# Correct relation between metallicity (z) and iron content ([Fe/H])

The Wikipedia entry on Metallicity states that:

$$\log_{10}\left(\frac{Z/X}{Z_\mathrm{sun}/X_\mathrm{sun}}\right) = [\mathrm{M}/\mathrm{H}]$$

where $$[M/H]$$ is the star's total metal abundance (i.e. all elements heavier than helium) defined as a more general expression than the one for [Fe/H]:

$$[\mathrm{M}/\mathrm{H}] = \log_{10}{\left(\frac{N_{\mathrm{M}}}{N_{\mathrm{H}}}\right)_\mathrm{star}} - \log_{10}{\left(\frac{N_{\mathrm{M}}}{N_{\mathrm{H}}}\right)_\mathrm{sun}}$$

The iron abundance and the total metal abundance are often assumed to be related through a constant A as [citation needed]:

$$[\mathrm{M}/\mathrm{H}] = A\times[\mathrm{Fe}/\mathrm{H}]$$

where $$A$$ assumes values between 0.9 and 1. Using the formulas presented above, the relation between $$Z$$ and [Fe/H] can finally be written as:

$$\log_{10}\left(\frac{Z/X}{Z_\mathrm{sun}/X_\mathrm{sun}}\right) = A\times[\mathrm{Fe}/\mathrm{H}]$$

This all sounds reasonable, but I have two questions:

1. Where does the $$[\mathrm{M}/\mathrm{H}] = A\times[\mathrm{Fe}/\mathrm{H}]$$ relation come from? I could not find a proper source.
2. This article actually says (see Eq 9) that the general relation is $$\log_{10}\left(\frac{Z/X}{Z_\mathrm{sun}/X_\mathrm{sun}}\right) = [\mathrm{Fe}/\mathrm{H}]$$ i.e., they seem to equate [M/H] with [Fe/H]. I assume that it is because the aforementioned A parameter is between 0.9 and 1, but again that leaves me with the need for a proper source to state that.
• As someone who works in the field, I have not seen your middle equation (with the $A$). [Fe/H] is often lazily conflated with [M/H], but obviously it depends on the detailed composition of the star whether that is approximately true. I suspect it is approximately true for Pop I stars, but perhaps not for older, metal poor stars which have a larger predominance of alpha elements. Commented Apr 23, 2018 at 18:33
• I have never seen that $A$ parameter either, I just saw it on WP an thought is sounded reasonable. Since I could not find a "proper" general definition for the relation between the $z$ metallicity and the $[Fe/H]$ iron content (aside from the one I show above, from the Bertelli article), I was not sure about its validity. You say that "[Fe/H] is often lazily conflated with [M/H]", so you agree that the A parameter sounds reasonable? Commented Apr 23, 2018 at 18:40
• Maybe I'll work out some $A$ values. Commented Apr 23, 2018 at 18:52

$$[\mathrm{Fe}/\mathrm{H}] = \log_{10} \left[\frac{f_\mathrm{Fe}(\alpha)}{f_\mathrm{Fe}(0)} \frac{m_z(0)}{m_z(\alpha)} \left(\frac{Z}{X}\right)_\ast \left(\frac{X}{Z}\right)_\odot \right]$$

You can use this relation to estimate [Fe/H] from [M/H] of a star. From [M/H] you can get the (Z/X) ratio.

• Is this the second part to your first answer? You can use the "edit" button to correct or add to your answers. Each post should be a self-contained answer, rather than have answers split over multiple posts. Commented Feb 6, 2020 at 15:59
• It is better to include formulae using the LaTeX markup rather than as images. I've converted the images on this answer and your other answer, but you will still need to provide more information as to what the quantities mean.
– user24157
Commented Feb 6, 2020 at 18:48
• Explain the symbols! Commented Feb 7, 2020 at 7:20

$$\frac{f_{Fe}(\alpha)}{f_{Fe}(0)} \frac{m_z(0)}{m_z(\alpha)}$$

This is the Value of the constant that u get. The above values you can get from any standard paper. Check fergusion et al 2005

• What constant is this? Also, would you have a link to the article you mention? Commented Feb 6, 2020 at 15:56