IB物理HL 天体物理 光度 赫罗图 恒星演化
IB 物理的天体物理内容这几年换过一次版面。旧版指南把它放在 Option D，编号从 12.1 排到 12.4；现行指南取消了 Paper 3 的选做模块，天体物理挪进主题 E，成为 E.5 Fusion and stars，与核物理、量子物理放在一起考。很多学生以为换了版面内容也跟着换，其实光度、视差、赫罗图这几块的计算逻辑没动，只是从独立选项变成了主线里的一节。带来的直接后果是复习顺序要调：主题 E 前面几节先讲核的质量亏损与结合能，之后再进恒星内部的聚变，衔接比旧版自然。IBO 官网的物理指南 2023 年发布，2025 年 5 月首考，SL 与 HL 都覆盖 E.5。这篇文章把这一节涉及的量、公式与常见错误拆开写，数值一律按 IB 数据手册的取值。
Astrophysics sits in a different place in the current IB Physics guide than in its predecessor. What used to be Option D, with four numbered sub-topics, is now folded into theme E as E.5 Fusion and stars, examined alongside nuclear and quantum physics. The physics has not moved much: luminosity, parallax, the Hertzsprung-Russell diagram and stellar evolution still carry the marks; the sequence has changed, because students now meet mass defect and binding energy before fusion inside a core. The numbers below follow the IB data booklet.
质量亏损与每核子结合能曲线
恒星的能量账本从一件反直觉的事开始：四个质子合成一个氦核之后，总质量变小了。四颗质子的质量是 4.0291 u，氦核是 4.0015 u，差值 0.0276 u 乘上 931.5 MeV/u，得到约 26.7 MeV。缺掉的质量按质能关系变成能量放了出去。要判断哪种核反应放能，看的是每核子结合能曲线而不是反应温度。这条曲线随质量数上升，在铁附近达到每核子约 8.8 MeV 的最高点，之后缓慢下降。所以轻核聚变与重核裂变都放能，铁附近的核最稳定，再往上走就只能靠吸收能量。很多学生以为聚变放能是因为温度高，温度只是让质子靠到强核力能起作用的距离。IB 数据手册已经把原子质量单位与换算系数印好，考场上不需要记。
The energy budget of a star begins with a counter-intuitive piece of bookkeeping. Four protons weigh 4.0291 u and the helium nucleus they build weighs 4.0015 u; that 0.0276 u difference, multiplied by 931.5 MeV per u, comes out at about 26.7 MeV. Deciding whether a reaction releases energy depends on the binding energy per nucleon curve rather than on temperature; the curve climbs to roughly 8.8 MeV per nucleon near iron and then declines, which is why both fusion of light nuclei and fission of heavy nuclei pay out. Temperature only sets how closely protons can approach for the strong force to act, since Coulomb repulsion has to be overcome first.
质子-质子链与 CNO 循环
太阳核心温度约 1500 万开尔文，这个量级下质子有机会穿过库仑势垒。主序星的能量八成以上来自质子-质子链：两个质子先合成氘，同时放出一个正电子和一个中微子；氘再吃进一个质子变成氦-3；两个氦-3 最后拼成氦-4，并把两个质子还回去。净反应就是四个质子换一个氦核。质量更大的恒星里 CNO 循环的份额上升，碳氮氧只当催化剂，进出数目不变。中微子几乎不与物质作用，从核心一路穿出太阳，也把核心的信息带了出来。早期探测器只测到理论值三分之一的中微子数目，缺口后来由中微子振荡解释。常见的错误是把净反应写成四个质子直接合并，中间步骤在 HL 的推导题里是给分点。
At fifteen million kelvin the protons in the solar core can tunnel through their mutual Coulomb barrier often enough to sustain fusion. The proton-proton chain supplies most of a main sequence star's power: two protons fuse into deuterium and release a positron and a neutrino, deuterium captures another proton to form helium-3, and two helium-3 nuclei combine into helium-4 while returning two protons. Heavier stars run a larger fraction of their output through the CNO cycle, where carbon, nitrogen and oxygen act purely as catalysts. Neutrinos leave the core almost without interaction, and early detectors recording only a third of the expected flux opened the way to neutrino oscillation.
光度与视亮度
光度 L 是恒星每秒辐射出去的总能量，单位瓦特，跟观测者站在哪里无关。视亮度 b 是地球上每平方米每秒收到的能量，随距离平方衰减，关系写成 b 等于 L 除以 4 pi d 平方。考这类题基本只做两件事：给 b 和 d 求 L，或者给 L 和 b 求 d。太阳的光度约 3.83 乘 10 的 26 次方瓦，地球轨道处的视亮度约 1370 瓦每平方米，把日地距离 1.50 乘 10 的 11 次方米代进去，结果正好吻合。距离用米算，别和秒差距混在一道题里。有的恒星看上去暗，只是因为远，并不表示它本身弱。题目里出现标准烛光这个词，意思就是 L 已知，测 b 反推距离。
Luminosity is the total power a star radiates in watts, a property of the star alone, while apparent brightness is the power arriving per square metre at the observer and falls off with the square of distance. Rearranging b equals L over four pi d squared answers almost every question in this section, either for L or for d. The Sun radiates about 3.83 times ten to the twenty-sixth watts, and at the Earth's orbit that works out at roughly 1370 watts per square metre from a distance of 1.50 times ten to the eleventh metres. A faint appearance therefore says little about intrinsic power, and the phrase standard candle simply flags a source whose luminosity is already known.
秒差距与三角视差
天体距离的量级太大，米和千米都不好用，天文学改用秒差距。定义很干脆：当地球轨道半径 1 天文单位在恒星处张角为 1 角秒时，那颗星的距离就是 1 秒差距，等于 3.09 乘 10 的 16 次方米，约 3.26 光年。公式是 d 等于 1 除以 p，d 用秒差距，p 用角秒。地球绕日公转带来的视差半年一换边，测的是角位移的一半。最近恒星比邻星约 1.3 秒差距，角秒级的视差已经很小，地面望远镜在 0.01 角秒附近就吃力了。依巴谷与盖亚卫星做的正是这件事，把视差测到毫角秒，距离才能推到几千秒差距以外。常见错误是拿两个月之间的位移当 p，实际要用半年基线的总位移除以二。
Astronomers avoid metres and kilometres because stellar distances are too large for them, so the parsec is defined through geometry instead. One parsec is the distance at which the Earth's orbital radius of one astronomical unit subtends one arcsecond, equal to 3.09 times ten to the sixteenth metres or about 3.26 light years. The working relation is d equals one over p, with d in parsecs and p in arcseconds, and the baseline is half a year of orbital motion rather than one season. Proxima Centauri lies about 1.3 parsecs away, and measuring parallax at that scale needs satellites such as Hipparcos and Gaia.
视星等、绝对星等与距离模数
星等是希腊人留下的反向标尺，数字越小越亮，每差 5 个星等亮度差 100 倍。视星等 m 是在地球上看到的亮度等级，绝对星等 M 是把恒星搬到 10 秒差距处应有的视星等。两者由距离联系起来，m 减 M 等于 5 倍 log 的 d 除以 10，d 用秒差距。太阳的视星等约负 26.7，绝对星等约正 4.8，把它搬到 10 秒差距外就只是一颗肉眼勉强可见的暗星。算出来的 m 减 M 叫距离模数，已知 M 就能反推距离。符号容易被忽略，m 比 M 大表示比 10 秒差距远。这类题最常见的手滑是把 d 直接填成千米，公式里的距离只能是秒差距。
Magnitudes run backwards from everyday intuition: smaller numbers mean brighter stars, and five magnitudes correspond to a factor of one hundred in brightness. Apparent magnitude describes what an observer records, whereas absolute magnitude is the value the same star would show from ten parsecs away. The link between them is the distance modulus, m minus M equals five times the logarithm of d over ten, with d again in parsecs. The Sun's apparent magnitude of about minus 26.7 falls to roughly plus 4.8 at ten parsecs, which is how a brilliant nearby star becomes an unremarkable one.
维恩位移定律与恒星颜色
恒星的颜色不是任意挑的，峰值波长与表面温度成反比。维恩位移定律写成 lambda max 乘 T 等于 2.9 乘 10 的负 3 次方米开尔文，波长用米，温度用开尔文。太阳表面约 5800 开尔文，峰值波长落在 500 纳米上下，正好是可见光的黄绿。蓝白星温度高，峰值波长短，参宿七这类蓝超巨星表面可以到 1 万开尔文以上；红色星相反，表面只有三四千开尔文。测出一条吸收谱里最亮的波长，就能估温度，这也是光谱巡天能批量给出温度的原因。常见的错误是把峰值波长当成整颗星辐射最强的可见颜色，其实峰值以外还有大量能量在红外。计算时先统一单位，纳米先换成米再代公式。
Colour tells you temperature because the peak wavelength of a star's thermal spectrum moves inversely with surface temperature. Wien's displacement law, lambda max times T equals 2.9 times ten to the minus three metre kelvin, puts the Sun's 5800 kelvin peak near 500 nanometres, in the yellow-green part of the visible band. Blue-white stars such as Rigel sit well above ten thousand kelvin, while red stars are cooler objects in the three to four thousand kelvin range. Reading a peak out of a spectrum therefore gives a temperature for every star in a survey, provided nanometres are converted to metres before substitution.
斯特凡-玻尔兹曼定律与恒星半径
光度不只与温度有关，还跟恒星表面积挂钩。黑体辐射的总功率按斯特凡-玻尔兹曼定律写成 L 等于 4 pi R 平方 sigma T 的四次方，sigma 是 5.67 乘 10 的负 8 次方瓦每平方米开尔文四次方。温度是四次方，所以温度翻一倍，同样大小的星亮度变成 16 倍。两条恒星光度一样而温度差一倍时，冷的那颗半径要大 4 倍，因为面积按平方补回来。这条关系把赫罗图上的位置和物理尺寸连了起来：红巨星温度低却亮，只能是半径巨大。做题时一次只改一个变量，把两式相除最省事。阅卷时经常看到学生把 R 写成直径，四次方算完却没有开平方，答案差一倍。
Radiated power depends on surface area as well as temperature, and the Stefan-Boltzmann law states it as luminosity equals four pi R squared sigma T to the fourth, with sigma equal to 5.67 times ten to the minus eight. Because temperature enters to the fourth power, doubling it raises the output of a star of fixed size sixteenfold. Two stars of equal luminosity but a factor of two in temperature must differ in radius by a factor of four, since area has to make up the difference. That link explains why cool red giants can outshine hot dwarfs, and ratioing the two expressions is quicker than substituting numbers twice.
赫罗图上的三条主带
赫罗图把光度放在纵轴、温度放在横轴，温度轴反向，左边热右边冷。绝大多数恒星落在一条从左上到右下的斜带上，这条带叫主序，氢在核心稳定聚变。右上角是巨星与超巨星，温度低、光度高，半径必然大。左下角是白矮星，温度可以很高但光度很低，因为半径只有地球量级。把一颗星放进图里，等于同时读出它的温度和光度，再由斯特凡-玻尔兹曼定律推出半径。星团是最好用的样本，同一星团里的恒星距离相同、年龄相同，主序转折点的位置直接对应年龄。常见的错误是把白矮星画到右下角，那是冷而暗的红矮星所在的位置。
A Hertzsprung-Russell diagram plots luminosity vertically against temperature horizontally, with the temperature axis reversed so that hot stars sit on the left. Most stars gather along a diagonal band running from hot and bright to cool and faint, the main sequence, where core hydrogen fusion is steady. Giants and supergiants occupy the upper right, cool but luminous, and white dwarfs sit lower left as hot but tiny objects. Placing a star on the diagram fixes its temperature and luminosity at once, so the Stefan-Boltzmann law then yields its radius, which is why clusters with a shared distance provide such clean age estimates.
光谱型 OBAFGKM 与吸收线
按温度把恒星排成序列，得到 O、B、A、F、G、K、M 七个光谱型，从三万开尔文以上一直排到三千开尔文以下。吸收线的强弱由温度决定：O 型星里氦被电离两次，谱线出现在氦离子上；A 型星温度约一万开尔文，氢的巴尔末线最强；太阳这类 G 型星金属线丰富；M 型星太冷，钛氧化物分子带占了上风。看谱线判断光谱型，再用光谱型读温度，是天体物理最基本的操作。很多学生以为氢线最强的星含氢最多，其实几乎所有恒星成分接近，差别在激发态粒子数。温度太高氢已经电离，温度太低激发不足，两头都不出强线。
Ordering stars by temperature produces the spectral sequence O, B, A, F, G, K, M, running from above thirty thousand kelvin down to below three thousand. Line strengths follow temperature rather than abundance, so ionised helium dominates the hottest class, the Balmer lines of hydrogen peak near ten thousand kelvin in class A, and molecular bands of titanium oxide appear in the coolest class. Using lines to assign a class and then reading temperature from the class is the standard first step in stellar work. Hydrogen lines peak at intermediate temperatures because hotter stars ionise the gas and cooler ones leave too few atoms excited.
主序寿命与质量的关系
恒星在主序上待多久，取决于它手上有多少燃料、烧得多快。燃料量与质量成正比，光度随质量上升得快得多，大致按质量的三点五次方走，结果寿命与质量的负二点五次方成正比。太阳的寿命约 100 亿年，质量十倍于太阳的星只有约三千万年，质量一半的星能烧上几百亿年。这解释了为什么星团里先消失的是大质量星，也解释了为什么亮星总是年轻。观测上验证这一点靠星团的主序截断，截断位置越高说明星团越年轻。算寿命比值时不必代入绝对数值，两式相除即可。常见的错误是把寿命写成与质量成正比，那样会得出大质量星活得更久的结论。
How long a star stays on the main sequence depends on how much fuel it holds and how fast it burns it. Fuel scales with mass while luminosity climbs far faster, roughly as the 3.5 power, so lifetime varies as mass to the minus 2.5. A star ten times the Sun's mass lasts only about thirty million years against the Sun's ten billion, and a half-solar-mass star can keep going for hundreds of billions. Cluster main sequence turn-off points test this directly, since the brightest surviving stars mark the cluster's age, and no absolute numbers are needed when lifetimes are compared as ratios.
演化终点与钱德拉塞卡极限
主序之后氢在核心耗尽，核心收缩升温，外壳膨胀，恒星成为红巨星。质量约在太阳八倍以下的星，氦聚变到碳氧就停住，外层抛出形成行星状星云，核心留下白矮星，靠电子简并压支撑。白矮星的质量上限是钱德拉塞卡极限，约 1.4 个太阳质量，超过就撑不住。更大的恒星一路烧到铁，铁核无法再靠聚变放能，核心塌缩触发超新星爆发，残核可能成为中子星，也可能塌成黑洞。中子星靠中子简并压支撑，半径只有十几千米。黑洞的边界用史瓦西半径描述，太阳质量对应的半径约 3 千米。判断归宿时先看初始质量，再看残核是否超过那两个上限。
Once core hydrogen runs out, the core contracts and heats while the envelope swells into a red giant. Stars below roughly eight solar masses stop at carbon and oxygen, shed a planetary nebula and leave a white dwarf held up by electron degeneracy pressure, with the Chandrasekhar limit near 1.4 solar masses marking the ceiling. More massive stars burn successive shells until iron builds up, and since iron fusion costs energy the core collapses and a supernova follows, leaving either a neutron star supported by neutron degeneracy or a black hole. A one solar mass black hole would have a Schwarzschild radius of about three kilometres.
标准烛光：造父变星与 Ia 型超新星
测远处星系的距离，靠的是光度已知的标准烛光。造父变星的光变周期与光度之间有稳定关系，周期越长光度越高，这条周期光度关系在二十世纪初由勒维特从麦哲伦云里的变星中发现。只要测出周期，就知道 L，再测视亮度 b，距离用平方反比公式反推。Ia 型超新星是另一类更好用的标准烛光：白矮星吸积伴星物质逼近钱德拉塞卡极限后热核爆炸，峰值光度相当一致，所以能看到极远的星系。上世纪九十年代用这类超新星测距，发现远星系比预期更暗，由此推出宇宙在加速膨胀。常见的错误是把造父变星的关系当成温度与周期，实际连的是光度与周期。
Distances to remote galaxies rest on standard candles whose luminosity is already known. Cepheid variables pulsate with a period tied tightly to luminosity, a relation Leavitt identified in Magellanic Cloud variables early in the twentieth century, so a measured period yields L and the inverse square law then converts apparent brightness into distance. Type Ia supernovae work even further out, because a white dwarf pushed past the Chandrasekhar limit detonates at a fairly consistent peak brightness. Comparing such supernovae across the sky in the late 1990s showed distant galaxies to be fainter than expected, the first evidence that expansion is accelerating.
哈勃定律、红移与宇宙组成
远处星系的光谱整体向长波方向移动，红移量 z 等于波长变化除以原波长，在速度远小于光速时约等于 v 除以 c。哈勃在 1929 年发现退行速度与距离成正比，写成 v 等于 H0 乘 d。H0 用公里每秒每兆秒差距，欧洲空间局 Planck 2018 官网数据给 67.4，近邻测量的 SH0ES 项目给约 73，两个值的差距就是常说的哈勃张力。把 1 除以 H0 换成年龄是 145 亿年量级，标准宇宙学模型结合暗能量后给出的宇宙年龄约 138 亿年。Planck 2018 的组分是普通物质 4.9%、暗物质 26.8%、暗能量 68.3%，空间几何平直到 0.4% 以内。1965 年彭齐亚斯与威尔逊测到各向同性的 2.7 开尔文微波背景，成为大爆炸模型的直接证据，两人因此获得 1978 年诺贝尔物理学奖。
Light from distant galaxies arrives stretched towards longer wavelengths, and the fractional shift z equals the change in wavelength over the original, which reduces to v over c for speeds far below light speed. Hubble's 1929 result, that recession speed grows in proportion to distance, is now written v equals H0 times d, with H0 around 67.4 kilometres per second per megaparsec from the Planck 2018 data published by the European Space Agency and near 73 from nearby SH0ES measurements, a disagreement known as the Hubble tension. Inverting H0 gives a timescale of roughly 14.5 billion years, while the standard model with dark energy settles on an age near 13.8 billion years. Planck 2018 also fixes the mix at 4.9 per cent ordinary matter, 26.8 per cent dark matter and 68.3 per cent dark energy, with geometry flat to within 0.4 per cent, and the 2.7 kelvin background measured by Penzias and Wilson in 1965 remains the strongest single piece of evidence for a hot early universe.
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