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- | ====== TALENT ====== | ||
- | ===== Training in Advanced Low Energy Nuclear Theory: EXERCISES ===== | ||
- | [[http://www.nucleartalent.org|{{:talent.png?650|Nuclear TALENT}}]] | ||
- | **[[RXNnetworkEX|Reaction Networks]]**, **[[HydroEX|Hydrodynamics]]**, **[[BBNEX|BBN]]**, **[[AnlyPlot|Abundances, Analysis and Plotting]]**, **[[NuclPhysInput|Nucl Physics Input]]** | ||
- | ==== Abundances, Analysis and Plotting ==== | ||
- | <box left 80% orange|**(Pre-Course Assignment)**> | ||
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- | **Analyze and plot abundances:** | ||
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- | - Grab the solar system abundances and plot them | ||
- | * Vs mass number (A) | ||
- | * Vs atomic number (Z) | ||
- | - Subtract the solar s-process abundances from solar | ||
- | * Plot the remaining abundance pattern | ||
- | - Grab stellar abundances from the [[http://saga.sci.hokudai.ac.jp/wiki/doku.php|SAGA Database]] for a handful of stars | ||
- | * Plot abundances in [A/H] notation | ||
- | * Translate [A/H] notation into mass fraction abundances | ||
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- | </box> | ||
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- | <box left 80% blue|**(During-Course Assignment)**> | ||
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- | **Abundance Measures** | ||
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- | 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 [[http://courses.atlas.illinois.edu/fall2009/astr/astr596npa/|Nuclear and Particle Astrophysics]] course.) | ||
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- | * 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. | ||
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- | * 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). | ||
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- | * 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. | ||
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- | * 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. | ||
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- | * 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. | ||
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- | * 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 $Y_\odot$? | ||
- | </box> | ||
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- | ~~DISCUSSION|Analysis and Plotting Discussion~~ |