Determining the gas density and temperature structures of protoplanetary disks is a fundamental task to constrain planet formation theories. This is a challenging procedure and most determinations are based on model-dependent assumptions. We attempt a direct determination of the radial and vertical temperature structure of the Flying Saucer disk, thanks to its favorable inclination of 90 degrees. We present a method based on the tomographic study of an edge-on disk. Using ALMA, we observe at 0.5$$ resolution the Flying Saucer in CO J=2-1 and CS J=5-4. This edge-on disk appears in silhouette against the CO J=2-1 emission from background molecular clouds in $rho$ Oph. The combination of velocity gradients due to the Keplerian rotation of the disk and intensity variations in the CO background as a function of velocity provide a direct measure of the gas temperature as a function of radius and height above the disk mid-plane. The overall thermal structure is consistent with model predictions, with a cold ($< 15-12 $~K), CO-depleted mid-plane, and a warmer disk atmosphere. However, we find evidence for CO gas along the mid-plane beyond a radius of about 200,au, coincident with a change of grain properties. Such a behavior is expected in case of efficient rise of UV penetration re-heating the disk and thus allowing CO thermal desorption or favoring direct CO photo-desorption. CO is also detected up to 3-4 scale heights while CS is confined around 1 scale height above the mid-plane. The limits of the method due to finite spatial and spectral resolutions are also discussed. This method appears to be very promising to determine the gas structure of planet-forming disks, provided that the molecular data have an angular resolution which is high enough, of the order of $0.3 - 0.1$ at the distance of the nearest star forming regions.
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