# Where does this equation for ionic abundances come from, and what does it mean?

So I've been attempting to find ionic abundances using visual emission spectra of planetary nebula, and I ran across an equation to do that in the paper Abundances of planetary nebulae NGC 40 and NGC 6153. I've tried looking up some of the definitions for the variables to get a sense of the information needed to be plugged in to find abundance, and I've found a large number of almost word for word reproductions of the same paragraph I found the equation in (1,2,3,4,5,6,7,8) but little success actually coming to a better understanding of it.

The ionic abundances have been determined using the following equation: $$\frac{N_{ion}}{N_p}=\frac{I_{ion}}{I_{H_β}}N_e\frac{λ_{ul}}{λ_{H_β}}\frac{α_{H_β}}{A_{ul}}\left(\frac{N_{u}}{N_{ion}}\right)^{-1}$$ where $I_{ion}/I_{H_β}$ is the measured intensity of the ionic line compared to Hβ, $N_p$ is the density of ionized hydrogen, $λ_{ul}$ is the wavelength of this line, $λ_{H_β}$ is the wavelength of Hβ, $α_{H_β}$ is the effective recombination coefficient for Hβ, $A_{ul}$ is the Einstein spontaneous transition rate for the line, and $N_u/N_{ion}$ is the ratio of the population of the level from which the line originates to the total population of the ion. This ratio has been determined using a five level atom.

I can understand some of it, but there are some pretty significant gaps too. Specifically:

• Is $N_{ion}$ the density of the ion in the nebula?
• What is $N_e$?
• How is the effective recombination coefficient for Hβ found?
• What is the Einstein spontaneous transition rate for the selected line?

I feel like these might be better served with their own individual questions, and I don't really expect them to be given answers here. I'll ask them separately if the need still ends up existing, but for now I'm concerned with understanding where this equation comes from as a whole. Any help towards that end would be much appreciated.

• The $N_{xx}$ variables are probably the number of $xx$ (e.g. ions), but since they're ratios the densities will be proportional. $N_e$ is then likely electrons. I don't know any details about the rate/coefficient calculations but they're based on radiative transfer in gases. Reading up on a radiative processes / radiative transfer textbook would probably be very helpful! Rybicki and Lightman is probably a good start. – DilithiumMatrix Mar 17 '17 at 17:43
• What the Einstein A coefficient is of course depends on what transition you are talking about. – Rob Jeffries Mar 19 '17 at 19:09
• It's the Einstein A coefficient. For the transition in question. en.wikipedia.org/wiki/Einstein_coefficients – Rob Jeffries Mar 19 '17 at 23:16
• I agree with @RobJeffries, it is the same thing. Unless my Ph.D. advisors and opponent and assessment committee all did an inexcusably poor job ;-) – Thriveth Mar 20 '17 at 14:17
• The units of $A_{21}$ are s$^{-1}$, so it is a rate, or rather it is the rate per atom. However, for a blob of gas, you might consider the "rate" to mean the number of photons per second in which case you would have to multiply by the number of atoms in state 2. Indeed in the equation in the OP you will see there is a term $A_{ul} N_u$, which is exactly that. Hopefully that resolves any confusion. – Rob Jeffries Mar 20 '17 at 14:43