A mathematical model is presented that describes the squeezing of solar wind plasma out along interplanetary magnetic field lines in the region between the bow shock and the effective planetary boundary (in the case of the earth, the magnetopause). In the absence of local magnetic merging the squeezing process should create a 'depletion layer,' a region of very low plasma density just outside the magnetopause. Numerical solutions are obtained for the dimensionless magnetohydrodynamic equations describing this depletion process for the case where the solar wind magnetic field is perpendicular to the solar wind flow direction. For the case of the earth with a magnetopause standoff distance of 10 RE, the theory predicts that the density should be reduced by a factor ?2 in a layer about 700--1300 km thick if MA, the Alfv¿n Mach number in the solar wind, is equal to 8. The layer thickness should vary as MA -2 and should be approximately uniform for a large area of the magnetopause around the subsolar point. Our computed layer thicknesses are somewhat smaller than those derived from Lees' axisymmetric model. Depletion layers should develop fully only where magnetic merging is locally unimportant. Scaling of the model calculations to Venus and Mars suggest layer thicknesses about 1/10 and 1/15 those of the earth, respectively, neglecting diffusion and ionospheric effects.