📚 A-Level WJEC Physics: Astrophysics Key Points | A-Level WJEC 物理：天体物理考点精讲
Astrophysics brings together the laws of mechanics, thermodynamics and electromagnetism to explain the behaviour of stars, galaxies and the universe as a whole. This revision guide covers the essential WJEC A-Level Physics astrophysics topics, from gravitational orbits to cosmology and exoplanets.
天体物理将力学、热力学和电磁学定律结合在一起，解释恒星、星系和整个宇宙的行为。这份复习指南涵盖了 WJEC A-Level 物理天体物理的核心主题，从引力轨道到宇宙学和系外行星。
1. Gravitational Fields and Orbits | 引力场与轨道运动
Newton’s law of universal gravitation states that any two point masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of their separation: F = G m₁ m₂ / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻².
牛顿万有引力定律指出，任何两个质点以与质量乘积成正比、与距离平方成反比的力相互吸引：F = G m₁ m₂ / r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²。
The gravitational field strength g at a distance r from a mass M is given by g = GM / r². In a radial field, g follows an inverse-square law and points towards the centre of the mass.
距质量 M 为 r 处的引力场强度 g 为 g = GM / r²。在径向场中，g 遵循平方反比定律并指向质量中心。
Kepler’s three laws describe orbital motion: (1) planets move in ellipses with the Sun at one focus; (2) a line joining a planet to the Sun sweeps out equal areas in equal times; (3) the square of the orbital period T is proportional to the cube of the semi-major axis r: T² ∝ r³. For circular orbits, combining Newton’s law and centripetal force yields v = √(GM / r) and T² = (4π² / GM) r³.
开普勒三定律描述了轨道运动：（1）行星沿椭圆轨道运行，太阳位于一个焦点；（2）行星与太阳的连线在相等时间内扫过相等的面积；（3）轨道周期 T 的平方与半长轴 r 的立方成正比：T² ∝ r³。对于圆轨道，结合牛顿定律和向心力可得 v = √(GM / r) 和 T² = (4π² / GM) r³。
2. Gravitational Potential Energy and Escape Velocity | 引力势能与逃逸速度
The gravitational potential V at a point is the work done per unit mass to bring a test mass from infinity to that point: V = -GM / r. Gravitational potential energy of two masses separated by distance r is U = -G M m / r. The negative sign indicates that work must be done against the field to separate the masses.
引力势 V 是将单位质量从无穷远移至该点所做的功：V = -GM / r。相距 r 的两个质量的引力势能为 U = -G M m / r。负号表示分离质量必须克服引力场做功。
Escape velocity is the minimum speed needed for an object to escape a planet’s gravitational field without further propulsion. By equating kinetic energy and gravitational potential energy, we get v_esc = √(2GM / R), where R is the planet’s radius. For Earth, v_esc ≈ 11.2 km s⁻¹.
逃逸速度是物体无需额外推进即可离开行星引力场的最小速率。通过动能与引力势能相等，得到 v_esc = √(2GM / R)，其中 R 为行星半径。地球的逃逸速度约为 11.2 km s⁻¹。
3. Black-Body Radiation and Stellar Luminosity | 黑体辐射与恒星光度
Stars behave approximately as black bodies. Wien’s displacement law relates the peak wavelength λ_max of emitted radiation to the surface temperature T: λ_max T = 2.9 × 10⁻³ m K. Hotter stars peak at shorter (bluer) wavelengths.
恒星近似视为黑体。维恩位移定律将辐射峰值波长 λ_max 与表面温度 T 联系起来：λ_max T = 2.9 × 10⁻³ m K。温度越高的恒星峰值波长越短（偏蓝）。
The Stefan–Boltzmann law gives the total power radiated per unit area: P/A = σ T⁴, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. For a star of radius R, its luminosity L is L = 4π R² σ T⁴. Luminosity depends strongly on both radius and temperature.
斯特藩–玻尔兹曼定律给出单位面积辐射的总功率：P/A = σ T⁴，σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴。对于半径为 R 的恒星，其光度 L 为 L = 4π R² σ T⁴。光度强烈依赖于半径和温度。
The absolute magnitude M is a logarithmic measure of luminosity; the apparent magnitude m depends on distance. The distance modulus links them: m − M = 5 log₁₀(d / 10), with d in parsecs.
绝对星等 M 是光度的对数量度；视星等 m 取决于距离。距离模数将它们联系起来：m − M = 5 log₁₀(d / 10)，其中 d 以秒差距为单位。
4. Stellar Classification and the Hertzsprung-Russell Diagram | 恒星分类与赫罗图
Stars are classified by spectral type (O, B, A, F, G, K, M) based on absorption line patterns, which reflect surface temperature. A simple summary:
恒星根据吸收线模式按光谱型（O、B、A、F、G、K、M）分类，吸收线反映了表面温度。简表如下：
| Spectral Class | Temperature / K | Colour |
|---|---|---|
| O | 30 000 – 50 000 | Blue |
| B | 10 000 – 30 000 | Blue-white |
| A | 7 500 – 10 000 | White |
| F | 6 000 – 7 500 | Yellow-white |
| G | 5 000 – 6 000 | Yellow |
| K | 3 500 – 5 000 | Orange |
| M | < 3 500 | Red |
The Hertzsprung-Russell (H-R) diagram plots luminosity (or absolute magnitude) against temperature (or spectral class). Most stars lie on the main sequence, where hydrogen fusion occurs in the core. Giants, supergiants and white dwarfs occupy distinct regions.
赫罗图（H-R 图）以光度（或绝对星等）对温度（或光谱型）作图。大多数恒星位于主序带，在那里核心发生氢聚变。巨星、超巨星和白矮星占据不同的区域。
The mass of a star determines its main-sequence lifetime; more massive stars are much more luminous and consume fuel rapidly, leading to shorter lives.
恒星的质量决定了其主序寿命；质量更大的恒星光度大得多，燃料消耗迅速，因此寿命更短。
5. Stellar Evolution: Low-Mass Stars | 恒星演化：小质量恒星
A star like the Sun forms from a collapsing cloud of gas and dust. Once core temperature reaches ~10⁷ K, proton–proton chain fusion begins: 4 ¹H → ⁴He + 2e⁺ + 2νₑ + energy. The star enters the main sequence, where it remains for about 10¹⁰ years.
类似太阳的恒星由坍缩的气体尘埃云形成。当核心温度达到约 10⁷ K 时，质子-质子链反应开始：4 ¹H → ⁴He + 2e⁺ + 2νₑ + 能量。恒星进入主序阶段，持续时间约 10¹⁰ 年。
When hydrogen in the core is exhausted, the core contracts and heats up, igniting hydrogen shell burning. The star expands into a red giant. Helium fusion can later occur via the triple-alpha process when core temperature reaches ~10⁸ K: 3 ⁴He → ¹²C.
当核心氢耗尽，核心收缩并升温，点燃氢壳层燃烧。恒星膨胀成红巨星。之后当核心温度达到约 10⁸ K 时，可通过三氦过程发生氦聚变：3 ⁴He → ¹²C。
Eventually, the outer layers are ejected as a planetary nebula, leaving behind a hot carbon–oxygen core that cools into a white dwarf, supported by electron degeneracy pressure. No further fusion occurs.
最终，外层被抛射形成行星状星云，留下一个热的碳氧核心，冷却成为白矮星，靠电子简并压支撑。不再发生聚变。
6. Evolution of Massive Stars | 大质量恒星的演化
Massive stars (M > 8 M_sun) follow a more rapid and violent evolution. After the main sequence, they undergo successive stages of fusion, producing heavier elements in shells: carbon, neon, oxygen and silicon burning. This builds up an iron core.
大质量恒星（M > 8 M_sun）经历更快速、更剧烈的演化。主序之后，它们依次经历碳、氖、氧和硅燃烧的壳层聚变阶段，生成更重的元素，最终形成铁核。
Iron fusion is endothermic; it cannot release energy to support the core. The core collapses under gravity, triggering a supernova explosion. The outer layers are blasted into space, enriching the interstellar medium with heavy elements. The remnant core forms either a neutron star (supported by neutron degeneracy pressure) or, if mass exceeds the Tolman–Oppenheimer–Volkoff limit, a black hole.
铁核聚变是吸热过程，无法释放能量支撑核心。核心在引力作用下坍缩，引发超新星爆发。外层被抛入太空，使星际介质富含重元素。残余核心形成中子星（由中子简并压支撑），若质量超过奥本海默极限则形成黑洞。
7. Stellar Nucleosynthesis | 恒星核合成
Stars are the factories of the elements. Hydrogen and helium were produced in the Big Bang, but elements up to iron are synthesised through fusion in stellar interiors. Elements heavier than iron are primarily created by neutron capture processes (s-process in AGB stars, r-process in supernovae).
恒星是元素的工厂。氢和氦产生于大爆炸，但从锂到铁的元素由恒星内部的聚变合成。比铁重的元素主要通过中子俘获过程产生（AGB 星中的慢过程、超新星中的快过程）。
The fusion stages release energy until iron; each stage requires progressively higher temperatures. The binding energy per nucleon peaks at iron-56, explaining why fusion beyond iron consumes energy.
聚变阶段一直释放能量直至铁；每一阶段需要越来越高的温度。铁-56 的每个核子结合能最大，这解释了为什么超过铁的聚变需要吸收能量。
8. Doppler Effect and Redshift | 多普勒效应与红移
When a light source moves relative to an observer, the observed wavelength shifts. For a source moving away with speed v, the redshift z is z = Δλ / λ₀ ≈ v / c (for v ≪ c). This is used to measure the radial velocity of stars and galaxies.
当光源相对于观察者运动时，观测到的波长会发生偏移。对于以速度 v 远离的光源，红移 z 为 z = Δλ / λ₀ ≈ v / c（在 v ≪ c 时）。这被用来测量恒星和星系的径向速度。
In binary star systems, periodic Doppler shifts reveal orbital motion. For receding galaxies, the redshift indicates cosmic expansion.
在双星系统中，周期性的多普勒频移揭示了轨道运动。对于远方星系，红移表明了宇宙膨胀。
9. Hubble’s Law and the Expanding Universe | 哈勃定律与膨胀的宇宙
Edwin Hubble discovered that the recession speed v of a galaxy is proportional to its distance d: v = H₀ d, where H₀ is the Hubble constant (≈ 70 km s⁻¹ Mpc⁻¹). This linear relationship is evidence of an expanding universe.
哈勃发现星系的退行速度 v 与其距离 d 成正比：v = H₀ d，其中 H₀ 为哈勃常数（≈ 70 km s⁻¹ Mpc⁻¹）。这种线性关系是宇宙膨胀的证据。
The age of the universe can be estimated as t ≈ 1 / H₀, giving about 13.8 billion years. Hubble’s law implies that galaxies further away are moving faster, consistent with the expansion of space itself.
宇宙年龄可估计为 t ≈ 1 / H₀，约 138 亿年。哈勃定律表明越远的星系退行越快，这与空间本身的膨胀一致。
10. Evidence for the Big Bang: Cosmic Microwave Background | 大爆炸的证据：宇宙微波背景
The cosmic microwave background radiation (CMBR) is a near-uniform glow of microwave radiation filling the universe, corresponding to a black-body temperature of 2.73 K. It is the cooled remnant of the hot, dense early universe.
宇宙微波背景辐射（CMBR）是几乎均匀的微波辐射，对应黑体温度 2.73 K。它是早期炽热致密宇宙冷却后的残余。
The CMBR’s black-body spectrum and tiny temperature fluctuations (anisotropies) match predictions of the Big Bang model. Together with the redshift of galaxies, it provides strong support for the theory.
CMBR 的黑体谱和微小的温度起伏（各向异性）与大爆炸模型的预测相符。它与星系红移一起，为这一理论提供了有力支持。
11. Dark Matter and Dark Energy | 暗物质与暗能量
Rotation curves of spiral galaxies and gravitational lensing show that visible mass is insufficient to account for observed gravitational effects. This suggests the existence of dark matter, non-luminous mass that interacts via gravity but not electromagnetism.
旋涡星系的旋转曲线和引力透镜现象表明，可见质量不足以解释观测到的引力效应。这暗示存在暗物质——一种通过引力但不通过电磁力相互作用的非发光物质。
Observations of distant Type Ia supernovae reveal that the universe’s expansion is accelerating. This is attributed to dark energy, a mysterious form of energy that permeates space and acts as a repulsive force. In the standard ΛCDM model, dark energy makes up about 68% of the total energy content of the universe.
对遥远 Ia 型超新星的观测显示宇宙膨胀正在加速。这归因于暗能量——一种充满空间并表现为排斥力的神秘能量形式。在标准 ΛCDM 模型中，暗能量约占宇宙总能量密度的 68%。
12. Exoplanet Detection Methods | 系外行星探测方法
The radial velocity (Doppler) method detects tiny periodic shifts in a star’s spectrum caused by the gravitational tug of an orbiting planet. The amplitude of the shift gives the planet’s minimum mass.
径向速度（多普勒）方法检测由行星引力拖曳引起的恒星光谱的微小周期性频移。频移幅度给出行星的最小质量。
The transit method measures the slight dimming of a star’s light when a planet passes in front of it. The depth of the transit reveals the planet’s radius, and the period gives the orbital distance. Combining transit and radial velocity data yields the planet’s density and possible composition.
凌星法测量行星从恒星前方经过时星光微弱的变暗。凌星深度揭示行星半径，周期给出轨道距离。结合凌星和径向速度数据可得到行星密度和可能的成分。
Direct imaging is challenging but possible for large, young planets far from their star. The main difficulties in exoplanet detection are the vast distances, the huge contrast between star and planet brightness, and the small angular separation.
直接成像具有挑战性，但对于离母星较远的大质量年轻行星是可能的。系外行星探测的主要困难在于遥远的距离、恒星与行星亮度的巨大反差以及微小的角间距。
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