Chapter 5, Thermonuclear Reactions in Stars Video Solutions, Stars and Stellar Processes

Problem 1

Using atomic mass tables, calculate the amount of energy released from the fusion of one gram of ${ }^1 \mathrm{H}$ into ${ }^4 \mathrm{He}$. What percentage is this of the original rest mass energy of the hydrogen?***

David Collins

Numerade Educator

02:41

Problem 2

The binding energy per nucleon for ${ }^4 \mathrm{He}$ is $7.074 \mathrm{MeV}$ and that of ${ }^{12} \mathrm{C}$ is $7.6802 \mathrm{MeV}$. What is the $Q$-value for the reaction ${ }^4 \mathrm{He}+{ }^4 \mathrm{He}+$ ${ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}$ in MeV? How much energy is released if one gram of ${ }^4 \mathrm{He}$ is converted to ${ }^{12} \mathrm{C}$ ?

Narayan Hari

Numerade Educator

02:11

Problem 3

According to certain model calculations the elemental composition of the Sun at formation was $X=0.71, Y=0.27$, and $Z=0.02$, while presently typical solar models indicate that the core has a composition $X=0.34, Y=0.64$, and $Z=0.02$. What are the mean molecular weights at these two times?***

Dominador Tan

Numerade Educator

10:34

Problem 4

Show that the maximum of the Gamow factor is located at an energy
$$
E_0=1.22\left(Z_\alpha^2 Z_X^2 \mu T_6^2\right)^{1 / 3} \mathrm{keV}
$$
where $Z$ denotes the atomic number for the relevant ion, $\mu$ is the mean molecular weight, and $T_6$ is the temperature in units of $10^6 \mathrm{~K}$. Show that for the reaction $\mathrm{p}+{ }^{12} \mathrm{C} \rightarrow{ }^{13} \mathrm{~N}+\gamma$, the Gamow peak is at
$E_0 \simeq 31 \mathrm{keV}$ for $T_6=22.5$, and that the height of the peak varies as $\sim T^{17}$ for this reaction in this temperature range. ${ }^{* * *}$

Ren Jie Tuieng

Numerade Educator

12:18

Problem 5

Derive an approximate expression for the width $\Delta$ of the Gamow peak and the velocity averaged cross section $\langle\sigma v\rangle_{\alpha \beta}$ in a nonresonant reaction by approximating the Gamow peak as a gaussian of the same height and curvature at the maximum as the actual Gamow peak.***

Khoobchandra Agrawal

Numerade Educator

02:42

Problem 6

Use the Caughlan and Fowler compilation in Ref. [88] to find a formula for the rate of the reaction ${ }^{12} \mathrm{C}(\mathrm{p}, \gamma){ }^{13} \mathrm{~N}$ as a function of temperature in units of $\mathrm{cm}^3 \mathrm{~s}^{-1}$, include resonant and nonresonant contributions. Hint: See $A$ ppendix D.

Niamat Khuda

Numerade Educator

03:30

Problem 7

If the $\mathrm{CN}$ part of the $\mathrm{CNO}$ cycle is running at equilibrium under central solar conditions, what is the ratio of abundances $Y_i$ for ${ }^{13}$ Cversus ${ }^{12} \mathrm{C},{ }^{14} \mathrm{~N}$ versus ${ }^{12} \mathrm{C}$, and ${ }^{15} \mathrm{~N}$ versus ${ }^{14} \mathrm{~N}$ ? Hint: Use the rates in Fig. 6.3 (a).

Salamat Ali

Numerade Educator

01:15

Problem 8

Show that for the rate-controlling step of the PP chain the temperature exponent is
$$
v_{\mathrm{pp}}=11.3 T_6^{-1 / 3}-\frac{2}{3},
$$
so that $v_{\mathrm{pp}} \simeq 4$ for $T_6=15$.***

Foster Wisusik

Numerade Educator

07:22

Problem 9

By summing net reactants and products around the cycles implied by Eqs. $(\underline{5.10})-(\underline{5.11})$ starting at different parts of the cycle, convince yourself that any of the carbon, nitrogen, or oxygen isotopes appearing in the $\mathrm{CNO}$ cycle may be viewed as a catalyst promoting the effective fusion reaction $4^1 \mathrm{H} \rightarrow{ }^4 \mathrm{He}$.***
5.10 Suppose the Sun were capable of converting all its hydrogen to iron by a sequence of nuclear reactions. How much energy would be released? To what radius would the Sun have to shrink in order to generate the same amount of energy by gravitational contraction? What would be the Sun's density following this contraction?
5.11 Look up the mass excesses of ${ }^4 \mathrm{He}$ and ${ }^{20} \mathrm{Ne}$ in Ref. [29] and use these to compute the corresponding masses in atomic mass units. Check your results against the tabulated values in Ref. [29].

Nicholas Majtenyi

Numerade Educator

01:57

Problem 10

Suppose the Sun were capable of converting all its hydrogen to iron by a sequence of nuclear reactions. How much energy would be released? To what radius would the Sun have to shrink in order to generate the same amount of energy by gravitational contraction? What would be the Sun's density following this contraction?

Penny Riley

Numerade Educator

02:02

Problem 11

Look up the mass excesses of ${ }^4 \mathrm{He}$ and ${ }^{20} \mathrm{Ne}$ in Ref. [29] and use these to compute the corresponding masses in atomic mass units. Check your results against the tabulated values in Ref. [29].

Amy Jiang

Numerade Educator

11:04

Problem 12

Show by two different methods that the energy production rate per unit volume in the core of the Sun is remarkably low - only several hundred watts per cubic meter.
(a) First, show this using the tabulated rates for the rate-controlling step of the PP chain and the average energy released by the PP chain, in conjunction with expected conditions in the core of the Sun.
(b) Second, show this using the observed luminosity of the Sun, the assumption that it is in equilibrium, and the assumption that most of the energy responsible for this luminosity is generated in approximately the inner $10 \%$ of the solar radius."

Vipender Rao

Numerade Educator

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Problem 13

For the reaction ${ }^{12} \mathrm{C}(\mathrm{p}, \gamma)^{13} \mathrm{~N}$ in Example 5.6, show that the excited state at $2.37 \mathrm{MeV}$ in ${ }^{13} \mathrm{~N}$ should give a resonance at a laboratory proton energy of $46 \mathrm{keV}$ (as observed in Fig. 5.8), assuming validity of the Breit-Wigner formula $(\underline{5.33})$.***

Lainey Roebuck

Numerade Educator