Cosmic ray induced ionisation of a molecular cloud shocked by the W28
  supernova remnant
  Cosmic rays are an essential ingredient in the evolution of the interstellar
medium, as they dominate the ionisation of the dense molecular gas, where stars
and planets form. However, since they are efficiently scattered by the galactic
magnetic fields, many questions remain open, such as where exactly they are
accelerated, what is their original energy spectrum, and how they propagate
into molecular clouds. In this work we present new observations and discuss in
detail a method that allows us to measure the cosmic ray ionisation rate
towards the molecular clouds close to the W28 supernova remnant. To perform
these measurements, we use CO, HCO$^+$, and DCO$^+$ millimetre line
observations and compare them with the predictions of radiative transfer and
chemical models away from thermodynamical equilibrium. The CO observations
allow us to constrain the density, temperature, and column density towards each
observed position, while the DCO$^+$/HCO$^+$ abundance ratios provide us with
constraints on the electron fraction and, consequently, on the cosmic ray
ionisation rate. Towards positions located close to the supernova remnant, we
find cosmic ray ionisation rates much larger ($\gtrsim100$) than those in
standard galactic clouds. Conversely, towards one position situated at a larger
distance, we derive a standard cosmic ray ionisation rate. Overall, these
observations support the hypothesis that the $\gamma$ rays observed in the
region have a hadronic origin. In addition, based on CR diffusion estimates, we
find that the ionisation of the gas is likely due to $0.1 -1$ GeV cosmic rays.
Finally, these observations are also in agreement with the global picture of
cosmic ray diffusion, in which the low-energy tail of the cosmic ray population
diffuses at smaller distances than the high-energy counterpart.
