  The spatial segregation between dSphs and dIrrs in the Local Group has long
been regarded as evidence of an interaction with their host galaxies. In this
paper, we assume that ram-pressure stripping is the dominant mechanism that
removed gas from the dSphs and we use this to derive a lower bound on the
density of the corona of the Milky Way at large distances (50-90 kpc) from the
Galactic centre. At the same time, we derive an upper bound by demanding that
the interstellar medium of the dSphs is in pressure equilibrium with the hot
corona. We consider two dwarfs (Sextans and Carina) with well-determined orbits
and star formation histories. Our approach introduces several novel features:
we use the measured star formation histories of the dwarfs to derive the time
at which they last lost their gas, and (via a modified version of the
Kennicutt-Schmidt relation) their internal gas density at that time; we use a
large suite of 2D hydrodynamical simulations to model the gas stripping; and we
include supernova feedback tied to the gas content. Despite having very
different orbits and star formation histories, we find results for the two
dSphs that are in excellent agreement with one another. We derive an average
particle density of the corona of the Milky Way at 50-90 kpc in the range
1.3-3.6 10^{-4} cm^{-3}. Including additional constraints from X-ray emission
limits and pulsar dispersion measurements, we extrapolate Galactic coronal
density profiles and we estimate the fraction of baryons that can exist within
the virial radius of the Milky Way. For an isothermal corona (T=1.8 10^6 K)
this is small, 10-20 % of the universal baryon fraction. Only a hot (T=3 10^6
K) and adiabatic corona can contain all of the Galaxy's missing baryons. Models
for the Milky Way must explain why its corona is in a hot adiabatic thermal
state or why a large fraction of its baryons lie beyond the virial radius.
