6 – The astronomical origin of tides

A reliable rhythm

Tides are a regular and ubiquitous phenomenon for those living near the sea.
The rhythmic rising and falling of the sea level can be as small as 0.5 m at mid-ocean locations such as Hawaii, or as big as 14 – 15 m in narrow bays such as the Bay of Fundy in Canada, or at Chausey (pictured below) near Mont Saint Michel [Wiki] in France.

However, regardless of the amplitude of the tides, they all have the same period; at virtually all seaside locations, there are two high tides and two low tides every 24 hours and 50 minutes.

In a more complete analysis, tides are generated by forces imposed on Earth that are caused by a combination of gravity and motion among Earth, the Sun, and the Moon. The latter body being the major cause of our tides.

It is a safe assumption that ancient mariners and those living near the coasts, see tidal history of the classical era [Wiki], will have tried to explain and forecast the tides.
Only around 330 BCE the Greek astronomer and explorer Pytheas [Wiki] discovered that the tides where somehow controlled by the Moon, according to the earliest written history of tidal theory [PDF].  
However, it wasn’t until Isaac Newton (1642 – 1727) developed the universal law of gravitation that the tides could adequately be explained.

For example, most of us are taught that “the Moon orbits Earth,” but it is not quite that simple. As shown in Figure 6.1, the two bodies actually rotate around a common center of mass called the barycenter, which is the balance point of the system, located 1600 kilometers beneath Earth’s surface. The wide green arrow in Figure 6.2 (a) shows the smooth, nearly circular path of the Earth – Moon barycenter around the Sun.

If the Moon and Earth and attracted to one another, why don’t the two collide? Moreover, the Earth – Moon system is involved in a mutual orbit held together by gravity and motion, which prevents the Moon and Earth from colliding. This is how orbits are established that keep objects at more or less fixed distances.

Newton’s work also allowed an understanding of why the tides behave as they do. Just as gravity and motion serve to keep bodies in mutual orbits, they also exert an influence on every particle of water on Earth, thus creating the tides.

Gravitational and centripetal forces in the Earth – Moon system

To understand how tide-generating forces influence the oceans, let’s examine how gravitational forces and centripetal forces affect objects on Earth within the Earth – Moon system.
(We’ll ignore the influence of the Sun for the moment.)

The gravitational force is derived from Newton’s law of universal gravitation, which states that every particle of mass in the universe attracts every other particle of mass. Stated mathematically, the attraction occurs with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between the masses. This can partly be expressed as

The inverse square law

The inverse square law [Wiki] stating e.g. that if two celestial bodies are 2 times as far away, the gravitational pull is 4 times less, or if 10 times farther away, the gravitational pull is 100 times less.

Gravitation:
$F=G\frac{m_1{m}_2}{r^2}$

Which means that gravity ∝ 1/distance² – the symbol ∝ stands for “is proportional to” – so gravity is inversely proportional to the distance squared.

Applying this inverse square proportionality to the Earth's tides, there is a high tide on the side nearest the Moon because the Moon pulls the water away from the Earth, and a high tide on the opposite side because the Moon pulls the Earth away from the water on the far side.

Figure 6.5 demonstrates that the tidal influence on the closer sphere is greater.
This is due to the larger drop in gravitational force [Wiki], which gives a greater ratio of the force on the near side of the sphere to that on the far side.

If mass increases ( ), then gravitational force increases ( ).

An example of this can be seen in objects with a large mass (such as the Sun), which produces a large gravitational force. The rest of Newton’s law can be expressed as

If distance increases ( ), then gravitational force greatly decreases ( ).

The equations of Newton’s law show that gravitational attraction varies with the square of distance, so even a small increase in the distance between two objects significantly decreases the gravitational force between them. This is why astronauts experience weightlessness in space when they get far enough from Earth’s gravitational pull. In summary, then, the greater the mass of the objects and (especially) the closer they are together, the greater their gravitational attraction.

Figure 6.3 shows how gravitational forces for points on Earth (caused by the Moon) vary depending on their distances from the Moon. The greatest gravitational attraction (the longest arrow) is at Z, the zenith (zenith = a path over the head), which is the point closest to the Moon. The gravitational attraction is weakest at N, the nadir (nadir = opposite the zenith), which is the point farthest from the Moon. The direction of the gravitational attraction between most particles and the center of the Moon is at an angle relative to a line connecting the center of Earth and the Moon, see Figure 6.3. This angle causes the force of gravitational attraction between each particle and the Moon to be slightly different.

The centripetal force required to keep planets in their orbits is provided by the gravitational attraction between each of them and the Sun.

A centripetal force “tethers” an orbiting body to its parent, pulling the object inward toward the parent, “seeking the center” of its orbit.
For example, if you tie a string to a ball and swing the tethered ball around your head, Figure 6.2 (a), the string pulls the ball toward your hand. The string exerts a centripetal force on the ball, forcing the ball to seek the center of its orbit. If the string should break, the force is gone and the ball can no longer maintain its circular orbit.
The ball then flies off in a straight line, tangent to the circle, Figure 6.2 (b), “tangent” meaning: a straight line touching a curve at a single point without crossing it there – “linea tangens“ is Latin for “a touching line”.

The Earth and Moon are tethered, too, not by strings but by gravity.
Gravity provides the centripetal force that holds the Moon in its orbit around Earth. If all gravity in the solar system could be shut off, centripetal force would vanish, and the momentum of the celestial bodies would send them flying off into space along straight-line paths, tangent to their previous orbits.

Resultant forces

Particles of identical mass rotate in identical sized paths due to the Earth – Moon rotation system, see Figure 6.4. Each particle requires an identical centripetal force to maintain it in its circular path. Gravitational attraction between the particle and the Moon supplies the centripetal force, but the supplied force is different than the required force (because gravitational attraction varies with distance from the Moon) except at the center of Earth. This difference creates tiny resultant forces, which are the mathematical difference between the two sets of arrows shown in Figure 6.3 and Figure 6.4.

Figure 6.6 combines Figures 6.3 and 6.4 to show that resultant forces are produced by the difference between the required centripetal (C) and supplied gravitational (G) forces.
However, do not think that both of these forces are being applied to the points, because (C) is a force that would be required to keep the particles in a perfectly circular path, while (G) is the force actually provided for this purpose by gravitational attraction between the particles and the Moon.

The resultant forces (purple arrows) are established by constructing an arrow from the tip of the centripetal (red) arrow to the tip of the gravity (black) arrow and located where the red and black arrows begin.

Tide-generating forces

It is crucial to realize that the resultant forces demonstrated in Figure 6.6 are minute, averaging about one-millionth the magnitude of Earth’s gravity.  

If the resultant force is vertical to the Earth’s surface, as it is at both the zenith and nadir (oriented upward) and along an “Equator” connecting all points halfway between the zenith and nadir (oriented downward), it has no tide-generating effect, see Figure 6.7

However, if the resultant force has a significant horizontal component — that is, tangential to Earth’s surface — it produces tidal bulges on Earth, creating what are known as the tide-generating forces. These tide-generating forces are quite small but reach their maximum value at points on Earth’s surface at a “latitude” of 45 degrees relative to the “Equator” between the zenith and nadir, see Figure 6.7.

As previously discussed, gravitational attraction is inversely proportional to the square of the distance between two masses.
The tide-generating force, however, is inversely proportional to the cube of the distance between each point on Earth and the center of the tide-generating body (Moon or Sun). Although the tide-generating force is derived from the gravitational force, it is not linearly proportional to it. As a result, distance is a more highly weighted variable for tide-generating forces.

The tide-generating forces push water into two bulges: one on the side of Earth directed toward the Moon (the zenith) and the other on the side directed away from the Moon (the nadir), see Figure 6.7.

The water flows horizontally, see Figure 6.8, following the green arrows – not fighting against the Earth's gravitational pull – literally being pulled by the Moon.
At the sub-lunar point (the zenith; the point where the Moon is directly above) and the nadir (opposite of the zenith aka the antipodal point) the water piles up and the sea level becomes slightly higher.

On the side directly facing the Moon, the bulge is created because the provided gravitational force is greater than the required centripetal force. Conversely, on the side facing away from the Moon, the bulge is created because the required centripetal force is greater than the provided gravitational force.
Although the forces are oriented in opposite directions on the two sides of Earth, the resultant forces are equal in magnitude, so the bulges are equal, too.  

The Moon’s tidal effect

It is easier to understand how tides on Earth are created if we consider an ideal Earth and an ideal ocean.
The ideal Earth has two tidal bulges, one toward the Moon and one away from the Moon (called the lunar bulges), as shown in Figure 6.9. The ideal ocean has a uniform depth, with no friction between the seawater and the sea floor. In fact, Newton made these same simplifications when he first explained Earth’s tides.

Since the Earth rotates on its axis there is an equatorial bulge of both earth and water, but this is treated as a “baseline” shape against which tidal variations are referenced, a discussion of tides does not need to mention centrifugal forces. That only invites confusion and misconceptions.
Centrifugal forces are not tidal (tide-raising) forces.   In a rotating coordinate system the centrifugal force of Moon on Earth is constant in size and direction over the volume of the Earth, therefore it has no tide associated with it.

If the Moon is “stationary” and aligned with the ideal Earth’s equator, the maximum bulges will occur on the equator on opposite sides of Earth. If you were standing on the equator, you would experience two high tides each day. The time between high tides, which is the tidal period, would be 12 hours. If you moved to any latitude north or south of the equator, you would experience the same tidal period, but the high tides would be less high, because you would be at a lower point on the bulge.

In most places on Earth, however, high tides occur every 12 hours 25 minutes because tides depend on the tidal day (aka lunar day), not the solar day.

Tidal day

An earth day or solar day is a 24-hour period: the time it takes for the Earth to rotate 360° on its axis.
Yet, the Moon – which is in reality not “stationary” – also revolves during this 24-hour period around the Earth in the same direction: a prograde direction.

A tidal day (sometimes called tidal cycle or lunar day) takes longer: 24 hours and 50 minutes, which is why high tide is 50 minutes later and the Moon rises 50 minutes later the next day, see Figure 6.10.
  Note that the term lunar day has several other meanings [Wiki],
hence the term tidal day is dominant.

The tidal day is measured from the time the Moon is on the meridian of an observer – i.e. directly overhead – to the next time the Moon is on that meridian and is 24 hours 50 minutes 28 seconds.
Similarly, the solar day is measured from the time the Sun is on the meridian of an observer to the next time the Sun is on that meridian and is 24 hours.

Tides are universal

The Moon and the Sun act on all nearby objects that have the ability to flow such as water in lakes or gasses in the atmosphere – and even deform the solid Earth.

In the ocean the height is just 1 meters, but that is large enough for water to move into the tidal bulges.
In smaller waters such as Lake Michigan can be raised about 5 cm. And even the water in a swimming pool experiences a tidal influence, although unmeaserable small.

In contrast, the tides in the atmosphere and the “solid” Earth have greater significance. Tides in the atmosphere – called atmospheric tides – can be kilometers high.
The tides inside Earth’s interior – called solid-body or Earth tides – cause a slight but measurable stretching of Earth’s crust, typically only several centimeters high.

Tides are everywhere: e.g. further away in our solar system there is the tidal heating of Jupiter's Io moon [Wiki].

The Sun's tidal effect

The Earth orbits around the Sun – with one revolution in one year – creating a gravitational interaction, which yields two solar tidal bulges per day (24 hours), inherently semidiurnal, half daily (meaning twice per 24 hours), due to the Earth's 24 hour day (revolution on its own axis).

The solar bulges, however, are much smaller than the lunar bulges. While the Sun is 27 million times more massive than the Moon, its tide-generating force is not 27 million times greater than the Moon’s. This is because the Sun is 391 times farther from Earth than the Moon.

Moon vs Sun tidal effects

Even though the Sun is 391 times farther away from the Earth as the Moon, its force on the Earth is about 175 times larger due to its greater mass, Figure 6.11.
Nevertheless, the Sun's tidal effect is smaller than that of the Moon, because tides are caused by the difference in gravity field across the Earth.

Earth’s rotation and the tides

The tides appear to move water in toward shore – the flood tide – and to move water away from shore – the ebb tide.
However, according to the nature of the idealized tides presented so far, Earth’s rotation carries various locations into and out of the tidal bulges, which are in fixed positions relative to the Moon and the Sun.
In essence, alternating high and low tides are created as Earth constantly rotates inside fluid bulges that are supported by the Moon and the Sun.

The monthly tidal cycle is 29 ½ or rather 29.53 days, because that’s how long it takes the Moon to complete an orbit around the Earth. During this orbit, the Moon’s changing position influences tidal conditions on Earth.

The 29 ½ - day monthly tidal cycle is also called a lunar cycle, a lunar month, a lunation or a synodic month.

Note that a tidal cycle (without the “monthly”) is synonymous with a tidal day: 24 hours 50 minutes 28 seconds.

Sun and Moon combined

Figure 6.12 shows the positions of the Earth, Moon, and Sun at various points during the 29 ½-day lunar cycle.
At the New 🌑︎ Moon (with the Moon in conjunction) and Full Moon 🌕︎ Moon (with the Moon in opposition), the Sun, Earth and Moon are in syzygy [Wiki]: three celestial bodies in a straight line.
Tidal forces act to reinforce each other, and the ocean both rises higher and falls lower than the average, see Figure 6.12 (a). At this time, the tidal range (the vertical difference between high and low tides) is large, resulting in very high high tides and quite low low tides, because there is constructive interference, see Figure 6.14 (left), between the lunar and solar tidal bulges.
The maximum tidal range is called a spring (springen = to rise up) tide (no connection with the spring season) because the tide is extremely large or “springs forth.”

Conversely, at the first 🌓︎ and third quarter 🌗︎, the Sun, Earth and Moon are at right angles, see Figure 6.12 (b), i.e. in quadrature [Wiki], and tidal forces counteract each other, i.e. there is destructive interference, see Figure 6.14 (right), and the tidal range is smaller than average.

It is the combined effect of the gravitational forces of the Sun and Moon – which varies during the 29 ½-day lunar cycle – that creates spring and neap tides. When both aligned (i.e. in syzygy) with the Earth they combine their attraction, and otherwise they counteract their attraction.

• Spring tides – word origin: Old English “Springere”, meaning to rise or spring up.
• Neap tides – word origin uncertain: possibly nep = “scarcely” or “barely touching”, as in Old English “nēpflōd”. Otherwise a silly acronym: near even as possible…
• Range is defined as the “difference in water level between high and low tide”.

Spring tides have higher high tides and lower low tides whereas neap tides have lower high tides and higher low tides, see Figure 6.13.
Hence, the range is larger in a spring tide than in a neap tide.

Interference

The lunar and solar tidal bulges can be seen as waves, causing constructive interference at spring tides. and deconstructive interference at neap tides. In the section on Moon vs Sun tidal effects it was demonstrated that – despite the Sun's immensely larger mass – the Moon exerts a 2.25 times larger gravitational attraction since the Moon is much closer to the Earth. As a result the solar bulges are much smaller than the lunar bulges and cannot wholly cancel out the lunar bulges – there is no complete deconstructive interference, which means that even at Neap tide there is a tidal range.

Phases of the Moon

The dramatic changes of the phase of the Moon are shown in Figure 6.15 as the Moon progresses during the monthly tidal cycle.
In between the phases of:

• New Moon 🌑︎ and First Quarter 🌓︎ waxing crescent (wax = “to increase”; crescere = “to grow”).
• First Quarter 🌓︎ and Full Moon 🌕︎ waxing gibbous (gibbus = “hump”).
• Full Moon 🌕︎ and Third Quarter 🌗︎ waning gibbous (wane = “to decrease”).
• Third Quarter 🌗︎ and New Moon 🌑︎ waning crescent.

Earth's rotation drags the position of the tidal bulge ahead of the position directly under the Moon's sublunar point or zenith, and tides are late by about 12 minutes.

The Moon has identical periods of rotation on its axis and revolution around Earth; a property called synchronous rotation. As a result, the same side of the Moon always faces Earth.

Intricacies

Besides the Earth’s rotation and the relative positions of the Moon and the Sun, many other factors influence tides on Earth – five of the most prominent complicating factors being:

• the declination of the Moon and Sun
• the axial tilt of the Earth
• the elliptical shapes of the orbits of the Earth around the Sun, and the Moon around the Earth.
• the apsidal precession of the Moon's orbit; ±8.85 years.
• the nodal precession of the Moon's orbit; ±18.6 years.

Declination of the Moon and Sun

Up to this point, we have assumed that the Moon and Sun have remained directly overhead at the equator, but this is not usually the case. Most of the year, in fact, they are either north or south of the equator.

The angular distance away from the equatorial plane is called declination; from Latin declinare = “to turn away”, “to bend”, “to deviate”. Declination is similar to geographic “latitude”. Note that in Chapter 3 we already saw Magnetic Declination defined as “the angle between Magnetic north and True north”.

Our planet revolves around the Sun along an ellipse in space, and the imaginary plane that contains this ellipse is called the ecliptic; from Ancient Greek ἔκλειψις (ékleipsis, “eclipse”), via ἐκλείπω (ekleípō, “I abandon”, “vanish”) – ἐκ (ek, “out”) and λείπω (leípō, “I leave behind”).

Crucially, the Earth’s axis of rotation is tilted 23.4° with respect to this ecliptic, see Figure 6.17, and it is this tilt that causes the Earth’s seasons as well as some significant tidal pecularities.
It also means the Sun's maximum declination is 23.4°.
The Tropic of Capricorn marks 23.4° S as the most southerly latitude at which the Sun can appear directly overhead at noon, which happens at the December Solstice. Similarly, the Tropic of Cancer marks 23.4° N as the most northerly circle of latitude at which the Sun can be directly overhead, which happens at the June Solstice.

Moon's orbital tilt

To complicate matters further, the plane of the Moon’s orbit is tilted 5.1° with respect to the ecliptic as shown in Figure 6.16. This leads to a maximum declination of the Moon’s orbit of 28.5° (namely, 5.1° plus the 23.4° of the Earth’s tilt), see Figure 6.18.
This declination changes from 28.5° south to 28.5° north and back to 28.5° south of the equator during the multiple lunar cycles within one year.

The Moon’s orbital path around the Earth is therefore not in line with the equator, except for two days in monthly tidal cycle is 29 ½ days. As the moon completes its orbit around the earth, it is north of the equator for ±14 days, and south of the equator for ±14 days.

This also explains the scarcity of eclipses, see Figure 6.17, since the Sun, Earth and Moon are only occasionally precisely in line.

The combined declinations of both the Moon and the Sun mean that tidal bulges are rarely aligned with the equator. Instead, they occur mostly north and south of the equator. The Moon affects Earth’s tides more than the Sun, so tidal bulges follow the Moon, ranging from a maximum of 28.5° south to a maximum of 28.5° north of the equator as shown in Figure 6.18.

Earth's axial tilt

As we saw above, the Earth's axis is tilted 23.4° against the ecliptic. Similar to the equator on Earth, we have the celestial equator on the celestial sphere. The ecliptic intersects the celestial equator at an angle of 23.4°, see Figure 6.19.

The two intersection points are the vernal Equinox and autumnal Equinox.
The two other points on the ecliptic – which are farthest away from the equator – are the summer solstice when the northern hemisphere is facing the Sun and the winter solstice when the southern Hemisphere is facing the Sun.

Usually, the Sun passes through:

• Vernal equinox on March 20
• Summer solstice on June 21
• Autumnal equinox on September 23
• Winter solstice on December 22

These terms have a northern hemisphere biase by pointing at the seasons on the northern hemisphere.

The Sun's movement in the sky is a like a pendulum

Though experiment :
Without the tilt of the Earth's axis of rotation in relation to our orbit – or equivalently, if the Earth's equatorial plane coincided with the ecliptic plane – every day and for everyone on Earth the path of the Sun would look like an equinox. Consequently, the Sun would rise due east, set due west, be up for 12 hours and down for 12 hours. The only difference for observers at different latitudes would be the noontime declination of the Sun.

Yet, with the actual tilt of 23.4°, there is the pendulating pattern in the movements of the Sun in the sky.
Like a pendulum, the sunrise direction changes from NE at summer solstice to SE at winter solstice, and back to NE, completing one full swing back and forth, or one full cycle, each year, see Figure 6.20.
Near the Equinox the sunrise direction changes rapidly from day to day, like the rapid motion of a pendulum at mid-swing, and there is a stopping or “standstill” at solstice, like the outer limit of the swing of a pendulum.
The Sun's declination changes little for the ~2 weeks around solstice. The sunrise and sunset directions change little over this time as well.

The Moon's movement in the sky is the second significant pendulum, and in the section Nodal precession we will discuss the “lunistice” or “lunar standstill“ when the Moon's declination is at its extremes, seemingly standing still.

“Equinox“ is derived from the Latin “aequinoctium“, from “aequus“ (equal) and “nox“ (night). On the day of an equinox, daytime and nighttime are of approximately equal duration all over the planet.

“Solstice“ is derived from the Latin “sol” and “sistere“ (to stand still). An accurate description since at the solstices, the Sun's declination appears to stand still; that is, the seasonal movement of the Sun's daily path (as seen from Earth) pauses at a northern or southern limit before reversing direction.

Within one lunar cycle the declination of the Moon turns from maximum north via the equator (zero degrees) to maximum south and back, via the equator, to maximum north.
Over the equator this movement is fast and at the extremes the slowest, apparently standing still: Lunar standstills
Lateron we will see that these maxima oscillate as well within a period of 18.6 years, see Figure 6.31.

Elliptic orbits

The orbits of the Moon around the Earth, and Earth around the Sun are elliptic, and therefore the changes in distances between the bodies cause variations in the tide-raising forces.
For the Moon this variation is significant (14%), yet for the Sun this variation is minimal (3%).

Apsides: Perigee & apogee and perihelion & aphelion

An apsis – from Greek ἁψίς, meaning “arch, vault”; plural “apsides” ἁψῖδες – is an extreme point in a body's orbit, which is nearly always an ellipse, see Figure 6.23

There are two apsides in an elliptic orbit, named with the Greek prefixes peri– (περί meaning “near”) and ap(o)– (ἀπ(ό) meaning “away from”) added to a reference to the celestial body being orbited.

•  –gee refers to Earth (Gaia)
•  –helion refers to the Sun (Helios)

These two orbits have two apsides each four points:

• perihelion: least distance between the Earth and the Sun, around January 3
• aphelion: greatest distance between the Earth and the Sun, around July 4
• perigee: least distance between the Moon and the Earth
• apogee: greatest distance between the Moon and the Earth

Although an ellipse, the Earth's orbit around the Sun is very nearly circular.
During the northern Hemisphere winter our planet is 147.1 million km from the Sun, a distance which increases just slightly – with 3.4% – to 152.1 million km in the summer.Tidal ranges are largest when Earth is near the perihelion in January, and smallest near the most distant point (aphelion) in July.

The Moon revolves around Earth in a more elliptical shaped orbit, in which the Earth–Moon distance varies by 14% between 356,500 km at the perigee to 406,700 km at the most distant point (apogee). The Moon cycles between perigee, apogee, and back to perigee in about 27.32 days, the sidereal month.

View the apparent size differences of apogee – perigee moon and
aphelion – perihelion sun.

Perigean spring tides

About three or four times a year – in spring and autumn – the New Moon or Full Moon coincides closely in time with the perigee of the Moon, resulting in a perigean spring tide also colloquially known as a king tide .
The difference between the ranges of a perigean spring tide and a normal spring tide is small: 15 centimeters or 2% maximum, yet large enough to result in the flooding of low-lying coastal areas during high tide. If a storm occurs during this time, damage can be extreme.

The perigee of the Moon can occur at any point in the orbit around the Earth due to apsidal precession [Wiki], (play around with an interactive animation); for mariners the Full Moon perigee shown in Figure 6.24 and the New Moon perigee shown in Figure 6.25 are of significance.

Apsidal precession

As we saw in the elliptic orbits section, the Moon's orbit is an ellipse and has therefore two apsides: a perigee (the point of orbit closest to Earth) and an apogee (the point of orbit farthest from Earth).

These two extreme points are connected by the line of apsides, which slowly rotates east, in the same direction as the Moon. This gradual rotation is known as lunar precession, apsidal precession, or cycle of lunar perigee, which completes a full circuit every 8.85 years, see Figure 6.26; causing tidal modulations on an interannual time scale, see Table 6.1.

Twice in this 8.85 year period – approximately every 4.5 years – the perigee points either towards or away from the Sun: so at either the March or September equinox the Moon is both at its closest point to the Earth, and is also overhead at the equator.

With the angle p measured in degrees longitude, see Chapter 1, maximum tide generating forces are at p = o° (perigean Full Moon at automnal equinox & perigean New Moon at vernal equinox) and p = 180° (perigean Full Moon at vernal equinox & perigean New Moon at automnal equinox).

• Vernal equinox, see Figure 6.19 (top left)
• Automnal equinox, see Figure 6.19 (bottom left)
• Perigean Full Moon spring tide, see Figure 6.24
• Perigean New Moon spring tide, see Figure 6.25

The 8.85 year perigean cycle / apsidal precession represents variation of lunar orientation relative to the Earth – Sun axis in a predominantly “longitudinal” motion.
Increase of tide generating forces occurs when there is Earth – Sun – Moon alignment in both longitude and declination, with the latter mechanism producing equinoctial peak tides for regions experiencing mainly semidiurnal tides.
As the solar declination is away from the equator during the solstices, apsidal precession does not produce a regular harmonic influence on high water levels in regions experiencing mainly diurnal tides.

The additional range provided by apsidal precession is largest (between 0.3 and 0.6 m) in regions dominated by semidiurnal tides where the tidal range is >6 m, notably in the eastern part of the Bering Sea, the Hudson Strait, the Gulf of Panama, on the Patagonian Shelf, the Irish Sea, the Bristol Channel and La Manche / English Channel as well as on the northwest Australian Shelf.
Proportionally, the additional range is at least 5% of the tidal range in most of the world's oceans. In the Caribbean Sea, the additional range is over 15% of the tidal range.

Sea level rise and floodings caused by global warming are exacerbated by apsidal precession, in addition to nodal precession, which described in the next section.

See Wikipedia for rotating of lines of apsides [in general] and lunar absidal precession [regarding our Moon].

Nodal precession

The precession of the line of nodes or nodal precession is the second “wobble” in the Moon's orbit that influences tides.
This motion is best envisioned by a plate or coin spinning on a table, yet pulling it through the table, cutting it in half, so that instead of one point touching the table it has two end points (nodes) of the nodal line sharing with the table.
While the plate or coin keeps spinning, this line of nodes rotates.

Obviously, the term “wobble” is polysemantic, since the Moon has several types of precessions, e.g. the apsidal precession we just covered.

Here be dragons 🐉︎

Figure 6.27 shows the lunar nodes, which are the points where the orbit of the Moon crosses the ecliptic, as well as the line of nodes (the nodal line) where both planes intersect.
The ascending node, where the Moon rises above the ecliptic, is historically referred to as the dragon's head ☊︎ (Latin: caput draconis).
Similarly, the descending node, where the Moon sets below the ecliptic, is known as the dragon's tail ☋︎ (cauda draconis).

The orientation of the line of nodes changes as the Earth – Moon system revolves around the Sun: twice a year the line of nodes points towards the Sun, and twice a year in the direction of the Earth's orbit; examine this nodal precession animation.

A full 360° precession of nodes takes ±18.61 years, see Figure 6.28, and the period of time between two consecutive passages of the Moon through the same node is known as a nodal month, draconic month or draconitic month (27 days, 5 hours, 5 minutes, 35.8 seconds).
Again, we are referring to dragons, like hic sunt draconis [Wiki] and not “draconian” (after Draco [Wiki], the first legislator of Athens).

Similar to a solstice, there is the “lunistice” or lunar standstill [Wiki] when the Moon is seemingly standing still with little change in the outer extremes of monthly range of rising / setting directions and declination.

• While the standstill at solstice is about 2 weeks, the lunar standstill is almost 3 years.
• Figure 6.29 shows the 2 lunar standstills:
  • Major lunar standstill – maximum range of Moon rising / setting, when the declinations of the Sun and the Moon amplify each other:
    23.4° + 5.1° = 28.5°.
  • Minor lunar standstill – minimum range of Moon rising / setting, when the declinations counteract each other:
    23.4° − 5.1° = 18.3°.

A half period of 9.3 years separates these two standstills, Table 6.2 lists the years when Major / Minor standstills are at their extremes.

The “” in front of the years 2015, 2034, 2052 forewarns us that the Moon’s orbital excursion above or below the equator takes the minimum value of circa 18.3 degrees, which occurs at the minor lunistice. This slightly increases the chances of the moon being directly overhead at the equator coinciding with the other factors that contribute to extreme tidal forces.
In some places, these extreme tidal conditions can cause water levels to be 0.5 m higher than a normal spring tide, see e.g. the National Oceanography Centre.

Looking back at Figure 6.21 (Moon at high declination) and Figure 6.22 (Moon at low declination) it will be come clear that the semidiurnal tide is strong when the lunar declination is small. The two high-water bulges are close to the equatorial plane during the Moon's revolution and the Earth's rotation then generates two similar tidal pulses every 24 h 50 m.
In contrast, if the lunar declination is large, the water bulges are on average further remote from the equator. Earth's rotation then generates a tidal signal with a significant diurnal component, i.e. with one tidal pulse every 24 h 50 m.
Consequently, the semidiurnal tide and the diurnal tide each have a maximum once in the lunar nodal cycle of 18.6 years, with an interval of about 9.3 years between both maxima.

• High declinations of the Moon strengthen diurnal tides.
• Low declinations of the Moon strengthen semi-diurnal tides.

Climate crisis and the 18.6 year cycle

Despite the heating up of the oceans, there is currently relatively little rise of sea level [Wiki], mainly because we are just past the middle of a 9.3-year dampening of the sea level rise, due to the effects of the moon’s orbital tilt and the precession of the orbital nodes.
By contrast, see table 6.2, in 2025 – 2035 we will face a 9.3-year-long enhancement of sea level rise.

The influence of the 18.6 year cycle on the tides is on average about 5 cm. That may not seem like much, however the total amount of global average sea level rise over the last decade is also about 5 cm. That means that nodal precession has roughly the same impact on the height of high tides as the most recent decade of sea level rise caused by climate change.
Moreover, the levels of spring and neap tides are about 20% higher or lower than average sea levels at and near the minor lunar standstill (2035).
Since most coasts have mesotidal ranges – see the section on tidal ranges – of just over 2 meters, 20% = 40 cm, and 5 cm is therefore significant.

The last time that the moon’s orbit exacerbated tidal flooding was in 2015, recall the Chausey archipelago photos. The National Oceanic and Atmospheric Administration recorded already than 600 tidal floods in the USA in 2019, even when the amplification effect from the moon’s orbit was less than it was in 2015. Global sea level rise – due to climate crisis [Wiki] – pushes high tides in only one direction: higher. Therefore, half of the 18.6-year lunar cycle counteracts the effect of sea level rise on high tides, and the other half increases the effect, see the Sea Level Projection Tool [NASA map]. Starting in the mid-2030s, human-made climate change on top of low lunar declinations will lead to high tide floods that can even occur in clusters lasting a month or longer.

References

• Global influences of the nodal cycle and cycle of lunar perigee – 18.61 year and 8.85 year on high tidal levels
• Change of High Water levels by up to 30 cm – the 18.61 year nodal tidal cycle
• Effect of the 18.6-year lunar nodal cycle on tides

Visualizations

Watch the following demystifying animations:

• Line of nodes regression [Youtube]
• The Moon's orbital plane precession in 18.6 years [Youtube, more detailed]

Idealized tide prediction

With an ideal ocean and ideal earth – where the declination of the Moon determines the position of the tidal bulges – we can make an idealized tide prediction.

Figure 6.32 illustrates that the Moon is directly overhead at 28° north latitude when its declination is 28° north of the equator. If you stand at this latitude when the Moon is directly overhead, it will be high tide, see Figure 6.32 (a).
Low tide occurs 6 lunar hours later (6 h 12 m 30 s solar time), as shown in Figure 6.32 (b).
Another high tide, but one much lower than the first, occurs 6 lunar hours later, as shown in Figure 6.32 (c).
Another low tide occurs 6 lunar hours later, see Figure 6.32 (d).
Six lunar hours later, at the end of a 24-lunar-hour period (24 h 50 m solar time), you will have passed through a complete lunar-day cycle of two high tides and two low tides.

The graphs in Figure 6.32 (e) show the heights of the tides observed during the same lunar day at 28° north latitude, the equator, and 28° south latitude when the declination of the Moon is 28° north of the equator.
Tide curves for 28° north and 28° south latitude have identically timed highs and lows, but the higher high tides and lower low tides occur 12 lunar hours later. The reason that they occur out of phase by 12 lunar hours is because the bulges in the two hemispheres are on opposite sides of Earth in relation to the Moon.

Summary of characteristics of the tides on the idealized Earth

• Any location (except the poles) will have two high tides and two low tides per lunar day.
• Neither the two high tides nor the two low tides are of the same height because of the declination of the Moon and the Sun (except for the rare occasions when the Moon and Sun are simultaneously above the equator).
• Monthly and yearly cycles of tidal range are related to the changing distances of the Moon and Sun from Earth.
• Each week, there would be alternating spring and neap tides. Thus, in a lunar month, there are two spring tides and two neap tides.

Tides offshore and in the ocean

If tidal bulges are wave crests separated by a distance of one-half Earth’s circumference – about 20,000 km – we would expect the bulges to move across Earth at about 1600 km/h.

Tides, however, are an extreme example of shallow-water waves, so their speed is proportional to the water depth. For a tide wave to travel at 1600 km/h, the ocean would have to be 22 km deep. Instead, the average depth of the ocean is only 3.7 km, limiting the speed of tidal bulges.

Based on the average ocean depth, the average speed at which tide waves can travel across the open ocean is only about 700 km/h.
Thus, the idealized bulges that are oriented toward and away from a tide-generating body cannot exist because they cannot keep up with the rotational speed of Earth. Instead, ocean tides break up into distinct units called cells.

Amphidromic points and cotidal lines

In the open ocean, the crests and troughs of the tide wave rotate around an amphidromic point – derived from Ancient Greek ἀμφίς (amphís, “around”) and δρόμος (drómos, “running”) – near the center of each cell.
There is essentially no tidal range at amphidromic points, but radiating from each point are cotidal (co = “with”, tidal = “tide”) lines, which connect all nearby locations where high tide occurs simultaneously. The labels on the cotidal lines in Figure 6.33 indicate the time of high tide in hours after the Moon crosses the Greenwich Meridian.

The times in Figure 6.33 indicate that the tide wave rotates counterclockwise in the northern Hemisphere and clockwise in the southern Hemisphere. The wave must complete one rotation during the tidal period (usually 12 lunar hours), so this limits the size of the cells. Low tide occurs 6 hours after high tide in an amphidromic cell. If high tide is occurring along the cotidal line labeled “10”, for example, then low tide is occurring along the cotidal line labeled “4”.

Effect of the continents

The continents affect tides, too, because they interrupt the free movement of the tidal bulges across the ocean surface. The ocean basins between continents have free standing waves set up within them. The positions and shapes of the continents modify the forced astronomical tide waves that develop within an ocean basin. In fact, two of the most important factors that influence tidal conditions along a coast are coastline shape and offshore depth.

In addition, increased turbulent mixing rates in deep water over areas of rough bottom topography are associated with internal waves created by tides breaking on this rough topography and against continental slopes. These tide-generated internal waves have recently been observed along the chain of Hawaiian Islands, have heights of up to 300 meters, and contribute to increased turbulence and mixing, which strongly affect the tides.

Other considerations

A detailed analysis of all the variables that affect the tides at any particular coast reveals that nearly 400 factors are involved, which are far more than can adequately be addressed here. One of the results of these factors, however, is that high tide rarely occurs when the Moon is at its highest point in the sky. Instead, the time between the Moon crossing the meridian and a corresponding high tide varies from place to place.

Because of the complexity of the tides, a completely mathematical model of the tides is beyond the limits of marine science. Instead, a combination of mathematical analysis and observation is required to adequately model the tides.

Types of tidal patterns

In theory, most areas on Earth should experience two high tides and two low tides of unequal heights during a tidal day of 24 h 50 m.
In practice, however, the various depths, sizes, and shapes of ocean basins modify tides so they exhibit four different tidal patterns in different parts of the world.
These four types, which are illustrated in Figure 6.34, are diurnal [rare], semidiurnal [most common], mixed diurnal [rare] and mixed semidiurnal [common].

• Semidiurnal most common
• Diurnal rare
• Mixed semidiurnal common

Large continents block the westward passage of the tidal bulges as the Earth rotates, and unable to move freely around the globe, these tides establish complex patterns within each ocean basin that often differ greatly from tidal patterns of adjacent ocean basins or other regions of the same ocean basin.

Furthermore, ocean currents, constrained waterways, friction, archipelagos, the Coriolis effect, and coastal topography (headlands, shoals, reefs) influence the rise and fall of the sea level.

As a result different tidal patterns can be experienced in different regions of the world.

Semidiurnal tidal pattern

A semi-diurnal or semidiurnal tidal pattern consists of two nearly equal high tides and two nearly equal low tides within the tidal day – “nearly”, since tides are always growing higher or lower at any location due to the spring-neap tide sequence, successive high tides and successive low tides can never be exactly the same at any location.
The tidal period is 12 hours and 25 minutes: from high tide to the next high tide. If the planet was covered entirely with water and without any continents obstructing the free motion of water, semidiurnal patterns would be default.
Most of the world's coastlines have semidiurnal tides.

The term “diurnal” can mean “during the day” – as an antonym of “nocturnal” – but with respect to tides it means “having a daily pattern that is completed every 24 hours”, and the prefix “semi” means “half”, see Semi–, Hemi–, Demi–.

Diurnal tidal pattern

A diurnal tidal pattern has a single high and a single low tide each tidal day. The tidal period is 24 hours and ±50 minutes and therefore twice as long as with the semidiurnal tidal pattern.
Diurnal tides can only be found in the Gulf of Mexico, short stretches of Australian coasts, parts of the Caribbean and on the east coast of the Kamchatka Peninsula.

Mixed semidiurnal tidal pattern

A mixed tidal pattern may have characteristics of both diurnal and semidiurnal tides, the former is rare, hence this pattern is often described as mixed semidiurnal.
Successive high tides and / or low tides will have significantly different heights, a condition called diurnal inequality. Mixed tides have a tidal period of 12 hours 25 minutes, but they may occasionally exhibit diurnal periods.
Mixed tides are the most common type in the world, including along the Pacific Coast of north America, the Caribbean, in parts of Australia and in south east Asia.

Key Concept

• Diurnal tidal pattern: one high and one low tide each tidal day (24 h ±50 m).
• Semidiurnal tidal pattern: two high and two low tides each tidal day of about the same height.
• Mixed tidal pattern: usually two high and two low tides of different heights each tidal day, but may also exhibit diurnal characteristics.

Tidal ranges

Besides the tidal pattern also the tidal range – the amplitude of the tide – is consequential to mariners.
  Explore this zoomable and adaptable world map of tidal ranges
In France, a unique way of expressing the amplitude of the tide is to use an indicator called the “tidal coefficient”, indicating the significance of the tide at a given time. It is a dimensionless number, usually between 20 and 120.

• Coefficient 120 – Extraordinary spring tides of equinox
• Coefficient 95 – Mean spring tides
• Coefficient 70 – Mean tides
• Coefficient 45 – Average neap tides
• Coefficient 20 – Lowest neap tides

The Bay of Mont Saint-Michel

For instance, on Saturday 21 March 2015 in the Bay of Mont Saint-Michel an extreme tide ocurred with an coefficient of 119, see the Chausey photos, with a tidal range of 14.15 meters. The next “Extraordinary spring tides of equinox” will take place in March 2033.
These tidal extremes are due in part to the low depth of the bay, the Coriolis forces combined with the barrier effect from the Cotentin Peninsula, see Figure 6.38 (c).
The intertidal zone – the coastal area affected by the tides – is exceptionally large: 200 km².
The bay is a UNESCO world heritage site for its aesthetic quality and its importance to the Christian tradition.

The Bay of Fundy

The largest tidal range in the world is found in Nova Scotia’s Bay of Fundy.
With a length of 258 km, the Bay of Fundy has a wide opening into the Atlantic Ocean.
At its northern end, however, it splits into two narrow basins, Chignecto Bay and Minas Basin, see Figure 6.38 (b).
The period of free oscillation in the bay – the oscillation that occurs when a body is displaced and then released – is very nearly that of the tidal period.
The resulting constructive interference – along with the narrowing and shoaling of the bay to the north – causes a buildup of tidal energy in the northern end of the bay.
In addition, the bay curves to the right, so the Coriolis effect in the northern Hemisphere adds to the extreme tidal range.
During maximum spring tide conditions, the tidal range at the mouth of the bay (where it opens to the ocean) is only about 2 meters. However, the tidal range increases progressively from the mouth of the bay northward. In the northern end of Minas Basin, the maximum spring tidal range is 17 meters, which leaves boats high and dry during low tide.

Mediterranean tides

Most people associate the Mediterranean with small tides. However, this is not to say there are no tides at all, as shown by this animation of measured Mediterranean tides.
Indeed, they generate a mean variation of about 40 centimeters, but atmospheric conditions often hide the rise and fall in sea level.
Headwinds or, more often, higher-than-normal atmospheric pressure attenuate the effect of these tides, sometimes making them virtually impossible to see.
However, for example, the Gulf of Gabes off the coast of Tunisia has a range of nearly two meters.
In other areas, such as the Adriatic and south of Sicily, tides are very small in the vicinity of amphidromic points where the tidal range is zero.
The Atlantic affects tides in the Strait of Gibraltar, but its influence soon declines further east.
Also see these sections of Greece:

• Ionian
• Saronic / Argolic
• Cyclades
• Dodecanese
• Northern Sporades

Tidal curves show neap and spring ranges

Figure 6.39 shows examples of monthly tidal curves for various coastal locations. Even though a tide at any particular location follows a single tidal pattern, it still may pass through stages of one or both of the other tidal patterns. Typically, however, the tidal pattern for a location remains the same throughout the year.
Also note that the tidal curves in Figure 6.39 clearly show the weekly alternating of the spring tide – neap tide cycle.

Tidal bore

Mascarets or tidal bores are atypical waves. These are tidal waves that travel up rivers as illustrated in Figure 6.40. At the time of the tidal shift, a wave is formed and then moves up the stream against the direction of the current. The tidal bore is a well-known phenomenon in the Bristol Channel (Severn) as well as the Gironde, where it delights surfers and kayakers. This phenomenon can generate impressive waves: e.g. up to 6 meters high in the Qiantang river, travelling at 40 km/h.
German: “Gezeitenwelle”, Spanish: “macareo”, Dutch: “vloedbranding”, Italian: “mascheretto”.

Coastal tidal currents

The current that accompanies the slowly turning tide crest in a northern Hemisphere basin rotates counterclockwise ↺, producing a rotary current in the open portion of the basin. Friction increases in nearshore shoaling waters, so the rotary current changes to an alternating or reversing current that moves into and out of restricted passages along a coast.

The velocity of rotary currents in the open ocean is usually well below 1 km/h. Reversing currents, however, can reach velocities up to 44 km/h in restricted channels such as between islands of coastal waters.

Reversing currents also exist in the mouths of bays (and some rivers) due to the daily flow of tides. Figure 6.41 shows that a flood current is produced when water rushes into a bay (or river) with an incoming high tide. Conversely, an ebb current is produced when water drains out of a bay (or river) because a low tide is approaching. No currents occur for several minutes during either high slack water (which occurs at the peak of each high tide) or during low slack water (at the peak of each low tide).

Reversing currents in bays can sometimes create a navigation hazard for ships. On the other hand, the daily flow of these currents often keeps sediment from closing off the bay and replenishes the bay with seawater and ocean nutrients.

Further reading

• Historical background – Lunar Motion
• Major / minor Lunar standstills
• Tides and centrifugal force
• Coriolis and friction & Coriolis
• Lunar sidereal and synodic months [short animation]
• History of tidal science [Wikipedia]
• What physics teachers get wrong about tides! [Youtube – PBS SpaceTime]
• Visualization of Tidal Forces [Youtube – (non)-inertial (pseudo)-forces]
• Archaeoastronomy Database [Youtube – Astronomy concepts]

Overview

• Tide: For yachtpersons in particular, the tides are defined as the rise and fall of the surface of a body of water caused primarily by the differences in gravitational attraction of the Moon, and to a lesser extent the Sun, upon different parts of the Earth when the positions of the Moon and Sun change with respect to the Earth.
• Centripetal force: A net force that acts on an object to keep it moving along a circular path.
• Cotidal lines: Lines that radiate out from the amphidromic point.
• Gravitational force: A force that attracts any two objects with mass, measured from the centers of the objects.
• Lunar / Solar / Tidal bulge: The tidal bulge created on the sides of the Earth toward and away from the Moon and Sun.
• Zenith: The point of the Earth closest to the Moon, i.e. where the Moon is overhead.
• Nadir: The point of the Earth furthest from the Moon and opposite of the zenith.
• Resultant force: The mathematical difference of the forces acting on an object, resulting in the single force that represents the effects of all of the other forces acting on that object. Therefore, it is a net force, being equal to the sum of all forces applied to the object.
• Gradient: The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.
• Syzygy : In astronomy this is a collinear configuration of three celestial bodies, from Greek Συζυγία “conjunction, yoked together”.
• Spring tide: The tidal effect of the Sun and the Moon acting in concert twice a lunar month, when the Sun, Earth and Moon are all in a straight line (Full Moon or New Moon) – syzygy. The range of tide is larger than average.

• Neap tide: This opposite effect occurs when the Moon is at right angles to the Earth – Sun line (First or Last Quarter) – no syzygy. The range of tide is smaller than average.

• Range: The vertical difference between the high and low tide water levels during one tidal cycle.
• Tidal cycle: One high tide plus a successive low tide.
• Barycenter: In astronomy, the barycenter is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit.
• A top view of the solar system is the norm: northern hemisphere bias, and from this perspective both the Earth and the Moon are orbiting counterclockwise ↺.
• Tidal day or Lunar day: 24 hours and ±50 minutes. The Moon orbits the Earth every month, and the Earth rotates (in the same direction as the Moon's orbit) on its axis once every 24 hours.

• Semidiurnal tide: The most common tidal pattern, featuring two highs and two lows each day, with minimal variation in the height of successive high or low waters.
• Diurnal tide: A tidal pattern with only a single high and a single low during each tidal day; successive high and low waters do not vary by a great deal, e.g. in the Gulf of Mexico, Java Sea and in the Tonkin Gulf.
• Mixed Tide or Mixed semidiurnal tide: This tidal pattern is characterized by a wide variation in heights of the successive high and low waters, and by longer tide cycles than those of the semidiurnal cycle, e.g. in the U.S. Pacific coast and many Pacific islands.
• Apsis, plural: Apsides: e.g. perigee, apogee, perihelion and aphelion.

• Declination: In astronomy this is the angle measured north or south of the celestial equator, much like the geographic “latitude”.
• Apogee: This is the point where the orbiting object (Moon) is farthest from it's parent (Earth).
  Lunar apogee: 406,697 km.
• Perigee: The point where the orbiting object is closest to its parent.
  Lunar perigee: 356,410 km.
• Line of apsides: A reference line passing through the orbital plane at perigee on one side and apogee on the other.
• Perihelion: The point in the elliptic orbit of a comet, planet, etc., where it is nearest to the Sun.
• Aphelion: The point in the elliptic orbit of a planet, comet, etc., where it is farthest from the Sun.
• Perigean spring tide: Colloquially known as a “king tide” and which typically occurs between 6 – 8 times each year; the New Moon or Full Moon coincides closely in time with the perigee of the Moon – the point when the Moon is closest to the Earth.

• Apsidal precession: All orbits have two points where the orbiting body is either closest to, or furthest away from, the object (i.e. “parent”) it is orbiting. These points are called apsides, and the imaginary line between them is called the line of apsides. Apsidal precession is the gradual rotation of this line over time, which occurs in cycles of around 8.85 years for the Moon.
  About every 4.45 years, the Sun is coincident with the line of apsides. Larger tidal ranges are experienced during these years, particularly around the time of the equinox when the Sun's declination is zero (which maximizes semidiurnal forces).
• Ecliptic: The plane of the Earth’s orbit around the Sun.
• Orbital tilt of the Moon or inclination of the Moon's orbit: The Moon's orbital plane is inclined by about 5.1° with respect to the ecliptic.

• Earth's axial tilt: The angle that the Earth's rotation axis makes with the ecliptic. The Earth is currently tilted 23.4° from this plane, and varies between 22.1° and 24.5° over time.
• Lunation is the 29 ½-day monthly tidal cycle.
  Other synonyms are: lunar cycle, lunar month, and synodic month.
  During the monthly tidal cycle, the phase of the Moon changes dramatically.
  When the Moon is between Earth and the Sun, it cannot easily be seen at night, and it is called the New Moon.
  When the Moon is on the side of Earth opposite the Sun, its entire disk is brightly visible, and it is called a Full Moon.
  A Quarter Moon – a moon that is half lit and half dark as viewed from Earth – occurs when the Moon is at right angles to the Sun relative to Earth.

• Lunar nodes: There are two points, called nodes, at which the orbit of the Moon intersects the ecliptic.
• Ascending node, dragon's head, ☊︎, Caput Draconis: The ascending (or north) node is where the Moon moves into the northern ecliptic hemisphere.
• Descending node, dragon's tail, ☋︎, Cauda Draconis: The descending (or south) lunar node, the point at which the orbit of the Moon intersects the ecliptic entering the southern ecliptic hemisphere.
• Line of nodes: The line connecting the two lunar nodes of the Moon's orbit.
• Precession of the line of nodes or nodal precession: Because the orbital plane of the Moon precesses in space, the lunar nodes also precess around the ecliptic, completing one revolution – called a draconic or nodal period – in 18.6 years, i.e. the line of nodes of the Moon's orbit precesses 360° in this period of 18.6 years.

  

• Draconic month: This period is also known as Nodical month, Draconitic, Nodal month [Wiki] or Draconic period, which is 27.21 days (i.e. 27 days 5 hours 5 minutes), and is the time between the Moon’s passages through the same node, or intersection of its orbit with the ecliptic.
• Sidereal time: Time as measured by the apparent motion about the Earth of the distant, so-called fixed, stars, as distinguished from solar time, which corresponds to the apparent motion of the Sun.
  To be specific, in the time it takes the Earth to spin once about its axis, it also moves along its orbit by over 2.5 million kilometers. Because the Earth has moved, the Sun will not appear in the same part of the sky at the end of that rotation. To end up facing the Sun again, the Earth has to rotate for another four minutes. In other words, a solar day is how long it takes Earth to rotate once – and then some. A sidereal day – 23 hours 56 minutes and 4.1 seconds – is the amount of time needed to complete one rotation.
  Astronomers rely on sidereal time keeping because any given star will transit the same meridian at the same sidereal time throughout the year.
  Sidereal time may be defined for any place on the Earth, but in the international system used by astronomers each sidereal day begins at the instant the vernal equinox transits the prime meridian.
• Sidereal month: The time it takes for the Moon to make one full orbit of the Earth relative to the stars.
  One Sidereal month = 27.3 days
  = 27 days 7 hours 43 minutes.
  
  From Latin “sidereus”, from sider-, sidus star, constellation. Hence, “sidereal”: relating to, or expressed in relation to stars or constellations, see sidereal period / month [Wiki].
• Synodic month: A synodic day is like the solar day on Earth, and a synodic month is defined as the time required to go through all of the lunar phases.
  • This is different from a sidereal month because the Earth – Moon system is orbiting the Sun.
  • At the end of a sidereal month the Earth has moved around the Sun and the Moon has further to go until it reaches the New phase.
  • A synodic month is therefore longer than a sidereal month.
  One synodic month = 29.53 days.
  Because of the motion of the Moon around the Earth,
  it rises about 50 minutes later each day.