Spectrum and ionization rate of low energy Galactic cosmic rays
  We consider the rate of ionization of diffuse and molecular clouds in the
interstellar medium by Galactic cosmic rays (GCR) in order to constrain its low
energy spectrum. We extrapolate the GCR spectrum obtained from PAMELA at high
energies ($\ge 200$ GeV/ nucleon) and a recently derived GCR proton flux at
$1\hbox{--}200$ GeV from observations of gamma rays from molecular clouds, and
find that the observed average Galactic ionization rate can be reconciled with
this GCR spectrum if there is a low energy cutoff for protons at
$10\hbox{--}100$ MeV. We also identify the flattening below a few GeV as being
due to (a) decrease of the diffusion coefficient and dominance of convective
loss at low energy and (b) the expected break in energy spectrum for a constant
spectral index in momentum. We show that the inferred CR proton spectrum of
$\Phi \propto E_{kin}^{-1.7\pm0.2}$ for $E_{kin} \le$ few GeV, is consistent
with a power-law spectrum in momentum $p^{-2.45\pm0.4}$, which we identify as
the spectrum at source. Diffusion loss at higher energies then introduces a
steepening by $E^{-\alpha}$ with $\alpha \sim 1/3$, making it consistent with
high energy measurements.
