By considering martian-like planetary embryos inside the habitable zone of solar-like stars we study the behavior of the hydrodynamic atmospheric escape of hydrogen for small values of the Jeans escape parameter \beta < 3 , near the base of the thermosphere , that is defined as a ratio of the gravitational and thermal energy .
Our study is based on a 1-D hydrodynamic upper atmosphere model that calculates the volume heating rate in a hydrogen dominated thermosphere due to the absorption of the stellar soft X-ray and extreme ultraviolet ( XUV ) flux .
We find that when the \beta value near the mesopause/homopause level exceeds a critical value of \sim 2.5 , there exists a steady hydrodynamic solution with a smooth transition from subsonic to supersonic flow .
For a fixed XUV flux , the escape rate of the upper atmosphere is an increasing function of the temperature at the lower boundary .
Our model results indicate a crucial enhancement of the atmospheric escape rate , when the Jeans escape parameter \beta decreases to this critical value .
When \beta becomes \leq 2.5 , there is no stationary hydrodynamic transition from subsonic to supersonic flow .
This is the case of a fast non-stationary atmospheric expansion that results in extreme thermal atmospheric escape rates .