Abundance Measures

There is a whole thicket of abundance measures that one finds in the literature, which can unfortunately lead to confusion. The following are some exercises to get you used to the different notations and to translating among them. (I have cribbed Brian Fields' first question from his first homework from his Nuclear and Particle Astrophysics course.)

• Given the mass density $\rho_i$ of nuclide species $i$, what is $n_i$, the number density in $i$? Express this both with and without including the nuclear binding energy of $i$. Show that ignoring the binding energy introduces a $< 1\%$ error in $n_i$. Of course, neutral matter will also contain electrons; show that neglecting the electron contribution to the mass densities also gives a $< 1\%$ error–thus, we can take $\rho$ to be the total matter density as well as the baryonic density. Using these approximations, express $\rho_i$ in terms of $n_i$ and $A_i$, the mass number.

• One way to define an abundance is the mass fraction $X_i \equiv \rho_i/\rho$, where $\rho = \Sigma_j \rho_j$ is the total baryonic mas density summed over all species. Show that $X_i$ is unaffected by a bulk, chemically homogeneous expansion of the gas, and that the mass fractions obey $\Sigma_i X_i = 1$ (this also means that any given $0 \leq X_i \leq 1$, i.e., that these are really fraction).

• Another useful measure is the mole fraction $Y_i = n_i/n_B$. What is the relationship between $Y_i$ and $X_i$? That is, find an expression for $X_i$ in terms of $Y_i$ and physical/nuclear constants. Show that $\Sigma_i Y_i \neq 1$ in general. In general, is $\Sigma_i Y_i \leq 1$ or $\geq 1$? Prove your answer.

• It is sometimes useful to define the electron fraction $Y_e = n_e/n_B$, where $n_e$ is the number density of electrons, with each nuclide species $i$ contributing $Z_i$ electrons (assuming electrical neutrality). Find an expression for $Y_e$ in terms of the set of $Y_i$. What are the upper and lower limits to $Y_e$? What astrophysical systems have compositions for which $Y_e$ approaches these limits? Estimate $Y_e$ for the Earth, assuming that nuclei of heavy elements typically have half protons and half neutrons.

• Mass fractions are not in practice easy to measure directly. A more useful observable is $y_i = {\mathcal A}_i/H = n_i/n_H$. This is usually what observers mean when they say “abundance.” Like $X_i$, this is dimensionless, but the “normalization” condition does not hold (show this). What is the connection between ${\mathcal A}_i/H$ and $X_i$? That is, express $X_i$ in terms of the set of {$y_i$} ≡ {${\mathcal A}_i/H$}, and express $y_i$ in terms of $X_i$ and $X_H$. Explain why mass fractions are seldom reported by observers.

• Geologists and cosmochemists of course use different abundance scales, often setting Si to some arbitrary fiducial value like $10^2$ or $10^6$. Using your handy table of abundances from Anders & Grevesse (1989 Geochim. et Cosmochim. Acta) compute the solar values of: D/H, He/H, O/H, and La/H. Using the table and $Z_\odot \simeq 0.02$, what is the inferred $Y_\odot$?