Meaning and mechanism
The Moon’s half is usually illuminated by sunlight. When the Moon changes its position relative to the Earth’s circumference, the portion of the disk facing Earth changes in brightness.
At new moon, in roughly the Sun’s direction, the lit hemisphere mostly faces away from Earth. At full moon, opposite the Sun, the Earth-facing hemisphere is lit. First and last quarter occur at intermediate positions.
Relationships and conditions
With the simulation’s circular-orbit and parallel-light approximations, the illuminated fraction is (1−cosθ)/2. At θ=90°, half the disk is illuminated; at 180°, the whole disk is illuminated.
The tilt of the disk and the direction of the brighter side can vary depending on the observation position and viewing angle. The model is a frontal view for the Northern Hemisphere, and lunar eclipses require a separate alignment condition.
What to observe in the simulation
90° is first quarter with 50% illumination; 180° is full moon with 100%. The waxing Moon is lit on the right, and the waning Moon on the left in this view.
Assume a circular orbit, parallel solar power, and a positive disk in the northern hemisphere. Sizes and distances are to illustrative scale and phase names are divided into eight divisions spaced at 45° intervals. Disk rotation, tilt, and eclipse phenomena at the observation site are omitted. Phases are not created by the Earth's shadow.
Key relationships and symbols
f = (1−cos θ)/2
| Symbol | Meaning | Unit |
|---|---|---|
| θ | Orbital configuration angle measured from the simulation’s new-moon position | ° |
| f | Illuminated fraction of the visible disk | 0~1 |
Worked example
What is the direct reason why the phases of the moon change?
The moon is always bright in the hemisphere facing the sun. The direction from which the hemisphere is viewed from the Earth changes, creating a phase.
Common misconceptions
- The reason for a crescent moon appearing is not because the Earth’s shadow covers the moon.