  HD 209458b is an exoplanet with an upper atmosphere undergoing blow-off
escape that has mainly been studied using measurements of the Ly-alpha
absorption. Recently, high-resolution measurements of absorption in the He I
triplet line at 10830 angstroms of several exoplanets (including HD 209458b)
have been reported, creating a new opportunity to probe escaping atmospheres.
We aim to better understand the atmospheric regions of HD 209458b from where
the escape originates. We developed a 1D hydrodynamic model with spherical
symmetry for the HD 209458 b thermosphere coupled with a non-local
thermodynamic model for the population of the He triplet state. In addition, we
performed high-resolution radiative transfer calculations of synthetic spectra
for the He triplet lines and compared them with the measured absorption
spectrum in order to retrieve information about the atmospheric parameters. We
find that the measured spectrum constrains the [H]/[H$^{+}$] transition
altitude occurring in the range of 1.2 to 1.9Rp. H is almost fully ionised at
altitudes above 2.9Rp. We also find that the X-ray and EUV absorption takes
place at effective radii from 1.16 to 1.30Rp, and that the He triplet peak
density occurs at altitudes from 1.04 to 1.60Rp. Additionally, the averaged mmw
is confined to the 0.61-0.73 g/mole interval, and the thermospheric H/He ratio
should be larger than 90/10, and most likely approximately 98/2. We also
provide a one-to-one relationship between mass-loss rate and temperature. Based
on the energy-limited escape approach and assuming heating efficiencies of
0.1-0.2, we find a mass-loss rate in the range of (0.42-1.00)$\times 10^{11}$
g/s and a corresponding temperature range of 7125 to 8125K. The analysis of the
measured He triplet absorption spectrum significantly constrains the
thermospheric structure of HD 209458b and advances our knowledge of its
escaping atmosphere.
