题型概述：轨道与太阳系Question types: orbits and the Solar System
Astro Challenge 常把基础力学放进彗星、月球和行星的背景中。本组包含三道 Section A 选择题（各 1 分）和两道 Section B 简答题（各 5 分），共 13 分。最容易丢分的是混淆会合周期与公转周期、弄反距离比，以及把太阳日当作恒星日。先确定参考对象，再写比值或角速度差。Astro Challenge applies basic mechanics to comets, moons and planets. This set contains three Section A MCQs (1 mark each) and two Section B written questions (5 marks each), totalling 13 marks. Common losses come from confusing synodic and orbital periods, reversing distance ratios, and mistaking solar days for sidereal days. Identify the reference body before using ratios or relative angular speeds.
1同一中心天体：先用比值Same central body: use ratios
两条轨道若绕同一中心天体运行，开普勒第三定律中的引力参数相同。圆轨道再用周长除以周期，椭圆轨道则用半长轴。For orbits around the same central body, the gravitational parameter in Kepler’s third law is shared. Use circumference divided by period for a circular orbit and the semimajor axis for an ellipse.
太阳系常用简写：T 用年、a 用 au，则 T² ≈ a³。For solar orbits, T in years and a in au give T² ≈ a³.
2相对角速度决定再次对齐Relative angular speed sets alignment
会合周期和太阳日都是“相对转一圈”的问题。同向运动时角速度相减；先判断哪个更快，避免求出负周期。Synodic periods and solar days both involve one complete relative turn. Subtract angular speeds for prograde motion; identify the faster motion to avoid a negative period.
内轨道可能有短于一年的会合周期；外轨道的会合周期必大于一年。An inner orbit can have a synodic period shorter than a year; an outer orbit has one longer than a year.
3椭圆距离与观测角度Elliptical distances and apparent angles
近日点与远日点距离由偏心率决定，中心天体位于焦点。物体的视直径与距离成反比，观察角度则要明确角的顶点。Eccentricity sets periapsis and apoapsis distances, with the central body at a focus. Angular diameter varies inversely with distance; identify the vertex when interpreting an observed angle.
距离比可以消掉未知直径，轨道周期比可以消掉中心质量。Distance ratios eliminate an unknown diameter; period ratios eliminate the central mass.
| 子题型Sub-type | 识别特征Clue | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 半月几何Half-Moon geometry | half illuminated | 直角在月球Right angle at Moon | 2024 Q2 |
| 会合周期Synodic period | closest every 300 days | 角速度差 → 开普勒定律Relative rate → Kepler | 2022 Q9 |
| 彗星速率Comet speed | eccentricity, fastest | 定位远日点 → 半周期Locate aphelion → half-period | 2021 Q9 |
| 蓝月亮与视直径Blue moons and apparent size | 25%, 14.1% larger | 频率倒数；距离反比Reciprocal rate; inverse distance | 2023 Q12 |
| 自转公转共振Spin–orbit resonance | 3:2 resonance | 恒星频率 − 公转频率Spin rate − orbital rate | 2019 Q11 |