Accurate measurements of the physical structure of protoplanetary discs are critical inputs for planet formation models .
These constraints are traditionally established via complex modelling of continuum and line observations .
Instead , we present an empirical framework to locate the CO isotopologue emitting surfaces from high spectral and spatial resolution ALMA observations .
We apply this framework to the disc surrounding IM Lupi , where we report the first direct , i.e .
model independent , measurements of the radial and vertical gradients of temperature and velocity in a protoplanetary disc .
The measured disc structure is consistent with an irradiated self-similar disc structure , where the temperature increases and the velocity decreases towards the disc surface .
We also directly map the vertical CO snow line , which is located at about one gas scale height at radii between 150 and 300 au , with a CO freeze-out temperature of 21 \pm 2 K. In the outer disc ( > 300 au ) , where the gas surface density transitions from a power law to an exponential taper , the velocity rotation field becomes significantly sub-Keplerian , in agreement with the expected steeper pressure gradient .
The sub-Keplerian velocities should result in a very efficient inward migration of large dust grains , explaining the lack of millimetre continuum emission outside of 300 au .
The sub-Keplerian motions may also be the signature of the base of an externally irradiated photo-evaporative wind .
In the same outer region , the measured CO temperature above the snow line decreases to \approx 15 K because of the reduced gas density , which can result in a lower CO freeze-out temperature , photo-desorption , or deviations from local thermodynamic equilibrium .