# How to determine atomic number density of an element in a star based on equivalent width measurements

Given an equivalent width measurement $W$ of a spectral line of element $X$ and the effective temperature $T_{eff}$ of a star, how can you determine the atomic number density of $X$ in that star?

According to Warner 1965, for the 6707.8 Lithium line we can apply the approximation

$$\log{\frac{N^*(X)}{N^{\odot}(X)}} = \log{\frac{W^*(X)}{W^{\odot}(X)}} - 4.93 (\gamma -1)$$

where $\gamma = \frac{T^{\odot}}{T^*}$ and $\log{W^{\odot}(X)}$ must be measured. The same paper says that $\log{W^{\odot}}(Li)=-6.15$ (my goal is to solve for the abundance of Lithium given a measured W of the 6707.8 $\overset{\circ}A$ line.

Under what regime is this approximation valid? Also, I am unable to solve this equation because I don't know what $N^{\odot}(Li)$ is. How can I determine a reputable value for $N^{\odot}(Li)$?

## 1 Answer

The measurement of a chemical abundance is not a question of using a simple equation.

The simplest it gets is using a "curve of growth", which relates equivalent width to abundance and assumes you already know the temperature of the star and its surface gravity.

For the Li 6708A line, the relevant tables, that can be interpolated, are found in Soderblom et al. (1993). http://adsabs.harvard.edu/abs/1993AJ....106.1059SS . These are LTE curves of growth based on Atlas atmospheric models.

As an aside, the equivalent width of this line in the Sun is 3 mA. The formula you quote looks like some sort of linear approximation, but this resonance line rapidly enters the saturated (non-linear) portion of the curve of growth for equivalent widths of 50 mA or more. As the equation is couched in terms of ratios, then the units you use are entirely up to you.

• Thank you for this. I am wondering under what regime the linear approximation I wrote above applies. While I am familiar with the curve of growth abundance analysis, I am currently measuring the equivalent width in the spectra of sunlight-reflecting asteroids, so was thinking such analysis would not be necessary. Is the approximation valid for this? – user5341 Oct 28 '15 at 23:54
• @user44816 Have a look at the curve of growth I referenced. If you're looking at sunlight reflected from asteroids why do you think the equivalent width of the Li line will be different from the solar spectrum?? By definition, such a spectrum is used as a proxy for the Sun and calibration standard in most chemical abundance analyses. – ProfRob Oct 29 '15 at 0:09
• Thanks, will do. I did not expect the equivalent width of the Li line to be different from the solar spectrum, however my code is returning an value that is not 3 mA, which is why I was curious. Thanks again, I will definitely check out this paper. – user5341 Oct 29 '15 at 1:06
• @user44816 Your code? You mean your spectrum? 3 mA is very hard to measure unless you have extremely high S/N and resolution. – ProfRob Oct 29 '15 at 7:13