OCR物理 维恩定律 斯特凡定律 哈勃红移
OCR 的 A Level 物理 H556 把天体物理放在模块 5 的最后一节，编号 5.5，标题是 Astrophysics and cosmology。这一节看起来内容散，其实只围绕一个问题：怎样从地球上收到的一点光，反推出恒星的温度和半径，再推到整个宇宙的膨胀。
在 H556 的三份卷子里，模块 5 的内容主要落在 Paper 1 上，Paper 3 的综合题也常把恒星的光度与引力场拼在同一道题里。计算本身不长，扣分大多发生在单位换算和公式选择上，这也是这一节最值得提前练熟的地方。
Astrophysics closes Module 5 in OCR Physics A (H556), and it rewards one habit above all others: treating a spectrum as data rather than as a picture. The arithmetic is short, which is exactly why examiners can be strict about units and about which law belongs to which measurement. Work through the three-step chain once, from peak wavelength to temperature to radius, and the rest of the module becomes bookkeeping.
光度与到达地球的辐射强度
光度 L 指恒星每秒向四周辐射的总能量，单位是瓦特，它由恒星自身的半径与表面温度决定，跟观察者站在哪里没有关系。到达地球的辐射强度 F 是另一个量，指每秒穿过单位面积的辐射能量，单位是瓦特每平方米。光向四周均匀扩散，同样多的能量摊在越来越大的球面上，因此 F 与距离的平方成反比，写作 F = L / (4πd²)。很多学生以为光度就是我们看到的亮度，其实亮度是 F，光度才是恒星自己的 L。太阳的光度约为 3.8×10²⁶ W，地球与太阳相距 1.50×10¹¹ m，代入公式得 F 约 1.4×10³ W m⁻²，这个数值就是太阳常数。常见的错误是距离用秒差距却没有换成米，算出的结果会小十几个数量级。
Luminosity belongs to the star, while flux belongs to the measurement, and mixing the two is the fastest way to lose a mark. Because the same power spreads over a sphere of area 4πd², doubling the distance cuts the received flux to a quarter. A useful sanity check is the solar constant itself: the Sun delivers roughly 1.4 kW to every square metre just outside the atmosphere, so any stellar flux far above that value should raise a question about units.
维恩位移定律读峰值波长
恒星的连续谱不是一条平线，强度随波长先升后降，存在一个峰值波长 λmax。维恩位移定律说 λmax 与绝对温度成反比，用数据表给出的常数可以写成 λmax × T = 2.90×10⁻³ m K。温度越高，峰值越往短波方向移，所以蓝白色的星比橙红色的星热得多，这个趋势肉眼也看得出来。取一颗峰值波长 480 nm 的星，先把 480 nm 写成 4.80×10⁻⁷ m，再用常数相除，得表面温度约 6.0×10³ K，比太阳的 5800 K 高一些。天狼星 A 的峰值落在紫外一侧，表面温度接近一万开尔文，所以它看上去偏白。常见的错误是把纳米直接当成米代入，或者把峰值波长误当成某条谱线的波长。
Wien's law turns a colour into a temperature, and the constant is supplied on the data sheet, so no memorising is needed. The peak shifts towards shorter wavelengths as a star gets hotter, which is why a blue-white star must be hotter than an orange one of the same apparent brightness. Examiners see the nanometre slip every year: a peak quoted in nanometres has to be rewritten in metres before the division, or the answer comes out a billion times too large.
斯特凡定律求恒星半径
斯特凡定律描述黑体表面单位面积辐射的功率，写作 P / A = σT⁴，其中 σ 是斯特凡常数 5.67×10⁻⁸ W m⁻² K⁻⁴。把恒星当成半径为 r 的球，表面积是 4πr²，于是光度 L = 4πr²σT⁴，这条式子把 L、r、T 三个量绑在一起。温度出现在四次方上，所以半径不变而温度翻倍时，光度变成原来的十六倍；温度不变而半径翻倍时，光度只变成四倍。已知 L 与 T 就能反解半径，考试里通常写成 r 等于 L 除以 4πσT⁴ 之后的平方根。很多学生以为谱线宽度能直接给出半径，其实谱线给的是温度与成分，半径要绕斯特凡定律这一圈。代入之前温度必须换成开尔文，摄氏温度要加 273。
Stefan's law is the only route from luminosity and temperature to a physical size, and it is worth writing out in full before substituting. Because temperature enters to the fourth power, a modest error in T matters far more than an error in the measured flux. The radius that comes out is usually quoted as a multiple of the solar radius, which is a quick way to see whether the answer is plausible.
温度与半径的联用算例
拿一个完整例子把三步走通。某星的峰值波长测得 580 nm，按维恩定律得 T = 2.90×10⁻³ 除以 5.80×10⁻⁷，等于 5000 K，这一步只用光谱。该星距离 12 pc，先把秒差距换成米，12 乘以 3.09×10¹⁶ 得 3.71×10¹⁷ m。再把 F = 2.5×10⁻⁹ W m⁻² 代进 F = L / (4πd²)，得 L 约 4.3×10²⁷ W，大约是太阳的十一倍。最后把 L 与 T 代进斯特凡定律反解半径，得 r 约 3.1×10⁹ m，约为太阳半径的四倍多。整条链的顺序是光谱定温度、距离与流量定光度、光度与温度定半径，顺序反了就会卡住。阅卷时通常按公式、代入、答案三段给分，中间跳步丢的是自己的分。
The worked chain runs from a peak wavelength to a temperature, from distance and flux to a luminosity, and only then from luminosity and temperature to a radius. Each step uses a different observation, which is why the order matters more than the algebra. In practice, examiners award the formula mark, the substitution mark and the final answer mark separately, so a candidate who writes the rearranged equation first usually salvages most of the credit even after an arithmetic slip.
视差与秒差距的量程
恒星视差 p 是地球绕太阳运动造成的视角摆动，取相隔半年的两次观测，夹角的一半就是 p。距离越远摆动越小，二者成反比，定义式写作 p = 1 / d，其中 p 用角秒，d 用秒差距。秒差距的定义是一个天文单位张成一角秒时的距离，换算成 3.09×10¹⁶ m，也就是 3.26 光年。很多学生以为视差可以一直量到很远，其实地面望远镜的分辨极限大约在 0.01 角秒，对应百秒差距量级。欧空局（ESA）官网的资料显示，1989 年发射的 Hipparcos 把精度推到毫角秒级，2013 年发射的 Gaia 进一步测到十亿量级的恒星。常见的错误是把半年间隔的夹角直接当成 p，这样算出的距离只有真值的一半。
Parallax is pure geometry: the apparent swing of a nearby star against distant background objects, halved to give the angle p. The definition of the parsec follows from that geometry, since a star at one parsec subtends one arcsecond for a baseline of one astronomical unit. Anyone quoting a parallax distance should say which angle was measured, because using the full six-month swing instead of half of it halves the distance.
光谱线、能级差与光栅测波长
孤立气体原子的电子只能待在离散能级上，能级取负值，零对应电子刚好脱离原子。电子从高能级跳到低能级时放出一个光子，能量满足 hf = ΔE，换成波长写就是 hc / λ = ΔE。炽热稀薄气体发出明线光谱，连续谱穿过较冷的气体则留下暗线，同一种元素的明线与暗线位置完全对应。不同元素的能级结构不同，谱线组合就像指纹，这是证认星体成分最直接的手段。测波长用透射光栅，条件是 d sinθ = nλ，d 是相邻缝的间距，n 是级次。很多学生以为光栅公式里的 θ 可以随便取，实际上要取同一级明纹相对中央亮纹的偏角。
Stellar spectra do two jobs at once: they identify which elements are present and they record how far the lines have moved. Emission lines come from hot, low-density gas, while absorption lines appear when a continuous spectrum passes through cooler gas in front of the source. A transmission grating converts wavelength into angle through d sinθ = nλ, so measuring a diffraction angle with a known grating spacing is enough to recover the wavelength to three significant figures.
赫罗图上的四个区域
赫罗图把光度画在纵轴、温度画在横轴，温度习惯从左向右降低，所以最热的星在左边。主序是一条从左上到右下的斜带，核心在进行氢核聚变的星都在这里，太阳位于中段。右上角是红巨星与超巨星，温度低但光度高，按斯特凡定律只能是半径极大。左下角是白矮星，温度高而光度低，半径只有地球量级。读图时先找位置，再由位置判断半径，不要反过来推。常见的错误是把横轴当成线性刻度，其实横轴常用对数或光谱型 OBAFGKM 排列，等距并不代表等温差。
The Hertzsprung-Russell diagram is a map of luminosity against temperature, and every region on it corresponds to a different stage of stellar life. Position on the diagram fixes two quantities, so the third, radius, follows from Stefan's law without any extra observation. A star sitting above and to the right of the main sequence must be enormous, because it is cool yet far too bright for its temperature.
低质量恒星的结局与钱德拉塞卡极限
恒星从星际尘埃与气体的引力塌缩开始，核心温度升到足够高时氢开始聚变成氦。核聚变产生的辐射压与引力抗衡，星体进入稳定的主序阶段，停留时间取决于质量，质量越大燃烧越快。氢耗尽后核心收缩、外壳膨胀，恒星变成红巨星，太阳在约五十亿年后会走到这一步。低质量星的外层最终被抛出去形成行星状星云，露出的核心就是白矮星。白矮星靠电子简并压支撑，密度极高而体积只有地球大小，表面温度可达数万开尔文。支撑能力有上限，钱德拉塞卡极限约为 1.4 倍太阳质量，超过这个值简并压就顶不住。很多学生以为白矮星会继续塌缩成黑洞，其实质量不够的星根本走不到那一步。
A star's life is a contest between gravity pulling inwards and pressure pushing outwards, and every stage is one arrangement of that contest. Once hydrogen in the core is exhausted, the core contracts while the outer envelope expands, and the star leaves the main sequence for the giant region of the diagram. The white dwarf that remains is held up by electron degeneracy pressure, an effect with a hard ceiling, which is why the Chandrasekhar limit sits at about 1.4 solar masses.
大质量恒星、超新星与致密天体
质量较大的恒星能一路聚变到铁，铁核继续聚变不再放能，反而吸能。核心失去支撑后急剧塌缩，外层被反弹激波抛射出去，形成超新星爆发，短时间内的亮度可与整个星系相比。爆发后残留核心的质量决定结局，质量较小的部分压成中子星，半径只有十几公里，密度达到核物质量级，快速自转并发出射电脉冲的就是脉冲星。质量更大的核心连中子简并压也挡不住，塌缩成黑洞，引力强到逃逸速度超过光速。两者在观测上的共同点是极小的尺度配极大的质量，这也是考试里要求写清的特征。
Massive stars run through successive fusion stages until an iron core forms, and iron is where the energy release stops. Collapse then happens in under a second, the outer layers are blown off in a supernova, and what remains depends entirely on how much mass is left behind. A neutron star packs more than the Sun's mass into a city-sized sphere, while a black hole is the outcome when even neutron degeneracy pressure cannot hold the core up.
多普勒红移的符号与量纲
光源相对观察者运动时，谱线位置会移动，这是多普勒效应在光上的表现。近似式写作 Δλ / λ ≈ Δf / f ≈ v / c，只适用于速度远小于光速的情况。远离时波长变长，Δλ 为正，叫红移；接近时波长变短，叫蓝移。测量时先证认谱线属于哪种元素，把实验室波长当作基准，再量移动量，比值的正负就给出运动方向。很多学生以为红移一定是宇宙膨胀造成的，其实引力红移与自身运动也会造成移动，只有对遥远星系整体统计时才能谈膨胀。常见的错误是把 v 当成总速度，其实公式里的 v 只是沿视线方向的分量。
The Doppler formula for light is an approximation, valid only when the speed is small compared with c, and it compares a measured wavelength with a laboratory reference. Redshift means the source is receding, blueshift means it is approaching, and the sign is decided by the order of subtraction. Astronomers write the ratio as z, and for nearby galaxies z maps almost linearly onto recession speed, which is what makes Hubble's law usable at all.
哈勃定律与宇宙年龄估算
哈勃发现遥远星系的退行速度与距离近似成正比，写成 v ≈ H0 d，H0 叫哈勃常数。H0 有两种常用单位，写成 km s⁻¹ Mpc⁻¹ 时方便用兆秒差距算距离；换成 s⁻¹ 时方便与光速和年龄比较。取 H0 约 70 km s⁻¹ Mpc⁻¹，把 70 km/s 化成 7.0×10⁴ m/s，把 1 Mpc 化成 3.09×10²² m，得 H0 约 2.3×10⁻¹⁸ s⁻¹。取倒数得 1 / H0 约 4.4×10¹⁷ s，换算成约 1.4×10¹⁰ 年，也就是一百四十亿年量级。这个结果只是量级估计，前提是假设膨胀速率一直不变，真实数值需要按宇宙学模型修正。代入前要统一单位，c 用 3.0×10⁸ m/s 才能与米制距离配套。
Hubble's law links recession speed to distance through a single constant, and its two sets of units serve two different purposes. In kilometres per second per megaparsec it gives distances from measured redshifts; converted to reciprocal seconds it gives a timescale for the expansion. That timescale, one over H0, comes out at roughly fourteen billion years, and it is quoted as an estimate rather than a measurement because the expansion rate is assumed constant.
微波背景辐射与元素丰度
支持大爆炸的证据主要来自三条相互独立的观测线索。第一条是星系普遍红移，而且距离越远退行越快，这与均匀膨胀的图景一致，也说明地球不在宇宙的中心。第二条是宇宙微波背景辐射，温度约 2.7 K，谱形是极为干净的黑体谱，1965 年由 Penzias 与 Wilson 偶然测到，两人因此获得 1978 年诺贝尔物理学奖。第三条是轻元素丰度，宇宙中氢与氦的比例以及氘的含量，与原初核合成理论算出的值相当接近。三条证据指向同一个热而密的起点，这也是大爆炸模型至今仍是主流的原因。很多学生以为大爆炸是星系在已有的空间里四散飞开，其实膨胀的是空间本身，星系只是随空间一起被拉开。
Three independent observations carry the Big Bang model: the systematic redshift of galaxies, the microwave background, and the primordial abundances of the light elements. The background radiation is the strongest of the three because its spectrum matches a blackbody at about 2.7 K almost perfectly, which is hard to produce any other way. Element abundances add a second, entirely separate line of evidence, since the measured hydrogen-to-helium ratio agrees with what primordial nucleosynthesis predicts.
Paper 1 与 Paper 3 的答题顺序
模块 5 的内容在 Paper 1 里是重头戏，Paper 3 的综合题常把恒星光度与引力场放在同一道题里。拿到计算题先写公式，再单独一行写代入，最后才写答案与单位，这样即使算错也能保住方法分。遇到给出峰值波长和流量的题，按光谱定温度、距离定光度、光度与温度定半径的顺序走，不要在第一步就急着求半径。要求求恒星质量时，用 g = GM / r² 或圆轨道关系，把前面算出的半径与观测到的轨道数据结合起来。很多学生以为答案对了就不必写过程，阅卷时公式与单位才是给分点。写解释题要把因果关系写完整，例如温度升高导致峰值波长短移，而不是只写峰值波长短移。
Astrophysics questions on Paper 1 are marked for method as much as for the final number, so the rearranged equation should appear before any substitution. On Paper 3 the same content reappears inside synoptic contexts, often combined with gravitational fields or circular motion, which is why the radius calculated from Stefan's law is frequently only an intermediate step. A candidate who states the formula, substitutes with units shown, and comments on whether the answer is physically sensible will collect most of the available credit even when the arithmetic goes wrong.
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