题型概述：引力与天体物理长题Question type: gravitation and astrophysics
Section 2 的这类长题会把熟悉的圆周运动与陌生的天体模型连在一起。本模块三题均为 25 分：先用基础力学拿到前半部分，再用题给公式处理碰撞、辐射或双星。常见失分点是把高度当半径、混用单位、忽略矢量和能量负号，以及在小差值中提前舍入。These Section 2 long questions connect familiar circular motion with unfamiliar astronomical models. Each selected question is worth 25 marks: begin with basic mechanics, then apply the supplied models to collisions, radiation or binaries. Common losses come from confusing altitude with radius, mixing units, ignoring vector directions or energy signs, and rounding before subtracting close quantities.
1圆轨道：一个受力方程，三个结果Circular orbits: one force equation, three results
引力提供向心力。半径 r 总从中心量起，轨道越高，速度越小，周期越长。Gravity supplies centripetal force. Measure r from the centre: larger circular orbits are slower and take longer.
下面只改变半径，质量固定为一个太阳质量。The small interactive below varies radius at a fixed solar mass.
2碰撞与束缚：分清两个时刻Collisions and binding: distinguish the stages
碰撞瞬间守恒矢量动量；完全非弹性碰撞损失动能。此后在恒星引力场中，判断总机械能是否为负。Conserve vector momentum at impact; a perfectly inelastic collision loses kinetic energy. Afterwards, binding in the stellar field is determined by negative total mechanical energy.
临界 ε=0 是抛物线，不是闭合椭圆。The ε=0 threshold is a parabola, not a closed ellipse.
3辐射题：把能量率、粒子率和通量分开Radiation: distinguish power, particle rate and flux
光度 L 是每秒发出的能量，光子率 N 是每秒的光子数。传播到半径 r 的球面后，每单位面积的量还要除以 4πr²。Luminosity L is emitted energy per second, while N counts photons per second. To obtain a quantity per unit area at radius r, divide by 4πr².
随机游走的总路程是 Nℓ，净位移只有 ℓ√N。A random walk has total path length Nℓ but net displacement only ℓ√N.
| 子题型Sub-type | 识别特征How to spot it | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 碰撞后轨道Post-impact orbit | 太阳消失、两行星粘合Sun disappears; planets merge | 切线几何 → 动量 → 能量Tangent geometry → momentum → energy | 2024 Q5 |
| 恒星形成与内部物理Star formation and interiors | 题给新公式、辐射与扩散New formulae, radiation and diffusion | 单位分析 → 逐步代入 → 物理解释Units → substitution → interpretation | 2022 Q5 |
| 轨道衰减与双星Decay and binaries | 阻力、超新星失质量Drag and supernova mass loss | 圆轨道能量 → 新质心能量Circular energy → new barycentric energy | 2020 Q5 |