题型概述：引力与天体物理Question types: gravitation and astrophysics
Round 2 的这条专题常以长题逐步建立模型：从牛顿力学出发，推到轨道设计、材料强度或观测反演。本页选取三道完整题；顶部统计数值核对完成度，不代表官方得分。常见失分在参考系、半径定义、守恒量和近似条件。先手写证明，再用数值检查核对；页面自动分数不代表原卷证明题得分。Round 2 develops models through extended questions, from Newtonian mechanics to orbital design, structural strength or inference from observations. This module contains three complete questions: the header counts numerical checks, not official marks. Typical losses come from reference frames, radius definitions, conserved quantities and approximation limits. Write the proofs first and then use numerical checks; automatic scores are not proof marks.
1圆周运动与旋转参考系Circular motion and rotating frames
自由圆轨道由引力提供向心力；与地球固定角速度共转时，额外支撑力通常不为零。Gravity alone supplies a free circular orbit’s centripetal force; corotation at Earth’s fixed angular speed generally requires another force.
q 正值表示有效向内，只有同步半径处为零。Positive q points effectively inward; only the synchronous radius makes it zero.
2轨道能量与角动量Orbital energy and angular momentum
中心力无力矩，因此角动量守恒；势能以无穷远为零。椭圆轨道不能全程使用圆轨道的速度公式。A central force has no torque, conserving angular momentum; potential energy is zero at infinity. The circular speed formula does not hold around an ellipse.
点火的 Δv 是新旧速度之差。A burn’s Δv is the difference between new and old velocities.
3从观测量反推质量Infer mass from observations
周期与距离给出总质量；频率变化率给出啁啾质量。反演前必须确认频率对应公转还是引力波。Period and separation constrain total mass; frequency evolution constrains chirp mass. Identify whether a frequency describes the orbit or the gravitational wave.
这是题给近似模型，读图误差会传播到质量估值。This is the stated approximate model; graph-reading uncertainty propagates into inferred masses.
比较重力与同速共转所需的向心加速度，均以 GM/rg² 归一化。Compare gravity and the centripetal acceleration required for corotation, both normalised by GM/rg².
| 子题型Subtype | 识别特征Recognition | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 旋转系与缆绳Rotating frame and cable | 同步、自重、变截面Synchronism, self-weight, taper | 微元平衡后积分Integrate element balance | 2023 Q4 |
| 轨道设计Orbit design | 近拱点、远拱点、点火Apsides and burns | 守恒量与活力公式Conservation and vis viva | 2021 Q3 |
| 引力波反演Wave inference | 啁啾、频率、黑洞Chirp, frequency, black holes | 量纲分析与能量损失Dimensions and energy loss | 2018 Q2 |