Constraints on the Cosmic-Ray Density Gradient beyond the Solar Circle from Fermi gamma-ray Observations of the Third Galactic Quadrant

We report an analysis of the interstellar $\gamma$-ray emission in the third Galactic quadrant measured by the {Fermi} Large Area Telescope. The window encompassing the Galactic plane from longitude $210\arcdeg$ to $250\arcdeg$ has kinematically well-defined segments of the Local and the Perseus arms, suitable to study the cosmic-ray densities across the outer Galaxy. We measure no large gradient with Galactocentric distance of the $\gamma$-ray emissivities per interstellar H atom over the regions sampled in this study. The gradient depends, however, on the optical depth correction applied to derive the \HI\ column densities. No significant variations are found in the interstellar spectra in the outer Galaxy, indicating similar shapes of the cosmic-ray spectrum up to the Perseus arm for particles with GeV to tens of GeV energies. The emissivity as a function of Galactocentric radius does not show a large enhancement in the spiral arms with respect to the interarm region. The measured emissivity gradient is flatter than expectations based on a cosmic-ray propagation model using the radial distribution of supernova remnants and uniform diffusion properties. In this context, observations require a larger halo size and/or a flatter CR source distribution than usually assumed. The molecular mass calibrating ratio, $X_{\rm CO} = N({\rm H_{2}})/W_{\rm CO}$, is found to be $(2.08 \pm 0.11) \times 10^{20} {\rm cm^{-2} (K km s^{-1})^{-1}}$ in the Local-arm clouds and is not significantly sensitive to the choice of \HI\ spin temperature. No significant variations are found for clouds in the interarm region.