  The supernova remnant (SNR) G338.3-0.0 spatially correlates with HESS
J1640-465, which is considered the most luminous gamma-ray source associated
with a SNR in our Galaxy. The X-ray pulsar PSR J1640-4631 has been recently
discovered within the SNR shell, which could favor a leptonic origin for the
detected very-high energy (VHE) emission. In spite of this, the origin of the
VHE radiation from HESS J1640-465 has not been unambiguously clarified so far.
Indeed, a hadronic explanation cannot be ruled out by current observations. On
the basis of atomic (HI) and molecular (12^CO) archival data, we determine, for
the first time, the total ambient density of protons in the region of the
G338.3-0.0/HESS J1640-465 system, a critical parameter for understanding the
emission mechanisms at very high energies. The value obtained is in the 100-130
cm^-3 range. Besides this, we developed a new hadronic model to describe the
spectral energy distribution (SED) of the HESS J1640-465 source, which includes
the latest total gamma-ray cross-section for proton-proton collisions available
in the literature. By using the assessed ambient proton density, we found that
the total energy in accelerated protons required to fit the data is
5.4^{+4.7}_{-2.3} x 10^49 erg and 1.6^{+1.4}_{-0.7} x 10^50 erg for a source
distance of 8.5 and 13 kpc, respectively. The case where the source distance is
8.5 kpc agrees with the typical scenario in which the energy released is on the
order of 10^51 erg and ~10% of that energy is transferred to the accelerated
protons, whereas the case corresponding to a source distance of 13 kpc requires
either a higher value of the energy released in the explosion or a larger
energy fraction to accelerate protons.
