Know that it takes approximately one month for the Moon to orbit the Earth and use this to explain the periodic nature of the Moon's cycle of phases

6.1.1 The Earth–Moon System

1. Earth’s motions – the reference frame for the Moon

• Rotation (day‑night cycle) – Earth rotates eastward once every ≈ 24 h (23 h 56 min sidereal). This gives the daily rise and set of the Sun and Moon.
• Axial tilt – The rotation axis is inclined ≈ 23.5° to the ecliptic plane. Combined with the yearly orbit it produces the seasons (AO1 1.2).
• Orbit around the Sun (year) – One complete revolution takes ≈ 365.25 days. The Earth travels roughly 30 km s⁻¹ at an average distance of 1 AU ≈ 1.5 × 10⁸ km.

2. Moon’s orbital periods

Why the two periods differ (AO1 6.1.1):

• During one sidereal orbit the Earth moves ≈ 30° (≈ 1° day⁻¹) around the Sun.
• To line up again with the Sun‑Earth direction, the Moon must travel an extra ≈ 30° in its orbit.
• Extra angular distance ≈ 30° ÷ (12.2° day⁻¹) ≈ 2.5 days, giving the longer synodic month.

3. Phase angle θ and the periodic nature of the lunar phases

The phase angle θ is the Sun–Earth–Moon angle measured at the centre of the Earth (Figure 1). As the Moon orbits, θ increases almost uniformly.

Because ω is essentially constant, after every synodic month the phase angle returns to the same value and the same lunar phase repeats – the cycle is periodic (AO2 6.1.1).

4. Sequence of lunar phases

5. Linking the orbital period to the phase cycle

1. Average orbital speed of the Moon:
   

    \(v = \dfrac{2\pi r}{T{\rm sid}}\) with \(r≈384 400\) km and \(T{\rm sid}=27.32\) days → \(v≈1.02\) km s⁻¹.

2. Angular displacement per day: \(360°/29.53 ≈ 12.2°\) day⁻¹.
3. Earth also moves ≈ 1° day⁻¹ around the Sun, so the Sun‑Earth line rotates slowly. The net change in the Sun‑Moon angle each day is therefore ≈ 12.2° (the Moon’s motion dominates).
4. After one synodic month (≈ 29.5 days) the Moon has again the same angle θ relative to the Sun, so the same phase reappears – the phase cycle is periodic.

6. Why the Moon rises ≈ 50 minutes later each night

• Earth rotates 360° in 24 h → 15° h⁻¹.
• During the same 24 h the Moon moves eastward ≈ 12.2° in its orbit.
• To reach the same altitude above the horizon the Moon must “catch up’’ an extra 12.2°.
  

   Extra time = \(12.2° ÷ 15° h^{-1} ≈ 0.81\) h ≈ 48 min (rounded to ≈ 50 min).

7. Common misconceptions (AO1 6.1.1)

• Moonlight comes from Earth – The illuminated side of the Moon is lit directly by sunlight. “Earth‑shine” is a faint glow on a thin crescent when sunlight reflected from Earth reaches the Moon’s dark side.
• The Moon rises at the same time each night – Because the Moon advances ≈ 12.2° eastward each day, it appears about 50 minutes later each successive night.
• Phases repeat every 27 days – The sidereal month (27.3 days) is the time to return to the same position against the stars. Identical phases repeat after the synodic month (≈ 29.5 days).
• The Moon is a source of its own light – The Moon has no intrinsic luminosity; it only reflects sunlight.

8. Suggested diagram (for classroom use)

A side‑view sketch showing the Sun, Earth and Moon at the eight principal phases. Include:

• Arrows indicating the direction of the Moon’s orbital motion (eastward).
• The Sun‑Earth line and the phase angle θ labelled for each phase.
• Relative positions of Earth’s shadow (for completeness, not required for phases).

9. Quick‑check questions (AO2 6.1.1)

1. State the lengths of the sidereal and synodic months and explain why they differ.
2. Calculate the angular speed ω of the Moon in degrees per day and use it to find the phase angle after 7 days from a New Moon.
3. Explain why the Moon rises later each night and show the calculation that gives ≈ 50 minutes.
4. If the Moon were stationary relative to Earth, what would an observer on Earth see each night?
5. Describe Earth‑shine and indicate when it can be observed.

10. Key constants and formulas (AO1 6.1.1)