Moon Phases, Tides and Eclipses Explained

Phases: one half is always lit

The Moon makes no light of its own; the Sun always lights the half facing it. What changes over the month is how much of that lit half faces the Earth. At new moon the Moon is roughly between the Earth and the Sun and its lit side faces away; at first quarter it is 90° from the Sun and we see half of the lit half, a half-disk; at full moon it is opposite the Sun and the whole lit side faces us. The share of the disk lit is

k = (1 − cos E) ÷ 2

where E, the elongation, is the angle between the Moon and the Sun as seen from the Earth. Seven days after new moon E is about 85° and k = (1 − 0.087) ÷ 2 = 46%. The cycle from new moon to new moon, the synodic month, takes 29.53 days, two days longer than the Moon's 27.32-day orbit, because the Earth moves round the Sun meanwhile and the Moon has to catch up. The Earth's shadow plays no part in the phases.

Tides: a stretch, not a pull

Gravity weakens with distance, so the Moon pulls the ocean on the near side of the Earth a little harder than the Earth's center, and the center a little harder than the far ocean. Relative to the center, the near water is drawn toward the Moon and the far water is left behind: two bulges, on opposite sides. That difference in pull, the tidal force, falls with the cube of the distance. The Earth turns under the bulges, so most coasts get two high and two low tides a day, about 50 minutes later each day, because the Moon has moved on in its orbit.

The height of the bulges, if the ocean could take its ideal shape (the equilibrium tide), is

A = (M ÷ M_E) × (R_E ÷ d)³ × R_E

For the Moon, 0.0123 × (6,371 ÷ 384,400)³ × 6,371 km = 0.36 m. For the Sun, 27 million times heavier but 389 times farther away, 333,000 × (6,371 ÷ 149.6 million)³ × 6,371 km = 0.16 m, just under half the Moon's. The tide at a place is A times P₂(cos ψ) = (3 cos² ψ − 1) ÷ 2 for each body, where ψ is the angle from the place to the point directly under the Moon or Sun: highest at ψ = 0° and 180°, lowest at 90°.

Spring and neap tides

At new and full moon the Sun's bulges line up with the Moon's, and the tides add: spring tides, about 0.36 + 0.16 = 0.52 m in amplitude in the equilibrium theory. At the quarter moons the two sets of bulges are at right angles and partly cancel: neap tides, 0.36 − 0.16 = 0.20 m. So the range swings by a factor of about 2.6 every two weeks. "Spring" means springing up, not the season.

The Moon's distance also matters. Its orbit is an ellipse, from about 356,500 km at perigee to 406,700 km at apogee, and because the tidal force goes as 1 ÷ d³, the Moon's tide is about 49% stronger at perigee than at apogee. A spring tide at perigee is a perigean spring tide, or king tide, which together with a storm surge causes much coastal flooding.

Eclipses

The Moon's orbit is tilted 5.14° to the plane of the Earth's orbit. At most new moons it passes above or below the Sun as seen from the Earth, and at most full moons above or below the Earth's shadow. An eclipse needs a new or full moon close to one of the two nodes, where the orbits cross:

• Solar eclipse at new moon, if the Moon is within about 1.5° of the Sun. If the Moon's disk looks bigger than the Sun's (it does when the Moon is nearer than about 380,000 km) the eclipse can be total; if smaller, annular, with a ring of Sun left.
• Lunar eclipse at full moon, if the Moon is within about 1° of the center of the Earth's shadow. In a total lunar eclipse the Moon turns red, lit only by sunlight bent through the Earth's atmosphere, the light of all the world's sunsets.

The Sun lines up with the nodes every 173.3 days, an eclipse season about a month long, so there are at least four eclipses a year, solar and lunar. The node line itself turns once every 18.6 years, and the Sun returns to the same node every 346.6 days, the eclipse year.

Using the simulation

In the Moon Phases, Tides and Eclipses Simulator the Sun shines from the right. Play the month and watch the Moon's lit side and the phase icon on the chart. The blue shell shows the ocean bulges, exaggerated tens of thousands of times; the red dot is a tide gauge whose next month of tides is charted. Things to try:

• Watch the chart: the daily tides swell to spring tides near new and full moon and shrink to neap tides at the quarters.
• Untick the elliptical orbit: the spring tides become all the same height. Ticked, the ones near perigee are bigger.
• Move the gauge toward the pole: the tides shrink, because in this model the Moon and Sun stay over the equator.
• Move the node angle until the next eclipse is soon, and play up to it: the Moon turns red, or its shadow crosses the Earth.

What the model assumes

• Equilibrium tides. The ocean is a smooth shell covering the whole Earth that takes the shape of the tidal force instantly. This is Newton's model; it gets the cause, the two bulges and the spring-neap cycle right.
• The Sun, Moon and gauge are in one plane with the Earth's equator for the tides. Both really move up to 28.6° north and south of it, so the two daily tides are usually unequal.
• A simple lunar orbit: an ellipse of eccentricity 0.0549 with its perigee every 27.55 days, a node line that slides past the Sun once every 346.6 days, and the 5.14° tilt. The real orbit is disturbed by the Sun in many smaller ways.
• Approximate eclipse limits measured from the center of the Earth (1.5° for a partial solar eclipse somewhere, 0.94° for a total or annular one, 1.0° and 0.45° for partial and total lunar ones). Real limits vary by a few tenths of a degree with the distances.
• Not to scale. The Moon is drawn much closer and the Earth and Moon much larger than they are; the Sun is shown close but is really 390 times farther than the Moon. The Moon's tilt is drawn four times steeper so it can be seen.
• Phases are named by age, with new, quarter and full moon each given a day around the exact moment.

Edge cases

• At the poles this model's tide is a small, steady offset with no daily rise and fall, because the bulges stay over the equator.
• At 35.3° latitude, where cos² φ = 2/3, the average level of the equilibrium tide over a day is the same as with no tide at all; nearer the equator the average is raised, nearer the poles lowered. The twice-daily range itself shrinks smoothly toward the pole.
• New moon is not always invisible: a day or two either side, a thin crescent shows low in the sky at dusk or dawn, and Earthshine lights the dark part faintly.
• Total and annular eclipses hinge on the Moon's distance; near 380,000 km it can be a hybrid, total along part of the track and annular along the rest.

Where the model stops being right

• Real tides are set by the oceans' shapes. Water cannot keep up with the Moon round the globe; it sloshes in basins, resonating in some and cancelling in others. Tides swing around amphidromic points where there is almost no tide at all, and are amplified in funnels: over 15 m in the Bay of Fundy, under 30 cm in much of the Mediterranean. Equilibrium theory cannot predict a local tide.
• Timing. High water at most ports is hours away from the Moon's passage overhead, and spring tides usually come a day or two after new and full moon (the age of the tide). Tide tables come from decades of measurements split into dozens of harmonic constituents, not from this formula.
• Diurnal tides. Some coasts, such as parts of the Gulf of Mexico, get one tide a day, from the Moon's declination and the basin's shape; this model always gives two.
• Weather adds storm surges and pressure changes of a meter or more; wind and rivers change the levels too.
• Eclipses need precise ephemerides. Where on Earth an eclipse is seen, and for how long, depends on details far finer than this model; it says only whether one happens.

Related simulations

The Seasons and Day Length Simulator shows the Earth's tilt and orbit round the Sun, and the Orbit Simulator shows how elliptical orbits come from gravity.