# 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. – Rob Jeffries Apr 23 '18 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? – Gabriel Apr 23 '18 at 18:40
• Maybe I'll work out some $A$ values. – Rob Jeffries Apr 23 '18 at 18:52