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We re-examine the inviscid solar-wind equations with heat conduction from below, and establish a fundamentally new approach for finding solar and plane- tary solutions. Although the problem is fourth order, only two independent integration constants can be assigned, since two boundary conditions that are required to specify well-behaved supersonic solutions determine the values of the other two constants. The solar-wind models of Noble and Scarf (Ap. J., 1963-65), are essentially accurate for practical purposes, but in a fundamental sense they are not self-consistent. At the supersonic point, the ratio of thermal energy to gravitational potential, (kTr/GMm), must lie in a narrow range, between 0.4375 and about 0.40, to permit well-behaved supersonic solutions for ionized hydrogen. For a super- sonic solution, this ratio uniquely determines the escape flux, the energy flux per particle, and the temperature gradient at infinity. For blowoff of the atmospheres of planets with diatomic molecular atmo- spheres, this range is lower but still quite narrow. These limits seriously constrain the physical conditions wherein hydrodynamic "blowoff" of planetary atmospheres can develop. In addition, an atmosphere having blowoff conditions is adiabatically unstable just above the sonic level.
