Probing local cosmic rays using Fermi-LAT observations of a mid-latitude region in the third Galactic quadrant

The $\gamma$-ray observation of interstellar gas provides a unique way to probe the cosmic rays (CRs) outside the solar system. In this work, we use an updated version of Fermi-LAT data and recent multi-wavelength tracers of interstellar gas to re-analyze a mid-latitude region in the third Galactic quadrant and estimate the local CR proton spectrum. Two $\gamma$-ray production cross section models for $pp$ interaction, the commonly used one from Kamae et al. (2006) and the up-to-date one from Kafexhiu et al. (2014), are adopted separately in the analysis. Both of them can well fit the emissivity and the derived proton spectra roughly resemble the direct measurements from AMS-02 and Voyager 1, but rather different spectral parameters are indicated. A break at $4\pm1~{\rm GeV}\;c^{-1}$ is shown if the cross section model by Kamae et al. (2006) is adopted. The resulting spectrum is $\lesssim 20\%$ larger than the AMS-02 observation above $15~\rm GeV$ and consistent with the de-modulated spectrum within $2\%$. The proton spectrum based on the cross section model of Kafexhiu et al. (2014) is about $1.4-1.8$ times that of AMS-02 at $2-100~\rm GeV$, however the difference decreases to $20\%$ below $10~\rm GeV$ with respect to the de-modulated spectrum. A spectral break at $20\pm11~{\rm GeV}\;c^{-1}$ is required in this model. An extrapolation down to $300~\rm MeV$ is performed to compare with the observation of Voyager 1, and we find a deviation of $\lesssim 2.5\sigma$ for both the models. In general, an approximately consistent CR spectrum can be obtained using $\gamma$-ray observation nowadays, but we still need a better $\gamma$-ray production cross section model to derive the parameters accurately.