I've been trying to *find experiments where the mass density distribution $\mathrm{\Sigma = \Sigma(r)}$ is determined* for protoplanetary disks. For instance: 

* [Here][1] A. Miotello, S. Facchini, E. F. van Dishoeck, and S. Bruderer use radiative transfer and submm data to probe the surface density distribution with $^{13}\mathrm{CO}$ emission. 

* [Here][2] Jonathan P. Williams and Conor McPartland measured density profiles with ALMA. 

Seems like the method is similar. Despite that, I know that the available resolution in this band does not allow us to conclude much about the mass when $r < 1$ au. [ALMA's resolution][3] reaches $\sim 0.02''$ "in the most extended 12-m array configuration ($\sim \,16 \;\mathrm{km}$), when observing at $230 \;\mathrm{GHz}$. 



**[TW Hydrae][4]** is the nearest known protoplanetary disk. A resolution of $\theta \sim 3.5 \times 10^{-5}\;''$ would be needed to spatially resolve a diameter of $\mathrm{d_{TWH}} = 0.1 \;\mathrm{au}$. 

[![TW Hydrae][5]][5] 

**Questions**: Is it possible to measure $\Sigma$ at this distances? Is there any work exploring on this? Are we limited mainly to models? (e.g., [Philip J. Armitage's Astrophysics of Planet Formation][6] textbook)


  [1]: https://www.aanda.org/articles/aa/pdf/2018/11/aa33595-18.pdf
  [2]: https://iopscience.iop.org/article/10.3847/0004-637X/830/1/32/pdf
  [3]: https://almascience.nrao.edu/about-alma/alma-basics
  [4]: https://aasnova.org/2016/04/11/featured-image-a-gap-in-tw-hydrae/
  [5]: https://i.sstatic.net/bREQW.png
  [6]: https://www.amazon.com/-/es/Philip-J-Armitage/dp/1107653088