fusion energy research. The as-manufactured ablators contain many shallow domes (many 100's of nm high and a few 10's of mu m wide) on the outer surface which are undesirable due to contributions to instabilities during implosion. These capsules were polished (i.e., tumble finished) by rotating a cylindrical vial containing the capsule, many borosilicate glass or zirconia media, and an aqueous-based colloidal silica polishing slurry. During tumble finishing, the relative media/capsule motions cause multiple, random sliding spherical-spherical Hertzian contacts, resulting in material removal, and possibly plastic deformation, on the capsule. As a result, the domes were observed to locally planarize (i.e., converge to lower heights). Utilizing the correct kinematics (i.e., the characteristics of the media/capsule motions), as controlled by the vial rotation rate and the fill fraction of media and slurry, the high velocity downward circumferential media motions were avoided, preventing fracturing of the fragile capsules. Also, the resulting post-polished surface roughness on the capsule was found to scale with the initial media surface roughness. Hence, pre-polishing the media greatly reduced the roughness of the media and thus the roughness of the polished capsule. A material removal model is described based on the Preston model and spherical-spherical Hertzian contacts which shows reasonable agreement with measured average removal rates of 35 +/- 15 nm/day and which serves as a valuable tool to scale the polishing behavior with changes in process variables. Narrow domes were observed to planarize more rapidly than wider domes. A local planarization convergence model is also described, based on the concept of workpiece-lap mismatch where the local pressure, and hence removal, varies with the gap at the interface contact. The calculated rate and shape evolution of various size isolated domes compares well with the experimental data. (C) 2012 Elsevier B. V. All rights reserved.