# Extract surface density profile from a temperature profile?

I have a protoplanetary disk model which outputs a temperature vs. radius profile, based on radiative transfer. Looks something like this...

One of the inputs to the model is the surface density profile of the dust, which essentially sets up the structure of the material in the protoplanetary disk for the radiative transfer. It is given by this standard equation:

$$\displaystyle \Sigma_\text{gas} = \Sigma_\text{c} \cdot \big(\frac{r}{R_\text{C}}\big)^{-\gamma} \cdot \exp\big[-\big( \frac{r}{R_\text{C}}\big)^{2-\gamma}\big]$$

(where all the variables are model inputs)

Is there a simple way to recover the surface density profile $$\Sigma_\text{gas}$$ from the known temperature profile? (Ignoring the fact that I already know what it is, just wondering if theoretically it can be done?)

• If you are feeding the surface density profile to the model, then you have the files available? Just look into them? Or is the equation all you have and you don't know the parameters $\gamma$, $\Sigma_c$ and $R_C$? Mar 26, 2022 at 14:21
• Yes that's right, I already know what the surface density profile is since it's an input to the model. I'm wondering whether theoretically the surface density can be determined from any given temperature profile, via some simple formula? I want to use the temperature output of one model to determine the surface density profile that feeds into the input of a second model. Have edited the question to make this more clear. Mar 26, 2022 at 15:10
• If the heating is provided exclusively by viscous heating, which it looks like in your outer disc, as the temperatures there drop to unphysically low values, then there is a direct inversion from T to $\Sigma$. But if it's an active disc, i.e. heated by stellar radiation (in which case I wouldn't understand your T-profile), then there is no inversion, as the optically thin parts of the disc will have temperatures which are independent of the density. Mar 26, 2022 at 20:43