  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.
