# How do you project a Sersic profile into 3D?

I'm attempting to create a velocity profile for the M31. I have a Sersic profile from a paper by Sofue et al (2009), but I'm unable to project it into 3D space in order to get the actual 3D density that I'd use to calculate the velocity at a given radius. Here's the formula from a 2008 paper by Noordermeer (arXiv link).

$$\rho(m)=-\frac{1}{\pi}\sqrt{\sin^2i+\frac{1}{q^2}\cos^2i}\int_m^\infty\frac{dI}{d\kappa}\frac{d\kappa}{\sqrt{\kappa^2-m^2}}\tag{(9)}$$

Where $dI$ is the Sersic Profile, $κ$ is the measured radius along the line-of-nodes, $m$ is the 3D radius and $\rho(m)$ is the density in 3D. But how is $m$ in the formula above related to $κ$? I get the general idea that we're projecting a 2D profile into 3D space, but without knowing either the radius or the line of sight dimension ($ζ$ in the paper listed above), I don't see how the 3D radius, $m$, can be determined. What am I missing?

• Is your question specifically about the transition from equation (8) to equation (9), or is it something else? That section of the paper outlines a whole slew of assumptions and then appears to walk through the math going one way and gives a reference for the inversion. So, what part is the problem? – dmckee --- ex-moderator kitten Sep 21 '15 at 17:23
• It's specifically about equation (9). I have a function p(m) that has a single parameter m. Two variables are given: i is the inclination, q is the ellipse ratio (1 for a sphere). So where do I get κ to put in the formula above? – Quarkly Sep 21 '15 at 18:44
• I deleted some comments that were inappropriate and/or obsolete. – David Z Sep 24 '15 at 18:00

## 1 Answer

OK. Here we go. I turns out that the above integration can't be solved analytically. There is a HUGE body of science devoted to the subject, enough for a couple of Ph. Ds. Some people have tried to create a generalization of the formula (see http://adsabs.harvard.edu/abs/1987A%26A...175....1M) but I found these approximation lacked the detail needed for any serious galactic bulge modeling. Finally, I found the seminal work on the subject: Young 1974 (http://adsabs.harvard.edu/full/1976AJ.....81..807Y) which contains some tables of numerically calculated values for the 3D project of a 2D surface brightness profile. You can feed the table into an interpolation function and get a nearly perfect match for 3D density or 3D mass given R/Re (in sky coordinates).