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Cosmic-ray (CR) electrons and nuclei interact with the Galactic interstellargas and produce high-energy $\gamma$ rays. The $\gamma$-ray emission rate perhydrogen atom, called emissivity, provides a unique indirect probe of the CRflux. We present the measurement and the interpretation of the emissivity inthe solar neighborhood for $\gamma$-ray energy from 50~MeV to 50~GeV. Weanalyzed a subset of 4 years of observations from the Large Area Telescope(LAT) aboard the {\it Fermi Gamma-ray Space Telescope} ({\it Fermi}) restrictedto absolute latitudes $10^o<|b| <70^o$. From a fit to the LAT data includingatomic, molecular and ionized hydrogen column density templates as well as adust optical depth map we derived the emissivities, the molecular hydrogen toCO conversion factor $X_{CO}=(0.902\pm0.007) \times 10^{20}$ cm$^{-2}$ (K kms$^{-1}$)$^{-1}$ and the dust-to-gas ratio $X_{DUST}=(41.4\pm0.3) \times10^{20}$ cm$^{-2}$ mag$^{-1}$. Moreover we detected for the first time$\gamma$-ray emission from ionized hydrogen. We compared the extractedemissivities to those calculated from $\gamma$-ray production cross-sectionsand to CR spectra measured in the heliosphere. We observed that theexperimental emissivities are reproduced only if the solar modulation isaccounted for. This provides a direct detection of solar modulation observedpreviously through the anticorrelation between CR fluxes and solar activity.Finally we fitted a parametrized spectral form to the heliospheric CRobservations as well as to the {\it Fermi}-LAT emissivity and obtainedcompatible local interstellar spectra for proton and Helium kinetic energy pernucleon between between 1 and 100~GeV and for electron-positrons between 0.1and 100~GeV.

Local H~{\sc i} emissivity measured with the {\it Fermi}-LAT and implications for cosmic-ray spectra