We study the spatial intensity profiles of galactic cosmic ray protons (H) and &agr;‐particles (He) during the solar minimum periods of 1987 (the so‐called negative drift state) and 1977/1997 (both positive drift states) of the heliosphere. These intensities were measured with the Pioneer, Voyager and IMP spacecraft. The 1997 intensities were so low that one cannot readily explain them, even with the acceleration at the solar wind termination shock (SWTS) and modulation in the heliosheath included. Our heliospheric model is azimuthally symmetric with a spherical shock and heliopause, however, and we infer from its results that more realistic geometries may produce modulation effects that will explain the observations better. © 2004 American Institute of Physics