Theoretical study of ionization profiles of molecular clouds near supernova remnants

Context. Over the past few years, signatures of supernova remnants have been detected in gamma rays, particularly those that are known to be associated with molecular clouds. Whether these gamma rays are generated by cosmic-ray electrons emitting bremsstrahlung or experiencing inverse Compton scattering, or by cosmic-ray protons interacting with ambient hydrogen is usually not known. The detection of hadronic ionization signatures in spatial coincidence with gamma-ray signatures can help to unambiguously identify supernova remnants as sources of cosmic-ray protons. Aims. Our central aim is to develop a method to investigate whether the gamma rays are formed by cosmic-ray protons. To achieve this goal, we derived the position-dependent cosmic-ray-induced and photoinduced ionization rates. Methods. To calculate hadronic signatures from cosmic-ray-induced ionization to examine the origin of the observed gamma rays from molecular clouds associated with supernova remnants, we solved analytically the transport equation for cosmic-ray protons propagating in a molecular cloud, including the relevant momentum-loss processes, and determined the proton flux at any penetration depth into the cloud. Results. Because the solution of the transport equation is obtained for arbitrary source functions, it can be used for a variety of supernova remnants. We derived the corresponding theoretical ionization rate as a function of the penetration depth from the position-dependent proton flux, and compared it with photoinduced ionization profiles for available X-ray data in a case study with four supernova remnants associated with molecular clouds. Three of the remnants show a clear dominance of the hadronically induced ionization rate, while for one remnant, X-ray emission seems to dominate by a factor of 10. Conclusions. This is the first derivation of position-dependent profiles for cosmic-ray-induced ionization with an analytic solution for arbitrary cosmic-ray source spectra. The cosmic-ray-induced ionization has to be compared with photoionization for strong X-ray sources. For this purpose, measurements of X-ray spectra from supernova remnant shocks in the sub-keV to keV domain are necessary for a proper comparison. For sources dominated by cosmic-ray-induced ionization (e.g., W49B), the ionization profiles can be used in the future to map the spatial structure of hadronic gamma rays and rotation-vibrational lines induced by cosmic-ray protons. With instruments such as ALMA