Astronomy

Moon, tides, and eclipses

Lunar phases, tides, and solar and lunar eclipses.

Basics

Lunar phases

We see sunlight on half the Moon; the Earth–Moon–Sun angle sets new, first quarter, full, and last quarter. The synodic month (new to new) is about 29.5 days, longer than the sidereal month (~27.3 days) because Earth has moved around the Sun while the Moon orbits. The same face stays toward us because spin and orbit are tidally locked.

Tides

The Moon (and more weakly the Sun) pulls Earth’s near side harder than the far side. That difference — the tidal force — raises two ocean bulges; Earth rotating under them often gives two high tides a day. At new and full Moon the Sun lines up (spring tides); at quarters they compete (neap tides). Coastline and depth change real times and heights a lot.

Eclipses

A solar eclipse is the Moon between Sun and Earth; a lunar eclipse is Earth’s shadow on the Moon. Umbra is total, penumbra partial. They do not happen every new or full Moon because the Moon’s orbit tilts about 5° to the ecliptic — only near the nodes do the shadows meet. The total-solar path is a narrow track; a lunar eclipse is visible wherever the Moon is up.

Periods and distance

The Moon’s mean distance is about 3.8×10⁵ km, on an ellipse with perigee and apogee. That its angular size nearly matches the Sun is why we get both total and annular solar eclipses. The saros (~18 years 11 days) is a cycle on which similar eclipse geometry repeats. The Moon recedes a few centimetres a year, and Earth’s day lengthens very slowly.

Formulas

Synodic month

1/T_syn = 1/T_sid − 1/T_year

Sidereal month and Earth’s year together set the phase period.

Symbols

  • T_sid sidereal month
  • T_year sidereal year

Tidal acceleration (sketch)

a_tide ~ 2 G M d / r³

Body mass M, distance r, Earth-size d. Closer mass pulls much harder.

Symbols

  • d Earth-radius scale
  • r distance to Moon (or Sun)

Angular diameter

θ ≈ D / r (radian, D ≪ r)

Total solar eclipse needs similar θ for Moon and Sun.

Symbols

  • D body diameter
  • r distance to observer

Shadow cone (sketch)

L_umbra ≈ R / (R_⊙/r_⊙ − R/r)

How far an occulting body’s umbra reaches. If the Moon’s umbra hits the ground, totality.

Symbols

  • R occulting radius
  • R_⊙ solar radius

Key table

Synodic month about 29.53 days
Sidereal month about 27.32 days
Mean Earth–Moon about 3.84×10⁸ m

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