OCR物理 赫罗图 红移 哈勃定律 考点解析
天体物理是 OCR A-Level Physics (H556) 里最容易被低估的一个模块。它看起来远离实验室, 公式不多, 但每一步都要求你理解物理图像, 而不是套公式。这篇文章把 OCR 考纲里的恒星距离测量, 星等与光度, 黑体辐射, 赫罗图演化和宇宙学红线一步步拆开, 配上计算示例和考场常见扣分点, 中英对照, 适合考前一周集中复习。
Astrophysics is one of the most underestimated modules in OCR A-Level Physics (H556). It looks far removed from the laboratory: there are few formulas to memorise, yet every question demands a clear physical picture instead of blind substitution. This guide walks through the OCR specification step by step: measuring stellar distances, magnitudes and luminosity, black-body radiation, the Hertzsprung-Russell diagram, stellar evolution and the cosmological redshift. Worked calculations and examiner pitfalls are included throughout, in Chinese and English, so you can revise efficiently in the final week before the exam.
1 天体物理在OCR考纲里的位置
天体物理属于 Module 5 (Newtonian World and Astrophysics), 主要在 Paper 3 中考查, 也可能以短计算题出现在其他试卷。考纲要求的核心内容有六块: 视差与秒差距的定义, 视星等与绝对星等以及距离模数, 光度与斯特藩-玻尔兹曼定律, 维恩位移定律与黑体谱, 赫罗图与恒星演化路径, 最后是多普勒红移与哈勃定律。这些内容彼此连接: 维恩定律给你温度, 斯特藩-玻尔兹曼定律把温度变成光度, 赫罗图把光度与温度放到同一张图上, 而红移把恒星物理推广到整个宇宙。复习时不要把它们当成六块孤立知识, 而要画出一条从恒星表面到宇宙尺度的逻辑链。
Astrophysics sits inside Module 5, Newtonian World and Astrophysics, and is examined mainly in Paper 3, though short calculations can appear elsewhere. The specification names six core areas: parallax and the definition of the parsec, apparent and absolute magnitude with the distance modulus, luminosity and the Stefan-Boltzmann law, Wien's displacement law and black-body spectra, the Hertzsprung-Russell diagram with stellar evolution, and finally the Doppler redshift with Hubble's law. These topics connect. Wien's law gives you surface temperature, Stefan-Boltzmann turns temperature into luminosity, the H-R diagram places luminosity against temperature, and redshift extends stellar physics to the scale of the whole universe. Treat them as one chain of reasoning from a stellar surface to the size of the cosmos rather than six isolated facts.
2 视差与秒差距: 距离的第一把尺子
离地球最近的恒星用三角视差测量。地球绕太阳公转, 半年后位置移动了约 3 亿公里, 于是近处恒星相对于遥远背景会出现微小位移, 位移角的一半就是视差角 p。定义: 视差角为 1 角秒时恒星的距离就是 1 秒差距 (parsec)。因为角度很小, 可以直接用 d(pc) = 1 / p(arcsecond)。1 pc 约等于 3.09 乘 10^16 米, 约 3.26 光年。
计算示例: 比邻星 (Proxima Centauri) 的视差角是 0.772 角秒, 代入得 d = 1 / 0.772 = 1.30 pc, 换成光年就是 1.30 乘 3.26 = 4.24 ly。地面望远镜受大气抖动限制, 视差角最小只能测到约 0.01 角秒, 对应约 100 pc; 这也是为什么盖亚 (Gaia) 卫星要在太空中做巡天: 它把测量精度推到微角秒量级, 距离尺度扩展到几千秒差距。
Trigonometric parallax is the first rung on the cosmic distance ladder. As the Earth orbits the Sun, a nearby star appears to shift against distant background stars; half of that shift is the parallax angle p. The parsec is defined so that a star with a parallax of one arcsecond lies at one parsec, and because the angle is tiny we may write d(pc) = 1 / p(arcsecond). One parsec equals about 3.09 x 10^16 m, or 3.26 light years.
Worked example: Proxima Centauri has a parallax of 0.772 arcseconds, so d = 1 / 0.772 = 1.30 pc, which is 1.30 x 3.26 = 4.24 light years. Ground-based telescopes are limited by atmospheric blurring to roughly 0.01 arcseconds, about 100 pc. That is exactly why the Gaia satellite observes from space: microarcsecond precision pushes the method out to several thousand parsecs and anchors the whole distance ladder.
3 光度 视星等 绝对星等与距离模数
光度 L 是恒星每秒辐射的总能量 (单位 W), 与距离无关; 视星等 m 描述我们接收到的辐射通量 F, 与距离有关。距离越远, 同样的光度摊开在越大球面上, 因此 F 与距离平方成反比: F = L / (4 pi d^2)。从天文学历史定义出发, 星等差 5 等对应通量比 100 倍, 于是有 m1 - m2 = -2.5 log10(F1 / F2)。若把恒星放在标准距离 10 pc 处, 得到的视星等就是绝对星等 M, 两条式子联立即得距离模数关系:
m - M = 5 log10(d / 10), 其中 d 以秒差距为单位。这个公式是考试里出现频率最高的天体物理公式之一。计算示例: 某星的视星等 m = 6.0, 绝对星等 M = 1.0, 则 m - M = 5 = 5 log10(d/10), 于是 d/10 = 10, d = 100 pc。反过来, 已知距离求星等差同样简单: d = 1000 pc 时 m - M = 5 log10(100) = 10, 说明视星等比绝对星等暗 10 个星等。
Luminosity L is the total energy radiated per second in watts and is independent of distance, while apparent magnitude m describes the flux F we actually receive and therefore depends on distance. The same luminosity spreads over a larger sphere further away, so F = L / (4 pi d^2). From the historical definition of the magnitude scale, a difference of five magnitudes corresponds to a flux ratio of one hundred, giving m1 - m2 = -2.5 log10(F1 / F2). Placing a star at the standard distance of 10 pc yields its absolute magnitude M, and combining the two expressions gives the distance modulus:
m - M = 5 log10(d / 10), with d in parsecs. This is one of the most frequently examined equations in astrophysics. Worked example: a star has m = 6.0 and M = 1.0, so m - M = 5 = 5 log10(d/10), hence d/10 = 10 and d = 100 pc. Working the other way, at d = 1000 pc we get m - M = 5 log10(100) = 10, meaning the star appears ten magnitudes fainter than its absolute magnitude.
4 黑体辐射 维恩位移定律与斯特藩-玻尔兹曼定律
把恒星近似看成黑体, 它发出的连续谱形状只由表面温度决定。维恩位移定律给出峰值波长与温度的反比关系: lambda_max 乘 T = 2.9 乘 10^-3 m K。斯特藩-玻尔兹曼定律给出单位面积辐射功率: P / A = sigma T^4, 其中斯特藩常数 sigma = 5.67 乘 10^-8 W m^-2 K^-4。把两个定律与球面积 4 pi r^2 结合, 就得到恒星光度 L = 4 pi r^2 sigma T^4, 这条式子把可测的 T 与 L 同恒星半径联系起来。
计算示例一: 参宿四 (Betelgeuse) 表面温度约 3500 K, 则 lambda_max = 2.9 乘 10^-3 / 3500 = 8.3 乘 10^-7 m, 即 830 nm, 落在红外区, 因此它看起来偏红。计算示例二: 参宿七 (Rigel) 温度约 11000 K, lambda_max = 2.64 乘 10^-7 m = 264 nm, 已进入紫外区, 它的可见光部分偏蓝。结论: 温度越高, 峰值波长越短, 看起来越蓝; 温度越低, 看起来越红。这在赫罗图上直接决定横轴方向。
Treating a star as a black body, the shape of its continuous spectrum depends only on surface temperature. Wien's displacement law states lambda_max x T = 2.9 x 10^-3 m K, an inverse relationship between peak wavelength and temperature. The Stefan-Boltzmann law gives power per unit area as sigma T^4 with sigma = 5.67 x 10^-8 W m^-2 K^-4. Combining these with the surface area of a sphere, 4 pi r^2, gives stellar luminosity L = 4 pi r^2 sigma T^4, linking the measurable quantities T and L to the radius of the star.
Worked example one: Betelgeuse has a surface temperature near 3500 K, so lambda_max = 2.9 x 10^-3 / 3500 = 8.3 x 10^-7 m, that is 830 nm in the infrared, which is why it looks red. Worked example two: Rigel at about 11000 K gives lambda_max = 2.64 x 10^-7 m, or 264 nm in the ultraviolet, so its visible light looks blue. The rule to remember is simple: hotter stars peak at shorter wavelengths and look blue, cooler stars peak at longer wavelengths and look red, which fixes the direction of the temperature axis on the H-R diagram.
5 赫罗图与恒星演化路径
赫罗图把光度 (或绝对星等) 画在纵轴, 表面温度画在横轴, 而且温度从左向右递减。这张图上有三个主要区域: 左上方到右下方的对角线是主序带, 太阳位于主序带中部; 右上方是红巨星与超巨星; 左下方是白矮星。恒星一生的大部分时间在主序上度过, 靠核心氢聚变维持, 处于流体静力学平衡。
演化路径由质量决定。小于约 8 个太阳质量的恒星离开主序后膨胀成红巨星, 核心氦聚变产生碳和氧, 外层被抛射形成行星状星云, 留下碳氧核心的白矮星; 白矮星靠电子简并压支撑, 质量上限是钱德拉塞卡极限, 约 1.4 个太阳质量。大于 8 个太阳质量的恒星会成为超巨星, 依次点燃更重的元素直到铁核, 铁核无法通过聚变产能, 于是核心塌缩并发生超新星爆发, 残留物可能是中子星或黑洞。
The Hertzsprung-Russell diagram plots luminosity, or absolute magnitude, against surface temperature, with temperature decreasing from left to right. Three regions dominate. A diagonal band from upper left to lower right is the main sequence, where the Sun sits near the middle. The upper right holds red giants and supergiants, and the lower left holds white dwarfs. A star spends most of its life on the main sequence, fusing hydrogen in its core and remaining in hydrostatic equilibrium.
Mass decides the evolutionary path. Below roughly eight solar masses a star leaves the main sequence, expands into a red giant, fuses helium into carbon and oxygen, sheds its outer layers as a planetary nebula and leaves a white dwarf supported by electron degeneracy pressure, with a maximum mass given by the Chandrasekhar limit of about 1.4 solar masses. Above eight solar masses the star becomes a supergiant, ignites successively heavier elements up to an iron core, and because iron fusion cannot release energy the core collapses in a supernova, leaving a neutron star or a black hole.
6 红移 哈勃定律与宇宙年龄
远处星系的光谱线整体向长波方向移动, 这就是红移。定义 z = delta lambda / lambda, 当 z 远小于 1 时可以用非相对论近似 v = z c。哈勃发现星系退行速度与距离成正比: v = H0 d, 哈勃常数 H0 的常用值是 67 到 70 km s^-1 Mpc^-1, 约合 2.2 乘 10^-18 s^-1。
计算示例: 某星系的红移 z = 0.05, 则退行速度 v = 0.05 乘 3.0 乘 10^8 = 1.5 乘 10^7 m/s, 即 15000 km/s。距离 d = v / H0 = 15000 / 68 = 220 Mpc, 约合 7.2 亿光年 (1 Mpc 约 3.26 百万光年)。如果把哈勃定律反过来用, 星系从同一点出发的旅行时间就是 1 / H0, 代入 H0 得约 4.5 乘 10^17 s, 也就是约 143 亿年, 这就是哈勃时间给出的宇宙年龄估计, 与普朗克卫星测得的 138 亿年相当接近。
Light from distant galaxies is shifted towards longer wavelengths, which we call redshift. It is defined as z = delta lambda / lambda, and for z much less than one the non-relativistic approximation v = z c is accurate. Hubble found that recession velocity is proportional to distance, v = H0 d, with H0 commonly quoted between 67 and 70 km s^-1 Mpc^-1, about 2.2 x 10^-18 s^-1.
Worked example: a galaxy with z = 0.05 recedes at v = 0.05 x 3.0 x 10^8 = 1.5 x 10^7 m/s, that is 15000 km/s. Its distance is d = v / H0 = 15000 / 68 = 220 Mpc, roughly 720 million light years. Inverting Hubble's law gives the travel time from a common origin as 1 / H0, which is about 4.5 x 10^17 s or 14.3 billion years. This Hubble time agrees well with the 13.8 billion year age measured by the Planck satellite, and it is a favourite exam calculation.
7 考试技巧与常见扣分点
第一, 看清命令词。State 只要一句定义, explain 要求因果链, calculate 要求代入并给单位, show that 要求中间步骤能看出而不是直接写答案。第二, 单位必须写全: 距离可以用 pc, ly 或 Mpc, 光度用 W, 温度用 K, 波长用 m 或 nm, 但公式里的单位要统一, 例如把 Mpc 化成 m 再除以 H0 时, 记得 1 Mpc = 3.09 乘 10^22 m。第三, 有效数字通常给 2 到 3 位, 中间步骤多留一位。
常见的三个错误: 把光度与视亮度混为一谈, 直接说远处恒星光度小; 在斯特藩-玻尔兹曼定律里用恒星到地球的距离 d 代替恒星半径 r, 写成 L = 4 pi d^2 sigma T^4; 以及把 log10 与自然对数混用, 在距离模数计算中代入 ln。考前建议按四步复习: 先把六个核心公式默写一遍并标注每个符号的含义, 再做两道距离模数和两道哈勃定律计算, 然后画一遍赫罗图并标出三类演化终点, 最后用真题做一次四十分钟限时训练, 检查单位与有效数字。
First, read the command word carefully. State needs one precise sentence, explain needs a causal chain, calculate needs working with units, and show that needs visible intermediate steps rather than a final answer. Second, always carry units: distances in pc, ly or Mpc, luminosity in W, temperature in K, wavelength in m or nm, and convert consistently, remembering that 1 Mpc = 3.09 x 10^22 m. Third, give two or three significant figures, keeping one extra digit in intermediate steps.
Three mistakes cost the most marks. Confusing luminosity with apparent brightness, so claiming a distant star has low luminosity. Substituting the distance d from Earth instead of the stellar radius r into the Stefan-Boltzmann law, writing L = 4 pi d^2 sigma T^4. Mixing up log10 and natural logarithms in the distance modulus. A four-step revision plan works well: write out the six core equations with the meaning of every symbol, do two distance modulus and two Hubble's law calculations, sketch the H-R diagram with the three end points of stellar evolution, and finish with a forty minute timed past paper, checking units and significant figures at the end.
8 观测方法: 光谱分类 双星与距离阶梯
恒星光谱按吸收线把恒星分为 O B A F G K M 七类, 顺序从热到冷, 记住那句英文口诀就足够: Oh Be A Fine Girl Kiss Me。光谱型与温度一一对应, 因此只要拿到光谱, 就能推断温度, 再从赫罗图读出光度。若恒星位于双星系统中, 通过开普勒第三定律可以求质量, 这是天文学里少数能直接测质量的途径之一。另一种重要方法是造父变星: 它的光变周期与光度相关, 周期光度关系让它成为标准烛光, 可以把距离测量推到几十百万秒差距之外的星系。
距离测量因此分成阶梯: 视差负责最近的几百秒差距, 造父变星负责邻近星系, 而红移与哈勃定律负责最遥远的宇宙尺度。答题时看清题目给出的是哪一级的方法, 不要在视差题里讨论红移, 也不要在宇宙学题里回退到秒差距公式。
Stellar spectra classify stars as O B A F G K M, running from hottest to coolest, and the familiar mnemonic is enough to recall the order. Spectral class corresponds directly to temperature, so a spectrum gives you T, and the H-R diagram then gives you luminosity. If the star lies in a binary system, Kepler's third law yields its mass, one of the very few direct mass measurements in astronomy. Cepheid variables offer another powerful tool: their pulsation period correlates with luminosity, and that period-luminosity relation makes them standard candles, pushing distance measurement out to galaxies tens of megaparsecs away.
Distances are therefore measured in rungs: parallax for the nearest few hundred parsecs, Cepheids for nearby galaxies, and redshift with Hubble's law for the largest cosmic scales. Read the question to see which rung is being tested, and do not discuss redshift in a parallax question or fall back to parsec formulas in a cosmology question.
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