Unit E5 · Nuclear and Quantum PhysicsUnit E5 · 核物理与量子物理
Fusion and Stars聚变与恒星
IB-Style Practice QuestionsIB 风格练习题
MEDIUMHARDPaper 1Paper 1BPaper 2HL ONLY
Syllabus E5.1 to E5.6考纲 E5.1 至 E5.6PHYSICS HL
Name:姓名：Date:日期：
PART I · PAPER 1 STYLE第一部分 · 第一卷风格Short structured · calculator · 28 marks短结构题 · 可用计算器 · 28 分
Short Structured Items短结构题
Show all working in the space below each question. Marks are awarded for correct method as well as final answers. Quote astrophysical answers to an appropriate number of significant figures and always state units.在每题下方空白处写出全部解题过程。方法分（method marks）与最终答案同等重要。天体物理答案保留适当的有效数字并始终写明单位。
The binding energy per nucleon rises steeply for light nuclei and peaks near iron ($^{56}\mathrm{Fe}$).每核子结合能在轻核区急升，并在铁（$^{56}\mathrm{Fe}$）附近达到峰值。
(a)Using the shape of the binding-energy-per-nucleon curve, explain why the fusion of two light nuclei releases energy.利用每核子结合能曲线的形状，解释为何两个轻核聚变会释放能量。[3]
(b)State the two physical conditions required for fusion to occur in a stellar core, and identify the barrier each condition helps to overcome.写出恒星核心发生聚变所需的两个物理条件，并指出每个条件分别帮助克服什么。[2]
Q2MEDIUMPaper 1mass defect to energy质量亏损转能量[5 marks]
The Sun shines because hydrogen fuses to helium in its core through the proton-proton chain. Its luminosity is $L_\odot = 3.85\times 10^{26}\ \mathrm{W}$. Take $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$.太阳发光是因为其核心中氢通过质子-质子链聚变为氦。其光度为 $L_\odot = 3.85\times 10^{26}\ \mathrm{W}$。取 $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$。
(a)Write the overall (net) nuclear reaction of the proton-proton chain in the Sun, including the leptons emitted.写出太阳中质子-质子链的总（净）核反应，包括所放出的轻子。[2]
(b)Explain in one sentence why the Sun's energy source is described as nuclear, not chemical.用一句话解释为何太阳的能源被称为核能源而非化学能源。[2]
(c)Calculate the rate at which the Sun converts rest mass into energy.计算太阳把静止质量转换为能量的速率。[3]
Star A and star B are both modelled as black-body spheres. Star A has twice the radius of star B and three times its surface temperature.A 星与 B 星均视为黑体球。A 星半径是 B 星的两倍，表面温度是其三倍。
(a)Write the Stefan-Boltzmann law for the luminosity of a spherical star in terms of its radius $R$ and surface temperature $T$.用半径 $R$ 与表面温度 $T$ 写出球形恒星光度的斯特藩-玻尔兹曼定律。[1]
(b)Show that the ratio $L_A / L_B$ can be written purely in terms of the radius and temperature ratios, and calculate its value.证明比值 $L_A / L_B$ 可纯粹用半径比与温度比表示，并计算其值。[3]
(c)State which star is more luminous and identify which factor (radius or temperature) dominates the difference.指出哪颗星光度更大，并指明哪个因素（半径或温度）主导了这一差异。[1]
PART II · PAPER 1B / DATA ANALYSIS第二部分 · 第一卷 B / 数据分析Graphs · data · uncertainties · 22 marks图像 · 数据 · 不确定度 · 22 分
Graph and Data Questions图像与数据题
These items reward careful reading of gradients, correct use of the data-booklet relations, and clean handling of powers of ten and uncertainties. Quote uncertainties to one significant figure.这些题考查对斜率的细致读取、对数据手册关系式的正确使用，以及对 $10$ 的幂与不确定度的干净处理。不确定度保留 1 位有效数字。
Q6HARDPaper 1Binverse-square law: $b$ vs $1/d^2$平方反比律：$b$ 对 $1/d^2$[10 marks]
A class models a star of fixed luminosity $L$ by measuring the apparent brightness $b$ that would be received at several distances $d$. Their tabulated values of $b$ against $1/d^2$ are:某班级通过测量在若干距离 $d$ 处接收到的视亮度 $b$，来模拟一颗光度 $L$ 固定的恒星。他们将 $b$ 对 $1/d^2$ 列表如下：
$1/d^{2}\ /\ 10^{-34}\ \mathrm{m^{-2}}$
$1.0$
$2.0$
$3.0$
$4.0$
$b\ /\ 10^{-9}\ \mathrm{W\,m^{-2}}$
$2.0$
$4.0$
$6.0$
$8.0$
(a)Starting from the data-booklet relation $b = L/(4\pi d^2)$, show that a graph of $b$ against $1/d^2$ should be a straight line through the origin, and state what the gradient represents.从数据手册关系式 $b = L/(4\pi d^2)$ 出发，证明 $b$ 对 $1/d^2$ 的图应为过原点的直线，并说明斜率代表什么。[3]
(b)Calculate the gradient of the line and hence determine the luminosity $L$ of the modelled star.计算该直线的斜率，由此求所模拟恒星的光度 $L$。[3]
(c)A real star of the same luminosity is observed and then a second, identical star is found three times farther away. State how the apparent brightness of the second star compares with the first, justifying your answer.观测到一颗相同光度的真实恒星，又发现第二颗完全相同的恒星，其距离为前者的三倍。说明第二颗恒星的视亮度与第一颗相比如何，并说明理由。[2]
(d)The brightness reading at $1/d^2 = 4.0\times 10^{-34}\ \mathrm{m^{-2}}$ has an absolute uncertainty of $\pm 0.2\times 10^{-9}\ \mathrm{W\,m^{-2}}$. Calculate the percentage uncertainty in this reading.在 $1/d^2 = 4.0\times 10^{-34}\ \mathrm{m^{-2}}$ 处的亮度读数绝对不确定度为 $\pm 0.2\times 10^{-9}\ \mathrm{W\,m^{-2}}$。计算该读数的百分比不确定度。[2]
Q7HARDPaper 1BWien's law + HR diagram reading维恩定律与赫罗图读图[12 marks]
A star's spectrum peaks at a wavelength $\lambda_{\max} = 5.0\times 10^{-7}\ \mathrm{m}$. Wien's displacement law is $\lambda_{\max}\,T = 2.9\times 10^{-3}\ \mathrm{m\,K}$. On a Hertzsprung-Russell (HR) diagram, surface temperature increases to the left and luminosity increases upward.某恒星的光谱在波长 $\lambda_{\max} = 5.0\times 10^{-7}\ \mathrm{m}$ 处达到峰值。维恩位移定律为 $\lambda_{\max}\,T = 2.9\times 10^{-3}\ \mathrm{m\,K}$。在赫罗图（HR 图）上，表面温度向左增大，光度向上增大。
(a)Calculate the surface temperature of the star.计算该恒星的表面温度。[2]
(b)A second star has its peak at $\lambda_{\max} = 2.9\times 10^{-7}\ \mathrm{m}$. State which star is hotter and which appears bluer, justifying your answer.第二颗恒星的峰值在 $\lambda_{\max} = 2.9\times 10^{-7}\ \mathrm{m}$。说明哪颗更热、哪颗看起来更蓝，并说明理由。[2]
(c)The first star lies on the main sequence with a luminosity close to the Sun's. Describe where the main sequence sits on the HR diagram and state, in terms of mass, what determines a star's position along it.第一颗恒星位于主序，光度接近太阳。描述主序在赫罗图上的位置，并用质量说明是什么决定了恒星沿主序的位置。[3]
(d)A red giant has the same surface temperature as the first star but lies far higher on the HR diagram. Using $L = 4\pi R^2 \sigma T^4$, explain why the red giant must have a much larger radius.某红巨星与第一颗恒星表面温度相同，但在赫罗图上位置高得多。用 $L = 4\pi R^2 \sigma T^4$ 解释为何该红巨星的半径必然大得多。[3]
(e)State the broad end state of a low-mass star (like the Sun) and of a high-mass star after each leaves the main sequence.写出低质量恒星（如太阳）与大质量恒星离开主序后的大致归宿。[2]
PART III · PAPER 2 STYLE第三部分 · 第二卷风格Extended structured · calculator · 30 marks长结构题 · 可用计算器 · 30 分
Extended Structured Problems长结构问题
Set up each problem with the data-booklet relation clearly written before substituting. Method marks dominate the longer items; carry intermediate values to extra figures and round only the final answer.每题先清楚写出数据手册关系式再代入。长题中方法分占比最大；中间值多保留几位，仅在最终答案处取舍有效数字。
Q8HARDPaper 2radius from $L,T$ then distance from $b$由 $L,T$ 求半径再由 $b$ 求距离[12 marks]
A star has luminosity $L = 4.0\times 10^{27}\ \mathrm{W}$ and surface temperature $T = 6000\ \mathrm{K}$. It is observed from Earth with apparent brightness $b = 8.0\times 10^{-10}\ \mathrm{W\,m^{-2}}$. Take $\sigma = 5.67\times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$ and $1\ \mathrm{AU} = 1.5\times 10^{11}\ \mathrm{m}$.某恒星光度 $L = 4.0\times 10^{27}\ \mathrm{W}$，表面温度 $T = 6000\ \mathrm{K}$。从地球观测的视亮度 $b = 8.0\times 10^{-10}\ \mathrm{W\,m^{-2}}$。取 $\sigma = 5.67\times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$、$1\ \mathrm{AU} = 1.5\times 10^{11}\ \mathrm{m}$。
(a)Using the Stefan-Boltzmann law $L = 4\pi R^2 \sigma T^4$, calculate the radius $R$ of the star.用斯特藩-玻尔兹曼定律 $L = 4\pi R^2 \sigma T^4$，计算恒星半径 $R$。[4]
(b)Using $b = L/(4\pi d^2)$, calculate the distance $d$ of the star from Earth.用 $b = L/(4\pi d^2)$，计算恒星到地球的距离 $d$。[3]
(c)Express the distance found in (b) in astronomical units (AU).把 (b) 求得的距离用天文单位（AU）表示。[2]
(d)If the star's surface temperature were doubled while its radius stayed the same, state by what factor its luminosity would change, and justify this.若恒星表面温度翻倍而半径不变，写出其光度变为多少倍，并说明理由。[2]
(e)Distinguish, in one sentence, between the luminosity $L$ and the apparent brightness $b$ of a star.用一句话区分恒星的光度 $L$ 与视亮度 $b$。[1]
Q9HARDPaper 2p-p chain energetics from luminosity由光度求 p-p 链能量学[10 marks]
The Sun ($L_\odot = 3.85\times 10^{26}\ \mathrm{W}$) is powered by the proton-proton chain, in which four protons become one helium-4 nucleus, releasing about $26.7\ \mathrm{MeV}$ per helium-4 produced. Take $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$ and $1\ \mathrm{MeV} = 1.60\times 10^{-13}\ \mathrm{J}$.太阳（$L_\odot = 3.85\times 10^{26}\ \mathrm{W}$）由质子-质子链供能，其中四个质子变成一个氦-4 核，每生成一个氦-4 约释放 $26.7\ \mathrm{MeV}$。取 $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$、$1\ \mathrm{MeV} = 1.60\times 10^{-13}\ \mathrm{J}$。
(a)Calculate the rate at which the Sun converts rest mass into energy.计算太阳把静止质量转换为能量的速率。[2]
(b)Convert the energy released per helium-4 nucleus ($26.7\ \mathrm{MeV}$) into joules.把每个氦-4 核释放的能量（$26.7\ \mathrm{MeV}$）换算为焦耳。[2]
(c)Hence estimate the number of helium-4 nuclei produced in the Sun's core each second.由此估算太阳核心每秒生成的氦-4 核数目。[3]
(d)State why the energy that actually heats the Sun is slightly less than $26.7\ \mathrm{MeV}$ per helium-4 nucleus.说明为何实际加热太阳的能量略小于每个氦-4 核 $26.7\ \mathrm{MeV}$。[1]
(e)Explain why fusion in the Sun's core requires both high temperature and high density.解释为何太阳核心的聚变同时需要高温与高密度。[2]
Q10HARDPaper 2HL ONLYmass-luminosity and stellar lifetime质量-光度关系与恒星寿命[8 marks]
A main-sequence star has mass $M = 5.0\,M_\odot$. For main-sequence stars the luminosity scales as $L \propto M^{3.5}$ and the main-sequence lifetime scales as $t \propto M/L$. The Sun's main-sequence lifetime is about $1.0\times 10^{10}$ years.某主序星质量 $M = 5.0\,M_\odot$。对主序星，光度按 $L \propto M^{3.5}$ 变化，主序寿命按 $t \propto M/L$ 变化。太阳的主序寿命约为 $1.0\times 10^{10}$ 年。
(a)Calculate the luminosity of this star relative to the Sun, $L/L_\odot$.计算该星相对太阳的光度 $L/L_\odot$。[2]
(b)Show that the main-sequence lifetime scales as $t \propto M^{-2.5}$, and calculate the lifetime of this star relative to the Sun, $t/t_\odot$.证明主序寿命按 $t \propto M^{-2.5}$ 变化，并计算该星相对太阳的寿命 $t/t_\odot$。[2]
(c)Hence estimate the main-sequence lifetime of this star, in years.由此估算该星的主序寿命，单位为年。[2]
(d)Explain why a more massive main-sequence star has a shorter lifetime even though it begins with more nuclear fuel.解释为何质量更大的主序星尽管起初拥有更多核燃料，寿命却更短。[2]