  Aims. In this letter, we calculate for the first time the full transonic
hydrodynamic escape of an Earth-like atmosphere. We consider the case of an
Earth-mass planet with an atmospheric composition identical to that of the
current Earth orbiting at 1 AU around a young and very active solar mass star.
  Methods. To model the upper atmosphere, we used the Kompot Code, which is a
first-principles model that calculates the physical structures of the upper
atmospheres of planets, taking into account hydrodynamics and the main chemical
and thermal processes taking place in the upper atmosphere of a planet. This
model enabled us to calculate the 1D vertical structure of the atmosphere using
as input the high-energy spectrum of a young and active Sun.
  Results. The atmosphere has the form of a transonic hydrodynamic Parker wind,
which has an outflow velocity at the upper boundary of our computational domain
that exceeds the escape velocity. The outflowing gas is dominated by atomic
nitrogen and oxygen and their ion equivalents and has a maximum ionization
fraction of 20%. The mass outflow rate is found to be 1.8x10^9 g s^-1, which
would erode the modern Earth's atmosphere in less than 0.1 Myr.
  Conclusions. This extreme mass loss rate suggests that an Earth-like
atmosphere cannot form when the planet is orbiting within the habitable zone of
a very active star. Instead, such an atmosphere can only form after the
activity of the star has decreased to a much lower level. This happened in the
early atmosphere of the Earth, which was likely dominated by other gases such
as CO2. Since the time it takes for the activity of a star to decay is highly
dependent on its mass, this is important for understanding possible formation
timescales for planets orbiting low-mass stars.
