IB物理HL 恒星视差 光度 赫罗图 哈勃定律
IB 物理的新指南在 2023 年发布、2025 年 5 月首次评估，天体物理内容不再作为 Paper 3 的选做模块，而是并入主题 E 核物理与量子物理的 E.5 小节聚变与恒星。很多学生还在按旧版指南里的 Option D 找复习材料，翻到新版试卷才发现卷面上没有选做题。这一节要求把恒星当成可以测量的物理对象：先量出距离，再由距离推出光度，最后用光度与表面温度把恒星放上赫罗图。宇宙尺度的结论也来自同一条测量链条，哈勃定律与红移、宇宙微波背景都是它的延伸。本文按测量、恒星属性、恒星演化、宇宙学的顺序走一遍，公式同时交代适用条件。文中的官方口径来自 IB 物理指南（2023 年发布、2025 年首考）与欧洲空间局 Planck 卫星 2018 年公布的宇宙学参数。E.5 在 SL 与 HL 都要考，两个层级的差别主要在计算深度与图像分析上。
IB Physics was reworked in 2023 and first assessed in May 2025: astrophysics is no longer a Paper 3 option but sits inside Theme E, Nuclear and quantum physics, as subtopic E.5 Fusion and stars. Students who revise from the old Option D material often arrive at the paper looking for a module that has disappeared. What E.5 asks for is a measurement chain, beginning with how far away a star is, moving from distance to luminosity, and then placing stars on the Hertzsprung-Russell diagram with luminosity and surface temperature; the same chain carries through to Hubble's law, redshift and the cosmic microwave background. The figures quoted below come from the IB Physics guide (published 2023, first assessment 2025) and from the European Space Agency's Planck 2018 cosmological results.
恒星视差与秒差距
视差来自地球绕太阳的公转：同一颗近星在半年内相对背景恒星发生微小位移，这个位移角的一半就是视差角 p。把日地距离当作基线，秒差距的定义是视差角恰好为 1 角秒时的距离，写成 d = 1 / p，距离单位取秒差距、角度单位取角秒。1 秒差距约等于 3.26 光年，换成米是 3.09e16 米。地面望远镜受大气抖动限制，可靠测距范围大约到 100 秒差距；欧洲空间局的 Hipparcos 卫星在 20 世纪 90 年代把精度推进到毫角秒级，2013 年发射的 Gaia 卫星在 2022 年发布的第三版星表里给出了十多亿颗恒星的视差。很多学生以为这个公式直接给出光年，于是角度不换算、量级也不检查，算出的小数看不出对错。常见的错误是距离 200 秒差距的恒星照样套 1 / p，却忽略此时视差角只剩 5 毫角秒，已经落在测量误差里。距离越远，同样的角度误差被放大的倍数越大，误差从百分之几变成数量级偏差。
Parallax comes from the Earth's orbit: a nearby star shifts slightly against the background stars over six months, and half of that shift is the parallax angle p. The parsec is defined as the distance at which the parallax angle is exactly one arcsecond, which gives the tidy relation d = 1 / p with d in parsecs and p in arcseconds; one parsec is about 3.26 light years, or roughly 3.09e16 metres. Atmospheric blur limits ground-based measurement to about 100 parsecs, whereas ESA's Hipparcos satellite pushed accuracy towards the milliarcsecond level during the 1990s, and Gaia, launched in 2013, published parallaxes for more than a billion stars in its third data release in 2022. A frequent mistake is to plug numbers into the relation while treating the answer as light years, so a star at 200 parsecs, where the parallax is only 5 milliarcseconds, ends up quoted to a precision the instrument never had.
光度 视亮度与距离平方反比
光度 L 是恒星每秒辐射的总能量，与距离无关；视亮度 b 是地球上每平方米每秒收到的能量，随距离衰减。点源在距离 d 处的视亮度遵循平方反比关系 b = L / (4 pi d^2)，反过来写就能从实测视亮度和已知光度反推距离。太阳的光度约为 3.8e26 瓦，太阳质量约 2.0e30 千克，这两个数值常用来把恒星参数换成以太阳为单位的相对值。造父变星的特殊之处在于光变周期与光度之间存在稳定关系，这一关系由 Leavitt 在 1912 年从麦哲伦云的底片中整理出来，此后成为测量河外星系距离的标准烛光。很多学生把视亮度和光度当成同一个量，看到一颗星在图上很暗就断定它本身很暗，其实可能只是离得远。常见的错误是把两个量记反，处理距离平方时漏掉平方，答案差出几十倍。衡量亮度还有星等标度，每差 5 个星等对应亮度相差 100 倍，这个换算在计算题里经常出现。
Luminosity L is the total energy a star radiates per second and does not depend on distance, while apparent brightness b is the power received per square metre at the Earth and falls off with distance. For a point source the two are linked by the inverse-square relation b = L / (4 pi d^2), usually rearranged to turn a measured brightness into a distance once L is known; the Sun's luminosity is about 3.8e26 W and its mass about 2.0e30 kg, and both serve as yardsticks for stellar quantities. Cepheid variables are valuable because their pulsation period is tied to their luminosity, a relation compiled by Leavitt in 1912 from photographic plates of the Magellanic Clouds, and that period-luminosity law made them standard candles for distances far beyond parallax. Mixing up a faint-looking star with an intrinsically dim star is the usual slip, since brightness alone says nothing until distance is known, and the magnitude scale adds a second trap because five magnitudes correspond to a factor of 100 in brightness.
维恩位移定律与光谱分类
恒星表面温度可以从光谱峰值位置读出，维恩位移定律给出峰值波长与温度的乘积为常数，写成 lambda_max * T = 2.9e-3 米开尔文。温度高的恒星峰值落在蓝端，温度低的落在红端。恒星光谱按吸收线特征分成 O、B、A、F、G、K、M 七类，太阳属于 G2，表面温度约 5800 开尔文。光谱型与温度一一对应，做题时按温度顺序排列光谱型比死记字母顺序更稳。光度与温度还通过斯特藩玻尔兹曼定律联系：L = 4 pi r^2 sigma T^4，半径的平方与温度的四次方共同决定光度。很多学生拿维恩位移定律算出温度后，直接代入 L 公式却忘了 L 是单位时间的总能量，单位混用导致量级出错。常见的错误还有把峰值波长当成可见光的中心波长，红外峰值对应的是温度更低的恒星。温度相差一倍，单位面积的光度按四次方相差十六倍，这是半径比较题最常用的突破口。
Surface temperature is read off from where a star's spectrum peaks, and Wien's displacement law fixes the product of peak wavelength and temperature at about 2.9e-3 m K, so hot stars peak in the blue and cool ones in the red. Stellar spectra are sorted into the classes O, B, A, F, G, K and M, with the Sun sitting at G2 and a surface temperature near 5800 K; the sequence follows temperature, which is why ranking the classes by temperature is safer than memorising letters. Luminosity and temperature are connected through the Stefan-Boltzmann law, L = 4 pi r^2 sigma T^4, where radius squared and the fourth power of temperature together set the output. Students often compute a temperature from the peak wavelength and then substitute it into the luminosity equation without checking units, and a star twice as hot as another radiates sixteen times as much power per unit area, which is the standard route into radius-comparison questions.
赫罗图上的主序带
赫罗图把光度画在纵轴、表面温度画在横轴，温度轴习惯反向，从左到右温度下降。九成以上的恒星落在从左上到右下的主序带上，位置由质量决定：质量大的星温度高、光度高、寿命短。图的右上角是红巨星与超巨星，左下角是白矮星，两者都不在主序上但数量可观。把恒星的实测视亮度换成光度，需要先知道距离，这一步正是视差与标准烛光的用处。很多学生看图时把纵轴当作视亮度，于是同一颗星换个距离就从主序跑到巨星区，图上的位置完全失去意义。常见的错误是忽略温度轴的反向，把左右顺序读反，导致红巨星与蓝巨星互换。做演化题时先在图上一句定位：主序星在带状区域，脱离主序后走向右上角还是左下角取决于质量。图上位置的移动方向，就是恒星半径与表面温度这两个变量此消彼长的过程。
The Hertzsprung-Russell diagram plots luminosity against surface temperature, with the temperature axis conventionally reversed so that it decreases from left to right. More than nine stars in ten lie on the main sequence, a band running from the hot, bright upper left to the cool, dim lower right, and position along it is set by mass: heavy stars are hotter, more luminous and far shorter lived. Red giants and supergiants occupy the upper right, white dwarfs the lower left, and neither group sits on the main sequence even though both are common. Turning an observed apparent brightness into a luminosity requires a distance, which is exactly where parallax and standard candles earn their place; plotting raw brightness instead scatters a single star across the diagram and makes the position meaningless. Reading the reversed temperature axis the wrong way round swaps giants for blue supergiants, so it pays to fix a star's band on the diagram before attempting an evolution question.
恒星演化与钱德拉塞卡极限
恒星的一生由质量决定，主序阶段靠氢核聚变维持，核心氢耗尽后外层膨胀成为红巨星。质量与太阳相近的恒星最终抛掉外层形成行星状星云，留下由电子简并压支撑的白矮星，其质量上限约为 1.4 个太阳质量，这一上限与钱德拉塞卡的名字联系在一起。质量更大的恒星演化更快，核心坍缩时可能形成中子星，半径只有十几公里，密度达到每立方厘米上亿吨。若核心质量超过中子星的支撑极限，坍缩没有已知机制可以阻止，结果就是黑洞。太阳的寿命估算在百亿年量级，而 20 倍太阳质量的恒星主序寿命只有几百万年，这个反差是演化题里最常考的数量级。很多学生以为质量越大寿命越长，把恒星当成燃料越多烧得越久，方向刚好相反。常见的错误是把白矮星与中子星的质量门槛记混，或者忘记简并压而不是热压才是小质量恒星残骸的支撑来源。
A star's life is decided by its mass: the main sequence is sustained by hydrogen fusion, and once the core hydrogen is used up the outer layers swell into a red giant. A star like the Sun eventually sheds those layers as a planetary nebula and leaves behind a white dwarf supported by electron degeneracy pressure, with a mass ceiling near 1.4 solar masses that carries Chandrasekhar's name. Heavier stars evolve much faster and may collapse into a neutron star a few tens of kilometres across and dense enough to weigh hundreds of millions of tonnes per cubic centimetre; if the collapsing core passes the limit that neutron degeneracy can hold, nothing known stops it and a black hole forms. The Sun has a lifetime measured in billions of years, while a star twenty times heavier burns through its main sequence in a few million, and that contrast is the usual numerical question. Assuming that a bigger star lasts longer inverts the physics, because more fuel comes with a much steeper rise in luminosity.
哈勃定律与红移
星系光谱的吸收线普遍向长波方向移动，红移量 z 定义为波长变化量与原始波长之比。哈勃定律把退行速度与距离连成直线关系 v = H0 * d，H0 是哈勃常数，单位写作千米每秒每兆秒差距。哈勃常数并不是常数，它随时间变化，这个名字只是历史沿用。H0 的测量存在分歧：Planck 卫星用宇宙微波背景拟合得到的 2018 年结果是 67.4 千米每秒每兆秒差距，而利用造父变星与 Ia 型超新星做的本地测量在 2022 年给出约 73，两者的差异被称为哈勃张力。SL 与 HL 的考题通常给出固定的 H0，直接代入 v = H0 * d 求距离或速度，用红移求速度时要注意低速近似 z 约等于 v / c 的适用条件。很多学生把红移解释成星系在空间里运动的多普勒效应，忽略了更主流的解释是空间本身在膨胀，两种说法在低速下数值相近但物理图像不同。常见的错误是距离单位不统一，把兆秒差距和光年混在同一道题里。
Absorption lines in galaxy spectra are shifted towards longer wavelengths, and the redshift z is defined as the fractional change in wavelength. Hubble's law links recessional speed to distance through v = H0 * d, with H0 in kilometres per second per megaparsec; the name is historical, since the parameter changes as the universe expands. Measured values disagree, and that is the Hubble tension: fitting the cosmic microwave background gives 67.4 km/s/Mpc in the Planck 2018 results, while local measurements using Cepheids and type Ia supernovae reached about 73 by 2022. Exam questions usually supply one value of H0 and expect straightforward substitution, and at low speed the redshift relation reduces to z = v / c, so the approximation needs checking before it is used. Interpreting redshift purely as motion through space misses the mainstream picture in which space itself stretches, and unit slips between megaparsecs and light years are the most frequent arithmetic error.
宇宙微波背景与大爆炸证据
宇宙微波背景是早期高温等离子体冷却后遗留的辐射，今天测到的温度约 2.7 开尔文，落在微波波段。Penzias 与 Wilson 在 1965 年报告了无法消除的天线噪声，随后被确认为来自各个方向的背景辐射，这项工作获得了 1978 年诺贝尔物理学奖。它的高度各向同性说明早期宇宙在大尺度上均匀，而十万分之一的温度涨落提供了后来结构生长的密度种子。欧洲空间局的 Planck 卫星在 2018 年公布的最终结果把宇宙年龄定在约 138 亿年，并给出普通物质占 4.9%、暗物质占 26.8%、暗能量占 68.3% 的组成，宇宙几何在 0.4% 的精度内接近平直。这些数字支持热大爆炸模型，因为它要求宇宙曾经处于高温高密的状态。很多学生以为微波背景来自某个天体，答题时把它和恒星或星系的信号混在一起，忽略了它铺满整个天空。常见的错误是只写温度不写来源，把背景辐射当成假设，说不出它是被测量到的证据。
The cosmic microwave background is the relic radiation left when the early hot plasma cooled, and it now arrives at about 2.7 K in the microwave band. Penzias and Wilson reported unexplained antenna noise in 1965 which turned out to be radiation arriving equally from every direction, work recognised by the 1978 Nobel Prize in Physics; its near-perfect isotropy shows that the early universe was smooth on large scales, while temperature ripples of about one part in a hundred thousand seeded the structure that grew later. The European Space Agency's Planck mission, in its final 2018 results, put the age of the universe near 13.8 billion years and its contents at 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy, with the geometry flat to within 0.4%. Those numbers fit a hot Big Bang model because they require an earlier hot, dense state, and the usual error is to treat the background as a signal from one celestial object rather than from the whole sky.
聚变反应与临界密度
太阳的能量来自质子链反应，四个质子经过一系列步骤合成一个氦核，净释放约 26.7 兆电子伏特，同时放出正电子与中微子。质量较大的恒星核心温度更高，主要走碳氮氧循环，两条路径都靠质量亏损按照 E = m c^2 转成能量。聚变产生向外的压强抵抗引力收缩，两者平衡时恒星稳稳停在主序上。宇宙的整体命运由密度与临界密度的比值决定，临界密度写成 3 H0^2 / (8 pi G)，取 H0 约 70 千米每秒每兆秒差距时约为 9.5e-27 千克每立方米。密度大于临界值对应闭合宇宙，等于对应平直，小于对应开放，Planck 2018 结果支持接近平直的几何。很多学生把临界密度当成实测密度，忘了它是用哈勃常数算出来的参考值，做比较时两个量必须同时给出。常见的错误是聚变方程不配平电荷与质量数，或者把 26.7 兆电子伏特当成每个质子释放的能量，实际这是四个质子合成一个氦核的总输出。
The Sun is powered by the proton-proton chain, in which four protons are fused into one helium nucleus with a net release near 26.7 MeV plus positrons and neutrinos; more massive stars run hot enough for the carbon-nitrogen-oxygen cycle, and either route converts the mass defect into energy through E = m c^2. Fusion pressure outwards balances gravitational contraction inwards, and that balance is what keeps a star sitting quietly on the main sequence for most of its life. The universe's fate is set by the ratio of its density to the critical density, 3 H0^2 / (8 pi G), which is roughly 9.5e-27 kg per cubic metre for H0 near 70 km/s/Mpc; a density above that value closes the geometry, one equal to it keeps space flat, and below it opens the universe up, with the Planck 2018 results favouring a geometry flat to within 0.4%. The critical density is a reference value calculated from H0 rather than something measured directly, and the commonest calculation errors are unbalanced charge and mass number in a fusion equation, and treating 26.7 MeV as the output per proton when it is the total for four protons.
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