Chapter 6, Stellar Burning Processes Video Solutions, Stars and Stellar Processes

Problem 1

Derive the nuclear Saha equations $(6.4),(6.5)$, and $(6.9)$ for helium burning, beginning from Eq. (3.55) and the assumption of nuclear statistical equilibrium. Hint: The condition for concentration equilibrium in the reaction $\mathrm{a}+\mathrm{b} \leftrightharpoons \mathrm{c}$ is that the sum of chemical potentials on the two sides of the equation balance, $\mu_{\mathrm{a}}+\mu_{\mathrm{b}}=\mu_{\mathrm{c}^{* * *}}{ }^{* * *}$

Bin Chen

Numerade Educator

04:54

Problem 2

Under helium flash conditions in low-mass red giant stars (see Section 13.5.3), $T_9 \simeq 0.1$ and the density is $\rho \simeq 10^6 \mathrm{gcm}^{-3}$. Estimate the ${ }^8 \mathrm{Be}$ concentration and the ratio of ${ }^8 \mathrm{Be}$ to ${ }^4 \mathrm{He}$ concentrations assuming that the flash takes place in a pure helium core.

Willis James

Numerade Educator

05:12

Problem 3

It was stated in Section 6.4 that radiative alpha capture on ${ }^{16} \mathrm{O}$ to produce ${ }^{20} \mathrm{Ne}$ is inhibited relative to radiative alpha capture on ${ }^{12} \mathrm{C}$ to produce ${ }^{16} \mathrm{O}$ by a higher Coulomb barrier. Estimate the difference in Coulomb barrier penetration rates for these two reactions under conditions characteristic of helium burning. Does this estimate explain the basic rate difference between the two reactions exhibited in Fig. 6.6 under helium burning conditions?

Ren Jie Tuieng

Numerade Educator

02:36

Problem 4

Use the rates given in Fig. 6.3 (a) to estimate the mean life for each step of the main part of the CNO cycle (the CN cycle) illustrated in Fig. 5.3 (a), assuming a temperature of 20 million K and a density of $100 \mathrm{~g} \mathrm{~cm}^{-3}$. Based on these results, which isotope would you expect to become the most abundant after the $\mathrm{CNO}$ cycle has run long enough to reach equilibrium? Would you expect ${ }^{13} \mathrm{~N}$ to have a high or low abundance after the $\mathrm{CNO}$ cycle has been running for a while in a $\operatorname{star}$ ?

Gwendolyn Burke

Numerade Educator

02:56

Problem 5

For a $p+p$ reaction at a temperature of $T_6=15$, calculate the average energy of particles in the gas, the location of the Gamow peak, and its approximate width. For the second step of the triple- $\alpha$ process, ${ }^8 \mathrm{Be}+\alpha \rightarrow{ }^{12} \mathrm{C}^*$, estimate the location and width of the Gamow peak for a temperature of $T_8=3$.

Suzanne W.

Numerade Educator

02:50

Problem 6

Consider an idealized closed system in which ${ }^{12} \mathrm{C}$ can undergo a ( $p$, $\gamma$ ) reaction to produce ${ }^{13} \mathrm{~N}$ and the ${ }^{13} \mathrm{~N}$ can $\beta$-decay to ${ }^{13} \mathrm{C}$, with no other reaction or decay channels open. Write a differential equation expressing the time dependence of the ${ }^{13} \mathrm{~N}$ number density in terms of the number densities of the other species, the velocity-averaged $(p, \gamma)$ reaction rate $\langle\sigma v\rangle$, and the $\beta$-decay constant $\lambda$. Rewrite this expression in terms of the abundances $Y_i$ introduced in Eq. (3.18). Assuming that the abundances of protons and ${ }^{12} \mathrm{C}$ do not change significantly over a long period of time and that temperature and density are constant, show that the abundance of ${ }^{13} \mathrm{~N}$ approaches a constant for long times and derive an expression for this equilibrium concentration of ${ }^{13} \mathrm{~N}$.

Lottie Adams

Numerade Educator

13:48

Problem 7

Consider the burning of pure ${ }^{28} \mathrm{Si}$ as in Fig. 6.10, with an initial step corresponding to photodisintegration of silicon: $\gamma+{ }^{28} \mathrm{Si} \rightarrow a+{ }^{24} \mathrm{Mg}$. Using estimates from Figs. 6.9 and D.3 for the relevant rates, to what abundance does the $\alpha$-particle population have to grow before the rate of $\alpha$-capture on silicon, $\alpha+{ }^{28} \mathrm{Si} \rightarrow{ }^{32} \mathrm{~S}+\gamma$, becomes equal to the rate for photodisintegration of silicon, $\gamma+{ }^{28} \mathrm{Si} \rightarrow \alpha+{ }^{24} \mathrm{Mg}$, assuming a density of $10^6 \mathrm{~g} \mathrm{~cm}^{-3}$ and temperature of $T_9=3$ ? Hint: The relevant equations are given in Section D.2.3, and note that the rates in Figs. 6.9 and D. 3 are in different units.

Ren Jie Tuieng

Numerade Educator