Astronomy Stack Exchange is a question and answer site for astronomers and astrophysicists. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|improve this question
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 '14 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 '14 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 '14 at 22:45
up vote 1 down vote accepted

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.

share|improve this answer
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 '14 at 19:48

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.