Optimal fitting of Gaussian-apodized or under-resolved emission lines in Fourier transform spectra providing new insights on the velocity structure of NGC 6720

An analysis of the kinematics of NGC 6720 is performed on the commissioning data obtained with SITELLE, the Canada-France-Hawaii Telescope's new imaging Fourier transform spectrometer. In order to measure carefully the small broadening effect of a shell expansion on an unresolved emission line, we have determined a computationally robust implementation of the convolution of a Gaussian with a sinc instrumental line shape which avoids arithmetic overflows. This model can be used to measure line broadening of typically a few km s^-1 even at low spectral resolution (R < 5000). We have also designed the corresponding set of Gaussian apodizing functions that are now used by ORBS, the SITELLE's reduction pipeline. We have implemented this model in ORCS, a fitting engine for SITELLE's data, and used it to derive the [S II] density map of the central part of the nebula. The study of the broadening of the [N II] lines shows that the main ring and the central lobe are two different shells with different expansion velocities. We have also derived deep and spatially resolved velocity maps of the halo in [N II] and Hα and found that the brightest bubbles are originating from two bipolar structures with a velocity difference of more than 35 km s^-1 lying at the poles of a possibly unique halo shell expanding at a velocity of more than 15 km s^-1.