离太阳更近，才是夏天吗？Is summer when Earth is closer to the Sun?
地球绕太阳的轨道略呈椭圆，所以“近了就热”听起来很合理。左图把一年拆成四个视角：从斜上方看的轨道、侧面看的地球、照到地面的一束阳光，以及一整年每天收到的日照。拖动日期和纬度，看看到底是哪一样在变。
Earth’s orbit is slightly oval, so “closer means warmer” sounds reasonable. The figure splits a year into four views: the orbit from above at an angle, Earth from the side, one beam of sunlight hitting the ground, and the sunlight each day of the year receives. Drag the date and the latitude and find out what actually changes.
01预测Predict
02走过一年Travel through a year
先在北京看七月和一月，再换到悉尼看同一天。留意左下的年度曲线：实线是你选的纬度，虚线是南北对称的另一个半球。
Look at Beijing in July and in January, then switch to Sydney on the same days. Watch the year chart at the bottom left: the solid line is your latitude, the dashed line the mirror latitude in the other hemisphere.
- 北半球的夏天（6–8 月，北纬 10° 以北）A northern summer day (Jun–Aug, north of 10°N)
- 北半球的冬天（12–2 月，北纬 10° 以北）A northern winter day (Dec–Feb, north of 10°N)
- 南半球的同一个月份（南纬 10° 以南）The same months in the south (south of 10°S)
03换一颗行星Change the planet
把倾角调到 0°，年度曲线几乎变平。再把轨道改成“夸张椭圆”：季节回来了，但比较一下两条曲线，南北半球还是相反的吗？
Set the tilt to 0° and the year chart goes nearly flat. Then switch to the exaggerated oval: seasons come back. Compare the two curves. Are the hemispheres still opposite?
04解释Explain
按顺序看图上的五处，每一步只亮起一个部分。Look at five places in order; each step lights up one part.
- 先看地面上的光束：太阳越低，同一束光摊得越开，每平方米分到的越少。Start with the beam on the ground: the lower the Sun, the wider the same light spreads, so each square metre gets less.
- 再看白天长度：夏天的白天更长，收集阳光的时间也更长。Then the daylight bars: summer days are longer, so they collect light for longer.
- 看轨道：地球一月最近、七月最远，最近的时候恰好是北京的冬天。Now the orbit: Earth is closest in January and farthest in July, and closest falls in Beijing’s winter.
- 看年度曲线：实线和虚线正好相反，两个半球的季节互相对调。Look at the year chart: the solid and dashed curves are mirror images, so the hemispheres swap seasons.
- 转动三维视图看地球：地轴一年到头指向同一方向；六月北半球朝太阳倾，所在纬度圈亮的部分超过一半，白天就超过 12 小时。Turn the 3D view and watch the globe: the axis points the same way all year; in June the north leans sunward, so more than half of your latitude circle is lit and the day lasts over 12 hours.
- 最后看侧视图：倾斜的地轴让太阳直射点在南北回归线之间来回。Finally the side view: the tilted axis swings the overhead Sun between the two tropics.
1. 太阳低，同一束光就摊得更开
北京正午的太阳，6 月 21 日高 73.5°，12 月 21 日只有 26.7°。同一束阳光照到地面，面积要乘以 1/sin(高度角)：夏天是 1.04 倍，冬天是 2.23 倍。每平方米分到的光，冬天只有夏天的一半左右。
2. 白天更长，光照时间就更长
同样两天，北京的白天是 14.8 小时和 9.2 小时。角度和时长叠加起来，6 月 21 日每天收到的阳光（大气层顶）约是 12 月 21 日的 3.1 倍。
3. 用“距离”来解释，会被两个事实推翻
地球在 1 月 3 日前后离太阳最近（0.983 AU），7 月 4 日前后最远（1.017 AU）。阳光强度随 1/r² 变化，只有约 ±3.4%，而且最强的时候正是北京的冬天。更关键的是：同一天、同一个距离，悉尼却是相反的季节。如果是距离造成季节，两个半球应该同时变冷变热。
4. 倾角决定季节有多强
太阳直射点在南北纬“倾角”度之间来回，所以回归线在 ±23.4°，极圈在 90° − 23.4° = 66.6°。倾角为 0，就没有季节；倾角越大，季节越极端。
1. A low Sun spreads the same beam thinner
Beijing’s noon Sun stands 73.5° high on June 21 and only 26.7° on December 21. One beam of sunlight covers 1/sin(elevation) as much ground: 1.04× in summer, 2.23× in winter. Each square metre gets roughly half as much light in winter.
2. Longer days collect light for longer
On the same two days Beijing has 14.8 and 9.2 hours of daylight. Angle and duration together give about 3.1 times as much sunlight per day (top of the atmosphere) on June 21 as on December 21.
3. Two facts sink the distance explanation
Earth is closest to the Sun around January 3 (0.983 AU) and farthest around July 4 (1.017 AU). Sunlight scales as 1/r², so it changes by only about ±3.4%, and it peaks during Beijing’s winter. More decisively: on the same day, at the same distance, Sydney has the opposite season. If distance caused seasons, both hemispheres would warm and cool together.
4. The tilt sets how strong seasons are
The overhead Sun swings between “tilt” degrees north and south, which is why the tropics sit at ±23.4° and the polar circles at 90° − 23.4° = 66.6°. With no tilt there are no seasons; the bigger the tilt, the harsher they get.
cos h₀ = −tan φ · tan δ day = 24 h · h₀/π
Q = (S₀/π)(1/r²)(h₀ sin φ sin δ + cos φ cos δ sin h₀)
距离真的不重要吗？
对一年里的季节来说不重要，但在万年尺度上它很重要。地轴缓慢进动，让近日点在季节中移动，周期约 2.1 万年。大约 1 万年前，近日点落在北半球的夏天，北方夏季的日照比现在强得多。米兰科维奇认为，这类轨道变化调节了冰期的节奏。
Does distance never matter?
Not for the yearly cycle, but it does over millennia. Earth’s axis slowly precesses, moving perihelion through the seasons about every 21,000 years. Roughly 10,000 years ago perihelion fell in northern summer, and northern summers received noticeably more sunlight than today. Milankovitch proposed that such orbital changes pace the ice ages.
“七月北京热，是因为 ___。同一个七月悉尼冷，是因为 ___。”“Beijing is warm in July because ___. Sydney is cold in the same July because ___.”
迁移：换一颗行星Transfer question
05独立练习Independent practice
练习时，左图只保留形状：数字、年度曲线和昼夜条都会隐藏，控件也会锁住。用机制推出答案，核对之后再显示这个情形的数字。
During practice the figure keeps only its shapes: the numbers, the year chart and the daylight bars disappear, and the controls lock. Work each case out from the mechanism; after you check, the case’s numbers come back.
这个模型的适用范围Where this model applies
“每天的阳光”是大气层顶的日平均值，不含云、大气吸收、海洋和陆地的热容量。真实气温比日照晚约一个月达到峰值，所以最热的是七月下旬，而不是 6 月 21 日。
日出日落按太阳中心通过几何地平线计算，不含大气折射和日面半径，真实白天会长几分钟。日期按 365 天、近日点固定在 1 月 3 日计算，正午指太阳正午，不含时差。太阳的输出视为恒定（真实变化约 0.1%）。
轨道图为了看清，从斜上方投影，地球和太阳的大小都不按比例。
“Daily sunlight” is the daily mean at the top of the atmosphere; it leaves out clouds, absorption, and the heat capacity of oceans and land. Real temperatures peak about a month after sunlight does, which is why late July, not June 21, is hottest.
Sunrise and sunset use the Sun’s centre crossing a geometric horizon, without refraction or the Sun’s radius; real days run a few minutes longer. Dates use a 365-day year with perihelion fixed on January 3, and noon is solar noon, without the equation of time. The Sun’s output is held constant (it really varies by about 0.1%).
The orbit view is an oblique projection; Earth and Sun are not to scale.
离线可用。全部控件都是原生单选框、滑块、输入框和按钮，可以用 Tab、方向键和空格操作。没有快捷键会干扰你的写作。Works offline. Every control is a native radio, slider, input or button, reachable with Tab, arrow keys and Space. No keyboard shortcuts override your writing.
转动视角只改变你看的方向，不改变任何读数。地球上那圈纬度线：亮的部分占整圈的比例，就是这一天白天占 24 小时的比例。Turning the view changes only where you look from, never a readout. On the globe’s latitude circle, the lit share of the circle is the share of the 24 hours in daylight.