How is the maximum density in a circumstellar disk determined?

Question

I'm working with some equations to model the evolution of a circumstellar disk. One of the equations is $$\rho(r)=Ce^{-\frac{(r-r_{peak})^2}{2 \sigma ^2}}$$ where $\rho$ is density, $r$ is the distance from the center, $C$ is a constant, $\sigma$ is one standard deviation, and $r_{peak}$ is the radius at which the density is at a maximum.

If the function was of the form $$\rho(r)=Ce^{f(r)}$$ where $f(r)$ is a function of $r$, I could find the maximum easily by finding $$\rho'(r)=Cf'(r)e^{f(r)}=0$$ and solving for $r$. However, this appears to be impossible in the current case because $\rho(r)_{peak}$ is already in the equation, at $r_{peak}$.

How is $r_{peak}$ determined in a given scenario? Is it determined experimentally?

Answer 1

If your $r_{peak}$ is not known a priori then you have two unknowns and only one equation: this cannot be solved. At least not without a second equation or a measured $(\rho,r)$-pair (say the central density $\rho_0 = \rho(r=0)$).

In any case, this definition of radial density looks like an ad-hoc assumption to me, and not something that came out of a proper hydrodynamic model. So without knowing why you chose this form or the context of your disk model I can't comment on what equation is best suited to determine $r_{peak}$. A well-informed guess seems like the way to go.