New constraints are placed on the electron and ion temperature gradients at Parker's sonic point, demonstrated here as the location for an ion where the force of gravity and the electric field parallel to the magnetic field are balanced. The equal-temperature, one-fluid versions of these results precisely recover Parker's necessary and sufficient conditions for the supersonic expansion. The two species constraints are largely within the Parker envelope. For all possible critical points, the electron partial pressure must exceed 0.2437 of the plasma pressure at this point. The location of the critical point relative to the isothermal critical point is determined by the partial pressure average of the electron and ion logarithmic derivatives of temperature at the critical point. The bulk acceleration at the critical point is controlled by (1) the ion (as distinct from the electron) temperature gradient at this point and (2) the ratio of the ion to electron pressure at the critical point. The sonic point is a velocity space deLaval nozzle rather than a traditional configuration space nozzle. Critical points are most frequently accompanied by positive radial ion and electron temperature gradients. The electron and ion temperature gradients required for a critical point are coupled and not arbitrary. Parker's one-fluid discussion requires, by this analysis, both electron and ion temperature gradients to have positive radial exponents &bgr;=3/4 near the critical point and a local effective gas polytrope behavior of &ggr;=10/13&ggr;e(r*)>1/3. The allowed critical points with &ggr;<1 are used to infer a range of nonthermal &kgr; distributions that permit this unusual &ggr;<1 behavior; ion nonthermal suprathermal tail indices ranging between 2<&kgr;i<5 are required, reflective of decidedly nonthermal distributions at the critical point and consistent with previous inferences of their size [Scudder, 1992b>. Even when unequal electron and ion temperatures are considered, the vast preponderance of the critical points delineated also is found with 1/3<&ggr;e<1, with the attendant range of electron nonthermal indices, &kgr;e, ranging from 2 to the Maxwellian limit of ∞. Critical points with maximal acceleration invariably occur for a positive electron temperature power law exponent of &bgr;e≂1.12 that corresponds to &kgr;e≂3.3, whether or not the ions and electrons have a common temperature at the critical point. ¿ American Geophysical Union 1996