Probing the protoplanetary disk gas surface density distribution with
  $^{13}$CO emission
  It is key to constrain the gas surface density distribution, Sigma_gas, as
function of disk radius in protoplanetary disks. In this work we investigate if
spatially resolved observations of rarer CO isotopologues may be good tracers
of Sigma_gas. Physical-chemical models with different input Sigma_gas(R) are
run. The input disk surface density profiles are compared with the simulated
13CO intensity radial profiles to check if and where the two follow each other.
There is always an intermediate region in the disk where the slope of the 13CO
radial emission profile and Sigma_gas(R) coincide. At small radii the line
radial profile underestimates Sigma_gas, as 13CO emission becomes optically
thick. The same happens at large radii where the column densities become too
low and 13CO is not able to efficiently self-shield. If the gas surface density
profile is a simple power-law of the radius, the input power-law index can be
retrieved within 20% uncertainty if one choses the proper radial range. If
instead Sigma_gas(R) follows the self-similar solution for a viscously evolving
disk, retrieving the input power-law index becomes challenging, in particular
for small disks. Nevertheless, it is found that the power-law index can be in
any case reliably fitted at a given line intensity contour around 6 K km/s, and
this produces a practical method to constrain the slope of Sigma_gas(R).
Application of such a method is shown in the case study of the TW Hya disk.
Spatially resolved 13CO line radial profiles are promising to probe the disk
surface density distribution, as they directly trace Sigma_gas(R)profile at
radii well resolvable by ALMA. There, chemical processes like freeze-out and
isotope selective photodissociation do not affect the emission, and, assuming
that the volatile carbon does not change with radius, no chemical model is
needed when interpreting the observations.
