The remarkable polarization structure of the Na I D_1 and D_2 lines that is due to coherent scattering has remained an enigma, since it has not yet been possible to find an explanation that is consistent with both current understanding of quantum mechanics and the astrophysical properties of the Sun’s atmosphere. To guide future theoretical efforts we have here explored the detailed center-to-limb variation of the linearly polarized profiles in non-magnetic regions. In particular we find that the unexplained narrow polarization peaks in the Doppler cores of the two lines become even more pronounced with respect to the relative profile shape as we move away from the limb towards the center of the solar disk.