Are the Moon’s phases Earth’s shadow?

01预测Predict

在左边的轨道上，月球离太阳方向转过了 90°。从地球（北半球）看，它是什么样子，什么时候在天上最高？On the orbit at left, the Moon has moved 90° from the Sun’s direction. Seen from Earth (northern hemisphere), what does it look like, and when is it highest in the sky? 右半边亮；日落前后最高。The right half is lit; it is highest around sunset. 一弯细细的月牙：大部分被地球的影子挡住了。A thin crescent: Earth’s shadow covers most of it. 满月：地球的影子没挡着它，阳光照亮了整个月亮。Full: Earth’s shadow is not in the way, so sunlight lights the whole Moon. 一半亮；半夜最高，因为月亮属于夜晚。Half lit; highest at midnight, because the Moon belongs to the night.

02绕一个月Go around one month

拖动月球的位置，同时看两个视角。然后转动太空视角：拖动图形，或用图下方的滑块、方向键。从正上方看，再从太阳那边看，月球亮着的部分有没有变？

Move the Moon along its orbit and watch both views. Then turn the space view: drag the figure, or use the sliders below it or the arrow keys. Look from straight above, then from the Sun’s side. Does the lit part of the Moon itself ever change?

月球离太阳方向的角度Moon’s angle from the Sun

观察者在Observer in the

北半球North 南半球South

换个角度看太空View space from

一个月牙（亮的不到 40%）A crescent (less than 40% lit)
一个满月（亮的超过 97%）：它在地球影子里吗？A full Moon (over 97% lit): is it in Earth’s shadow?
从正上方看轨道（视角抬到 70° 以上）The orbit from above (tilt the view past 70°)

03解释Explain

按顺序看图上的五处，每一步只亮起相关的部分。Look at five places in order; each step lights up only what it needs.

先看太空里的月球：朝太阳的一半永远是亮的，无论它走到哪里。Start with the Moon in space: the half facing the Sun is always lit, wherever it is.
再看地球上的视角：我们只看到朝向地球的那一半，月相就是“亮的一半”里有多少转向了我们。Then the view from Earth: we only see the half facing us, and the phase is how much of the lit half is turned our way.
看轨道上的角度：离太阳方向 90°，亮的一半正好有一半朝向我们。Now the angle on the orbit: at 90° from the Sun, exactly half of the lit half faces us.
看地球的影子：它总是背对太阳。月球只有在满月附近才可能碰到它。Look at Earth’s shadow: it always points away from the Sun. The Moon can only reach it near full Moon.
最后看时间条：月球每离太阳 15°，升起就晚 1 小时。Finally the time bar: every 15° the Moon is from the Sun delays its rising by one hour.

1. 月球永远有一半是亮的

阳光照亮月球朝向太阳的那一半，就像地球的白天。从正上方或从太阳那一侧看，你会发现每个位置的月球都是“半亮”的。

2. 月相是“亮的一半”里，转向我们的那一部分

月球离太阳方向转过的角度叫 E。我们看到的亮面比例是 (1 − cos E) ÷ 2：0° 时是 0（新月），90° 时是 50%（上弦），180° 时是 100%（满月）。它不是线性的：135° 时已经亮了 85%，不是 75%。

3. 地球的影子指向另一边

地球的影子总是背对太阳。月牙出现时，月球几乎在太阳和地球之间，离影子很远。只有满月时月球才在影子那一侧，可那正是它看起来最圆的时候。月球真的进入影子时（月食），整个圆面会变暗、发红，而不是变成月牙。

4. 月相决定月出时间

月球在太阳东边 E 度，就比太阳晚 E ÷ 15 小时升起。上弦月中午升起、日落时最高；满月日落时升起；下弦月半夜升起，早上还挂在天上。月球每天在轨道上走约 12.2°，所以每天晚升起约 49 分钟。

1. Half the Moon is always lit

Sunlight lights the half of the Moon that faces the Sun, just like Earth’s day side. Seen from straight above, or from the Sun’s side, the Moon is half lit at every position.

2. The phase is how much of the lit half faces us

Call the Moon’s angle from the Sun’s direction E. The lit fraction we see is (1 − cos E) ÷ 2: zero at 0° (new), 50% at 90° (first quarter), 100% at 180° (full). It is not linear: at 135° the disk is already 85% lit, not 75%.

3. Earth’s shadow points the other way

Earth’s shadow always points away from the Sun. A crescent appears when the Moon is nearly between Earth and the Sun, far from the shadow. Only at full Moon is the Moon on the shadow’s side, and that is exactly when it looks fully round. When the Moon really does enter the shadow (a lunar eclipse), the whole disk dims and turns reddish; it does not become a crescent.

4. The phase sets the moonrise

A Moon E degrees east of the Sun rises E ÷ 15 hours after the Sun. A first-quarter Moon rises at noon and is highest at sunset; a full Moon rises at sunset; a third-quarter Moon rises at midnight and is still up in the morning. The Moon moves about 12.2° along its orbit each day, so it rises about 49 minutes later each day.

lit = (1 − cos E) / 2
moonrise ≈ 6:00 + E / 15° h
daily delay = 24 h / 29.53 ≈ 49 min

迁移：为什么不是每个月都有月食？Transfer: why isn’t there an eclipse every month?

每个满月，地球的影子都在月球那一侧。可是月食一年只有一两次。月球轨道相对地球轨道倾斜约 5.1°，倾斜的“交线”方向（交点）随时间相对太阳转动。调整下面两个控件，看新月和满月什么时候会发生日食、月食。

At every full Moon, Earth’s shadow is on the Moon’s side, yet lunar eclipses happen only once or twice a year. The Moon’s orbit is tilted about 5.1° to Earth’s, and the line where the two planes cross (the nodes) turns relative to the Sun over the year. Change the two controls below and see when the new and full Moons give eclipses.

月球轨道倾角Tilt of the Moon’s orbit

真实 5.1°Real 5.1° 不倾斜 0°Flat 0°

交点方向与太阳方向的夹角Angle from the Sun to the node line

用你自己的话解释。Explain it in your own words.

“月牙缺掉的部分是 ___。满月的时候不在地球影子里，是因为 ___。”“The missing part of a crescent is ___. A full Moon usually misses Earth’s shadow because ___.”

04独立练习Independent practice

练习时，地球视角的月亮、时间条和读数会隐藏，月球位置也会锁住；太空视角仍然可以转动。用机制推出答案，核对之后再显示。提示会一题比一题少。

During practice the Moon seen from Earth, the time bar and the readouts are hidden, and the Moon’s position locks; you can still turn the space view. Work each case out from the mechanism; after you check, the answer comes back. The hints thin out as you go.

这个模型的适用范围Where this model applies

太空视角不按比例：真实的地月距离约是地球直径的 30 倍，太阳还要远约 390 倍。为了看得见，图中把月球轨道的倾角放大了 4 倍，影子也画得更粗。是否发生日食、月食，由模型用真实角度判断：月球离影子中心不到 0.96°（本影半径加月球半径）算月食，离太阳方向不到 1.5°（含视差）算日食；不含半影月食。

月出时间按赤道、春分秋分、圆轨道计算，不含大气折射和月球轨道的离心率；真实时间会差几十分钟，高纬度差得更多。一个朔望月取 29.53 天。

The space view is not to scale: the real Earth–Moon distance is about 30 Earth diameters, and the Sun is about 390 times farther still. So it is visible, the drawing exaggerates the Moon’s orbital tilt 4× and draws the shadow wider. Whether an eclipse happens is decided by the model with true angles: a lunar eclipse when the Moon is within 0.96° of the shadow’s centre (umbra plus the Moon’s radius), a solar eclipse within 1.5° of the Sun’s direction (including parallax); penumbral eclipses are left out.

Moonrise times assume the equator, an equinox and a circular orbit, without refraction or the orbit’s eccentricity; real times differ by tens of minutes, more at high latitudes. A lunar month is taken as 29.53 days.

离线可用。全部控件都是原生单选框、滑块、输入框和按钮。太空视角获得焦点时，方向键旋转视角，Home 键复位；其他地方没有快捷键。Works offline. Every control is a native radio, slider, input or button. When the space view has focus, the arrow keys turn it and Home resets it; there are no other shortcuts.