A recent axisymmetric magnetohydrodynamic (MHD) model for the Earth's magnetosheath near the Sun-Earth line, with boundary conditions given by Rankine-Hugoniot (RH) equations at the bow shock and the Chapman-Ferraro (CF) condition at the magnetopause, is extended by including smaller terms in the CF condition that become important at low Mach numbers. Three conclusions are predicted for the magnetosheath. (1) The magnetosheath thickness Δms decreases with decreasing Alfven Mach number MA for MA>M* (in which only the gasdynamic root of the RH equations for the bow shock boundary is physical), where M*~2 is the root-transition value for the RH equations at which all 3 roots coalesce. When MA* switch-on shock roots (purely MHD) are also physical, and Δms associated with those roots shows a rapid increase to a large value, followed by a rapid decrease back to a very small value as the decreasing MA approaches 1. Thus Δms associated with these switch-on roots may give more distant bow shocks, and may rapidly vary with small MA changes. (2) Δms is very sensitive to the value for the CF constant kCF at the magnetopause. For larger MA and sonic Mach numbers Ms the ratio of Δms to the geocentric radial distance amp of the magnetopause varies from 0.44 for 90% coupling efficiency of the solar wind momentum density to the magnetopause down to 0.18 for 60% coupling efficiency. This suggests a balance between the solar wind coupling efficiency to the magnetopause and the thickness of the magnetosheath, in which an increase (decrease) in the coupling efficiency increases (decreases) the thickness of the magnetosheath to counter that efficiency by moving the bow shock further upstream (downstream), hence minimizing the variation in time of that coupling efficiency. (3) The linear relation between Δms and the density ratio &rgr;sw/&rgr;bs observed in simulations breaks down for MA or Ms≲2. The slope steadily drops as Ms→1, but increases as MA→1.¿ 1997 American Geophysical Union