Lunar Motions Expounded

Illustration from Encyclopædia Britannica. Ptolemaic diagram of a geocentric system, from the star atlas Harmonia Macrocosmica by the cartographer Andreas Cellarius, 1660.

O! swear not by the moon, the inconstant moon,
That monthly changes in her circled orb,
Lest that thy love prove likewise variable.

Shakespeare, Romeo and Juliet, Act II, Scene 2

[Note: Because I live in the United States, the following discussion is written from the viewpoint of an observer in the northern hemisphere. Readers south of the equator, please stand on your head or exchange the words north and south in the appropriate places, whichever is more convenient.]

W e all know the moon goes through a cycle of phases every month, from new moon to full and back again. But that’s just one of its intricate set of interrelated cycles. Begin with the sidereal month (Latin sider = “star”), the period in which the moon makes one full rotation/revolution relative to the fixed stars. (The periods of axial rotation and orbital revolution are the same, which is why the moon always presents essentially the same face toward the earth. What I mean by “essentially the same” will be elucidated later.) As measured against the background of fixed stars, this period is about 27 days and a fraction. In that time, however, the earth will have progressed approximately one-thirteenth of its orbit around the sun (365 ÷ 27 ≈ 13.5). So in order to come back into the same alignment relative to the earth and sun, the moon must travel an extra thirteenth of its own orbit around the earth.

Of course, during this time the earth continues to move (though more slowly) in its orbit, so by the time the alignment catches up, it’s actually closer to a twelfth than a thirteenth of a year: about 29½ days, the synodic month (Greek synod = “meeting”), the interval between meetings of the sun and moon. This is the period commonly referred to as a “lunation,” the cycle of the moon’s visible phases. The actual ratio is about 12⅓ lunations per solar year, accounting for the twelve signs of the zodiac plus the fugitive thirteenth sign, Ophiuchus the Serpent-Bearer.

(This is also the reason the Jewish lunisolar calendar varies between twelve and thirteen months per year. The Muslims use a strict lunar calendar of twelve (synodic) months, with the result that their holidays and festivals migrate steadily backward, about 11 days per year, relative to the solar seasons; the holy month of Ramadan this year (2021) roughly coincides with April on the civil calendar, but 17 years from now it’ll be in October instead. Of course from the Muslim point of view it’s the seasons that migrate forward through the calendar rather than the calendar migrating backward through the seasons. That’s called relativity.)

As the earth revolves around the sun, the sun’s apparent position in the sky as seen from Earth shifts by about one degree of arc per day against the background of the fixed stars. The path it follows, known as the ecliptic, is defined by the plane of the earth’s orbit projected onto the stellar background; the twelve constellations (and a sliver of a thirteenth) through which it passes are the ones we call the zodiac.

Illustration from Andrew Fraknoi, David Morrison, and Sidney C. Wolff, Astronomy, ©OpenStax.

Because the earth’s axis of rotation diverges from the perpendicular by about 23½ degrees relative to its orbital plane, the ecliptic is inclined to the celestial equator (the projection of the earth’s own equatorial plane onto the celestial sphere) by this same angle.

Illustration from Arthur Berry, A Short History of Astronomy.

And since the moon circles the earth in roughly the same plane as the earth around the sun, it too traces nearly this same apparent path among the stars. In fact, the entire solar system moves in more or less the same plane, which is why the other planets also appear to move through this same band of zodiacal constellations.

But since the diurnal (daily) motion of objects across the sky, from rising in the east to setting in the west, is caused by the earth’s axial rotation, it parallels the plane of the equator rather than that of the ecliptic. For observers in the terrestrial northern hemisphere, this means that objects located north of the celestial equator remain above the horizon for more than twelve hours each day, those south of the equator for less than twelve hours (and the opposite, of course, in the southern hemisphere).

Illustration from Nathaniel Bowditch, The American Practical Navigator, ©Copyright 2002 by the National Imagery and Mapping Agency, U. S. Government.

The farther north an object lies, the earlier it rises and the later it sets; the farther from the equator, the more exaggerated the difference. At the extremes, any object closer to the celestial pole than the local latitude (like the line labeled “Declination 60° N” in the figure) never sets below the horizon, while anything within an equivalent “circumpolar circle” around the opposite pole (“Declination 60° S”) never rises above the local horizon. This is why those of us in the northern hemisphere can always see the Big Dipper and never the Southern Cross.

Now, because the ecliptic is tilted relative to the celestial equator, half of it lies in the northern celestial hemisphere and half in the southern. Thus any object (sun, moon, or planet) traveling along the ecliptic will spend half its time north of the equator (and hence above the horizon more than twelve hours a day for observers in the terrestrial northern hemisphere) and half south of it (above the horizon less than twelve hours each day). This principle, applied to the sun itself, is of course what gives us our long days and short nights in summer (when the sun is at the northern end of the ecliptic) and the opposite in winter.

Illustration from Andrew Fraknoi, David Morrison, and Sidney C. Wolff, Astronomy, ©OpenStax.

Note, however, that the moon revolves around the earth about thirteen times faster than the earth around the sun. This means that it also travels the ecliptic thirteen times faster — thirteen degrees per day compared to only one — so that its position in the sky elongates eastward from the sun by about twelve degrees per day. Thus the moon is north of the equator, and visible above the horizon more than twelve hours a day, for half of every month rather than half of every year.

But the phase at which the moon reaches its northernmost and southernmost points changes from month to month over the course of the year. At first quarter, for instance, the moon is seen in the same region of the zodiac where the sun will be three months later; at full moon, six months later; at last quarter, nine months later or three months ago. Thus in December, when the sun is in the southernmost part of the zodiac, the full moon, 180 degrees around the ecliptic from it, is at the northern end and travels a high arc across the sky like the summer sun, staying above the horizon nearly 14½ hours at New York’s latitude. The June full moon, by contrast, is at the southern end of the zodiac like the winter sun, crossing low across the southern sky and spending only about 9½ hours above the horizon.

This monthly north-and-south oscillation of the moon along the ecliptic, incidentally, is what gives rise to the “harvest moon” effect. At the time of the autumnal equinox, the full moon occurs halfway around the zodiac at the vernal (spring) equinox point, where the ecliptic crosses the equator from south to north. If you plot declination (the astronomers’ ten-dollar word for celestial latitude north or south of the equator) along the ecliptic, you’ll find that it traces a sinusoid or sine curve, crossing the zero axis at the equinox points in Virgo and Pisces (where the ecliptic crosses the equator) and reaching its extremes at the solstice points in Gemini and Sagittarius. (The solstices once fell in Cancer and Capricorn — hence the names of the Tropics — but have precessed over time because of the wobble of the earth’s axis.)

The slope, or rate of change, of the sine curve reaches its maximum when crossing the axis and flattens out to zero at the positive and negative peaks. (This is why the cosine, which is the rate of change of the sine, lags it in phase by 90 degrees.) Thus a body moving along the ecliptic experiences its greatest change in declination from day to day when crossing the equator at the equinox points, and least when near the solstice points, where it slows down to zero and eventually reverses direction. In particular, the September full moon, which falls near the point of the vernal equinox, is gaining declination northward at its fastest rate, and thirteen times faster than the sun at the vernal equinox in March.

Now recall that the farther north a body is, the earlier it rises and the later it sets. So although the autumn full moon is culminating (reaching its highest point in the southern sky) by the usual 48 minutes later each day (24 hours ÷ 29.5 days), the change in rising time from one day to the next is much less than that — only about 20 minutes per day — because it is partially offset by the rapid northward gain in declination. The result is that the autumn full moon rises at close to the same hour for several days in a row, producing the “harvest moon” phenomenon. (The same effect actually occurs at some point during every month, but in varying relation to the synodic cycle; only in the fall does it coincide with the full moon. A month after the equinox, it occurs at the waxing gibbous phase just before full, producing the “hunter’s moon.”)

Although I said before that the moon’s orbital plane around the earth is “roughly the same” as that of the earth around the sun, the two planes do not, in fact, exactly coincide: there’s about five degrees of tilt between them. This means that the moon doesn’t exactly follow the sun’s path through the zodiac, but oscillates around it, reaching a maximum of five degrees north and five degrees south of the ecliptic each month. Twice a month the moon crosses the ecliptic, at a pair of points known as the nodes: the ascending node (☊) where it crosses from south to north and the descending node (☋) from north to south.

Illustration from Horoscopius.app.

Again, the relationship between nodes and phases shifts by about one zodiacal sign each month. Approximately once a year, each node crossing will happen to coincide with the new moon, and these are the times when eclipses happen. Think about it: solar eclipses always occur at new moon, when the moon is in the same direction from earth as the sun. If the moon’s orbital plane coincided exactly with that of the earth, there would be an eclipse every month; but because of the five-degree tilt, most new moons pass north or south of the sun without blocking it. But when the new moon happens to fall just at the time the moon is crossing the ecliptic — that is, when it coincides with a node — then the moon will pass directly in front of the sun, producing an eclipse. (This, of course, is precisely the reason the sun’s apparent path among the stars is called the “ecliptic.”) Similarly, when the full moon coincides with a node, it means the moon is crossing the earth’s orbital plane just at the time it’s directly opposite the sun, so it will pass through the earth’s shadow instead of missing it to the north or south, and a lunar eclipse will result.

Illustration from Wikimedia Commons, ©Pearson Scott Foresman.

The existence and significance of the nodes were well known in ancient times. Although they don’t shine visibly in the sky in their own right, they were counted among the planets because they circle the ecliptic regularly, receding at a rate of about 19 degrees per year or one full circuit every 18.6 years. They were often conceived as a pair of dragons (or the head and tail of the same dragon) pursuing the sun and moon around the zodiac and occasionally catching up and devouring them in an eclipse. (For this reason, the cycle of nodes is referred to as the draconic cycle, and the interval between the moon’s successive transits of the same node as the draconic month.) Even today, you’ll find the nodes’ positions charted in any astrological ephemeris, to be plotted along with the planets when casting a horoscope.

When my wife and I traveled to India to see an eclipse in 1980, the observing location chosen for us was a temple of the sun god, Surya, at Konarak in the state of Orissa. (What more perfect site could be imagined?)

The Sun Temple at Konarak. Illustration from HinduFAQs.com.

Off to one side of the main temple grounds was a small side building housing a row of seated figures of the nine planetary deities: the sun, moon, five naked-eye planets (Mercury, Venus, Mars, Jupiter, and Saturn), and at the end of the row, the ascending and descending nodes — one holding in his hands the orb of the sun, the other the lunar crescent.

The planetary deities at Konarak. Illustration from en.wikipedia.org.

I offered my puja (worship) to the god of the descending node, then went out to watch the spectacle he had provided for us that day.

These times when a node coincides with the full or new moon are known as eclipse seasons. In fact, lunar and solar eclipses frequently occur together in the same eclipse season, since when one node coincides with the new moon, the other node, 180 degrees around the ecliptic, will catch the full moon either two weeks before or two weeks after. Even though in two weeks’ time the nodes will have regressed about three-quarters of a degree or so along the ecliptic, there’s enough tolerance in the sizes of the disks involved (sun, moon, and earth-shadow) to produce another eclipse on the backswing — though depending on the precise geometry, one eclipse or the other (lunar or solar) may be only partial. Then six (draconic) months later the whole thing happens again with the ascending and descending nodes reversed.

Because the nodes precess backward around the ecliptic to meet the sun, the interval between successive solar transits of the same node, called an eclipse year, is a bit shorter than the standard tropical year: about 346.6 days instead of 365.2. This period interacts with the 29.5-day synodic month to yield a repeating cycle: nineteen eclipse years equals 223 synodic months, to within a difference of about 8 hours. This comes to 18 (tropical) years and 11 days, a period known as the saros. At the end of this interval, the lunar nodes and phases return to almost exactly the same relative configuration as before, producing a repetition of the earlier cycle of eclipses (though offset geographically by about 120 degrees of longitude because of the third-of-a-day discrepancy). For instance, our Indian eclipse of 2/16/80 was repeated in South America 18 years and 11 days later, on 2/27/98. This saros cycle was also known and used to forecast eclipses in ancient times.

The duration of totality for a given eclipse depends largely on the moon’s varying distance from the earth. Like all orbiting bodies, the moon travels an elliptical rather than a circular orbit. At perigee (nearest the earth), its apparent disk is about five percent larger than at apogee (farthest away).

The moon at perigee and apogee. Illustration from Universe Today.

There’s also a similar but smaller variation in the size of the sun’s disk caused by the elliptical shape of the earth’s orbit.

The moon’s apogee and perigee migrate forward along the ecliptic at about 40 degrees per year, or one circuit in a little under nine years. The interval between successive perigees, the anomalistic month, is about 27.5 days. Because this differs from both the synodic and draconic months, the apogee and perigee are constantly shifting with respect to both the cycle of phases and of nodal passages. The greatest possible duration of totality is for an eclipse occurring at perigee (when the moon is nearest to earth and its apparent disk the largest), at aphelion (when the earth is farthest from the sun, making the sun’s apparent disk the smallest), observed from a point on earth at the center of the moon’s shadow, where the center of the lunar disk passes directly through the center of the solar disk. At the opposite extreme, for an eclipse falling at apogee and perihelion, the lunar disk is too small to cover the entire sun; this produces an annular eclipse (Latin annulus = “ring”), in which the moon blocks out the center of the solar disk but leaves a ring of sunlight visible around the edge.

An annular solar eclipse. Illustration from Universe Today.

The elliptical eccentricity of the moon’s orbit also produces another interesting effect. As I mentioned above, the moon rotates on its axis at the same rate it revolves around the earth, which is why we always see the same face presented toward the earth. (The common expression referring to the opposite side as the “dark side of the moon” is a misnomer. The far side is completely dark only at full moon, when it’s turned away from the sun; at new moon, it’s the near side that’s dark and the far one that’s illuminated, though we still can’t see it because it’s forever turned away from us.) However, whereas the rate of axial rotation is more or less uniform relative to the fixed stars, the rate of orbital revolution is not: it speeds up at the near end of the orbit (perigee) and slows down at the far end (apogee). The result is a slight side-to-side wobble, or libration, in the face presented to the earth. (Libration = Latin for “balancing,” from the same root as equilibrium or the constellation Libra, the Scales.)

Actually, there are two librations: the longitudinal one just described, as if the Man in the Moon were slowly shaking his head “no,” and another latitudinal, as if he were nodding “yes.” The latter is caused by the slight (about 1.5˚) inclination of the moon’s axis of rotation relative to its orbital plane, which causes it to present sometimes more of its northern hemisphere to the earth, sometimes more of the southern — just as the earth itself does in relation to the sun, producing the annual cycle of seasons. Between the two libration effects, about 60 percent of the moon’s surface is visible from earth at one time or another during the course of a month (though never, of course, more than 50 percent at a time).

Lunar libration. Illustration from Wikimedia Commons, ©Tomruen.

In summary, here are some parameters of the moon’s motion from my Handbook of the Heavens:

  • Sidereal month (relative to stars): 27.32166 days
    Synodic month (relative to sun): 29.53059 days
    Tropical month (relative to equinox): 27.32158 days
    Anomalistic month (relative to perigee): 27.55455 days
    Draconic month (relative to nodes): 27.21222 days
    Revolution of perigee (direct): 3232.6 days
    Revolution of nodes (retrograde): 6793.5 days
    Mean orbital eccentricity: 0.054900
    Mean inclination of orbit to ecliptic: 5º 8' 43"
    Inclination of lunar equator to ecliptic: 1º 32'
    Mean apparent diameter: 31' 5.2"
    Geocentric parallax: 57' 2.5"
    Maximum longitudinal libration: 7º 54'
    Maximum latitudinal libration: 6º 50'
    Fraction of surface always invisible: 0.410

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“Lunar Motions Expounded” by Stephen Chernicoff is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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