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adding an explanation of the [Fe/H]->XYZ calculation in docs
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\documentclass{article} | ||
\usepackage{amsmath} | ||
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\begin{document} | ||
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Constraints: | ||
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\begin{equation} | ||
X+Y+Z = 1 | ||
\end{equation} | ||
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\begin{equation} | ||
Y = Y_{BBN} + \frac{\Delta Y}{\Delta Z} Z | ||
\end{equation} | ||
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\begin{equation} | ||
\frac{Z}{X} = \left(\frac{Z}{X}\right)_{\odot} 10^{[Fe/H]} | ||
\end{equation} | ||
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Where $Y_{BBN}$, $\left(\frac{Z}{X}\right)_{\odot}$, $\frac{\Delta Y}{\Delta Z}$, and [Fe/H] are given as input. | ||
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Even though we don't know X or Z, we know (Z/X) from (3) and the input [Fe/H]. So we treat (Z/X) as a single variable below. | ||
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Begin with (1) and isolate Y. | ||
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\begin{equation} | ||
X+Z=1-Y | ||
\end{equation} | ||
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Replace Y with (2). | ||
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\begin{equation} | ||
X +Z = 1 - (Y_{BBN} + \frac{\Delta Y}{\Delta Z} Z) | ||
\end{equation} | ||
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Replace Z with (Z/X)$\times$X. | ||
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\begin{equation} | ||
X ( 1 + \frac{Z}{X}) = 1 - (Y_{BBN} + \frac{\Delta Y}{\Delta Z} \frac{Z}{X} X) | ||
\end{equation} | ||
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Gather terms with X on the LHS. | ||
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\begin{equation} | ||
X ( 1 + \frac{Z}{X} ( 1 + \frac{\Delta Y}{\Delta Z})) = 1 - Y_{BBN} | ||
\end{equation} | ||
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Finally make X all alone on the RHS. | ||
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\begin{align} | ||
X = \frac{1-Y_{BBN}}{ 1 + \frac{Z}{X} ( 1 + \frac{\Delta Y}{\Delta Z})} \\ | ||
Z= \frac{Z}{X} \times X\\ | ||
Y= 1 - X - Z | ||
\end{align} | ||
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Done! | ||
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\end{document} |