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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?

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Note: Some more parts of the problem are in the Sandbox post I started on Meta; I've been preparing some math in case it was needed to explain the scenario, or in case I ask other future questions about the problem. –  HDE 226868 Nov 26 at 1:21
    
If $r$ is the spherical radius (which is the usual convention for this symbol), then this is not a disc at all... If it is cylindrical radius, it is not a disc either, but a cylinder, since there is no $z$ dependence. –  Walter Nov 28 at 20:57
    
@Walter This is a disk. While I didn't explain properly, this equation is valid for the areal density of the mean plane. There is a separate function for the density at a point on the $z$ axis which is also expressed exponentially; there is a drop off as the distance from the mean plane increases, meaning that after a short distance, that density becomes completely negligible. It is technically a cylinder, but it is, for all intents and purposes, a disk. Besides, a disk is really a very short cylinder, right? I can explain the $z$-dependent equation of you want. –  HDE 226868 Nov 28 at 22:45

1 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.

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This is pretty much what I wanted. The equation(s) came out of an appendix of Michael M. Woolfson's On the Origin of Planets. –  HDE 226868 Nov 26 at 19:48

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