Astronomy

Start here: geocentric views versus heliocentric reality, the scale of the solar system, and gravity as the organizing force.

Moon

Here

Why the Moon cycles through phases, why a month of phases is longer than one orbit, why we only ever see one face, and why eclipses are rare.

Solar System

Finish

Kepler's three laws, the scale of planetary orbits, why inner planets move faster, and retrograde motion as a frame effect.

Moon

The Moon does not change shape, and it makes no light of its own. Phases and eclipses are entirely a matter of geometry in the Sun-Earth-Moon system.

At any instant the Sun illuminates exactly one hemisphere of the Moon. The phase we observe is simply how much of that lit half can be seen from Earth.

Top-down view — drag the Moon

View from

First Quarter

50% illuminated · waxing

The Moon is a quarter of the way around its orbit, so we see the lit half edge-on as a half disk. Seen from the North Pole, the right half is the bright one.

We say the Moon is waxing when the lit fraction we see is growing from new toward full, and waning when it is shrinking from full back toward new.

Eclipses

If phases come purely from alignment, why isn’t there a solar eclipse (the Moon casts a shadow on Earth) at every new moon and a lunar eclipse (Earth casts a shadow on the Moon) at every full moon? Because the Moon’s orbit is tilted about $5^\circ$ relative to the plane of Earth’s orbit (the ecliptic). Most months the Moon passes a little above or below the Sun–Earth line, and its shadow misses. Eclipses happen only when a new or full moon lands near a node — one of the two points where the tilted orbit crosses the ecliptic plane. That double requirement, right phase and near a node, is why eclipses come in seasons rather than monthly.

Lunar month

The Moon completes one orbit relative to the distant stars in the sidereal month, about $27.3$ days. But one complete cycle of the phases is the synodic month, about $29.5$ days. This cycle takes longer because Earth is also moving: while the Moon circles us, Earth advances along its own orbit, so the Sun’s direction shifts and the Moon must travel a little extra each cycle to catch back up to the Sun–Earth line:

\frac{1}{T_{\text{syn}}} = \frac{1}{T_{\text{sid}}} - \frac{1}{T_{\text{Earth}}}.

This is the same “catching up” idea that explains the retrograde motion of planets like Mars.

Problem Solving

From Sidereal to Synodic Month

Problem

The Moon’s sidereal period is $T_{\text{sid}} = 27.3\,\text{days}$ and Earth’s orbital period is $T_{\text{Earth}} = 365.25\,\text{days}$ . Find the synodic month — the time from one new moon to the next.

Use

\frac{1}{T_{\text{syn}}} = \frac{1}{T_{\text{sid}}} - \frac{1}{T_{\text{Earth}}}.

Solve

\frac{1}{T_{\text{syn}}} = \frac{1}{27.3} - \frac{1}{365.25} \approx 0.03663 - 0.00274 = 0.03389\,\text{day}^{-1}.

T_{\text{syn}} \approx \frac{1}{0.03389} \approx 29.5\,\text{days}.

Check

The synodic month is about $2.2$ days longer than the sidereal month — the extra time the Moon needs to catch up with the moving Sun–Earth line, and it matches the familiar length of the cycle of phases.

Moon Phases Checkpoint

Question 1 of 3

What determines the phase of the Moon we see from Earth?

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