BABYLONIAN MOONS -------------- John Pazmino NYSkies firstname.lastname@example.org 2005 October 19
Introduction ---------- In early 2001, I explained how the ancestral astronomers of Babylonia sussed out the cycles of the planets. That article arose from a discussion among New York astronomers regarding the conjunction of Jupiter and Saturn. The Babylonians had nothing of a solar system or other model of the planetary motions in space. By merely noting the instances of such conjunctions they were able to foretell future ones. The ensuing banter thru spring 2001 shifted to the ancient observations of the Moon. The Moon was -- and is still in some cultures -- crucial for timekeeping. The month was historicly founded on the cycle of lunar phases. Careful watch of the Moon's changing shape was an integral part of ancient astronomy.
Some history ---------- Most of the peoples of ancestral importance to us in the western society lived in what we now call the Middle East. The center of culture in this region in the last couple millennia of the BC era was Babylon, near present day Baghdad, Iraq. The empire seated in Babylon, called Babylonia, underwent many mutations over the centuries, including being overrun by foreign peoples and being sacked many times. Despite the repeated calamities, Babylon managed to adhaere to a conscientious scheme of astronomy, including careful logging of the Moon's motions. In the earlier period, before about 700 BC, the whole effort was for observing the Moon. By about 400 BC enough data were in hand and the civilization was matured enough to make some awfully shrewd predictions of the Moon's behavior. Similar enterprises were established for observing the planets and weather, but here I work with only the Moon. Out of this legacy from Babylon the Greeks developed a true system of lunar motion. Modern cultures still today use this system (with suitable updates) for maintaining their calendars. In fact, very much of the skills and principles of contemporary stargazing are rooted in Babylonian astronomy.
New Moons ------- As the Moon circulates around the Earth she alters her shape due to our perspective on more or less of her sunlighted hemisphere. Of course, to the unaided eye, the Moon looks pretty much like a flat disc which by some magic grows and shrinks over the course of one circuit thru the zodiac. A complete circuit of the Moon from a one conjunction with the Sun to the next is one synodic cycle or one lunation. When the Moon is against the Sun in the sky we can not see her at all, save for the rare instance of a solar eclipse. The moment of actual conjunction, when the Moon is directly alongside the Sun, either a bit to the north or to the south, is the instant of new Moon. This situation of the Moon is from the ground under our light scattering atmosphere utterly unobservable. But the moment of new Moon was a cardinal event for Babylonians. For on the next day they could expect to catch sight of the merest thin crescent of Moon as she pulls away from the Sun into the evening sky at sunset. Seeing this Moon signals the start of a new month. They could anticipate the new Moon by catching the last thin crescent west of the Sun in the morning twilight before sunrise. Within a day the actual new Moon would occur. The Babylonians then set out observers on the second day to see the first Moon in evening. The idea sounds so simple. Indeed, today many authors of astronomy exercises implore you to go out and repeat this Babylonian task. The idea ends up being a most trying effort. The air near the horizon is almost always slicked with haze, mist, cloud. The crescent, so thin and obliquely lighted, is a low contrast target against the bright twilight sky. Observers with better eyesight may see the Moon sooner than others with more normal vision, throwing the start of the month one day off. Similar considerations apply to the Moon in morning twilight.
Full Moon ------- When the Moon rounds to the halfway point in her march thru the zodiac and stands opposite in the sky from the Sun, she is fully lighted by the Sun. The moment of crossing the halfway mark, 180 degrees downrange from the Sun, is the instant of full Moon. With the full Moon being in a dark night sky it should be trivial to observe it, right? Nope. It is like really tough with bare eye to tell when the Moon is completely round, as against having some slight defect of shape along one edge. The Moon looks pretty full within a day on either side of the geometric full Moon moment. In fact, in the modern calculation of Easter, a schematic full Moon, not the real astronomy-based one, is employed in some cultures. It being easier to see the first crescent of the month and knowing the period of one lunation, some Easter computations clock off 14 days from that first Moon and declare that moment the full Moon. The full Moon rises at sunset and sets at sunrise. This is approximately true and serves well for casual stargazing. Unless the Moon hits the opposition point right at sunset, she will not herself rise at sunset. Similarly for the event at sunrise. The Moon will become full at some arbitrary instant within the 24 hour span of a certain day and not exactly at sunrise or sunset. Hence, on the day of full Moon you may see the Moon rise a little before or a little after sunset. Or see it set before or after sunrise.
Other phases ---------- It is a curious fact of history that the other cardinal phases of the Moon, like half or a certain thickness of crescent, apparently were not chronicled by Babylonia. Nor were they observed by other peoples who followed them! It is only in very recent (by Babylonian standard) times that we as stargazers even note the half Moon phases in our almanacs. I suppose a half Moon could be assayed by a ruler held against the terminator but even strong-sighted folk tell me that it's hardly a simple feat. I tried this a couple times over the years. Apart from my weak eyes, the dark and light spots on the lunar disc distort the terminator into an crooked edge. Recall that Aristarchus, who used the half Moon to form the right triangle of Earth-Sun-Moon, had such trouble that he could only give the distance to the Sun as a lower limit. One intriguing suggestion for neglecting the half Moon is that it could be verified by the layman, thus potentially compromising the authority of the king's starwatchers. Only they had the special mission to look after the Moon's motion. I can see how a layman may challenge the declaration of exact half Moon. He could lay a ruler on the Moon in twilight (when the ruler is visible against the sky) and check up on the official sky watchers. By picking intrinsicly difficult phaenomena, the officials can claim a special skill or training to determine new or full Moon. This is mission protection.
Babylonian measurements --------------------- As fascinating as Babylonian astronomy is -- and I do mean fascinating! -- I have to pass over a detailed account of their scheme of measurement. I'll use all contemporary measures. I also will not go into the Babylonian names of the various astronomy terms. I use only modern terminology here.
Horizon events ------------ Normally the horizon is the worse place for a target to be studied. We would like to hold off serious study of the target until it is removed above the horizon as far as possible. Our views of Mars in the apparition of 2001 are complicated by the planet's low altitude, for one modern example. The Babylonians had no choice. The horizon was the only obvious and secure coordinate circle on the sky. All other coordinate lines were mathematical constructs not directly observable. A passage of the Sun or Moon thru the horizon was a definite and verifiable event which can be (in theory) confidently timed. But the air near the horizon is the thickest in the sky, full of varying densities and optical distortions. The grossest distortion is refraction. You would think that timings of the Sun and Moon on the horizon would be killed by refraction, unknown and unknowable to the astronomers in Babylon. For a single timing in the absolute sense you are correct. Sunset in reality essentially never is seen to happen at the geometric moment cited in our almanacs. The Babylonians used the interval between two horizon timings, not their actual values. This had the effect of netting out refraction! The cancellation of refraction was more perfect by timing the two events with an hourish span, so refraction was about the same in both instances. Refraction drifts in value due to changing temperature and other weather factors. So while the absolute clock readings of the two events may be shifted bodily by a couple minutes, the difference between them would be practicly unaltered. And this is a good thing. The other problem on the horizon was the timing of sunset or sunrise. Geometricly we like to define these events as the passing of the Sun's center over the horizon, as is usual in the rest of astronomy. It is impossible, specially with a brilliant Sun, to tell when the disc is bisected. The Babylonians used a far simpler and more precise definition of sunset and sunrise. The Sun rises when the first bit of his upper edge breaks thru the horizon. Sunset is when the last bit of the Sun's upper edge is covered by the horizon. Guess what? These definitions are the very same ones we still use today! This is why when you calculate sunset geometricly for the Sun's center, the answer is a minute or so earlier than the value given in almanacs.
Horizontal parallax ----------------- We think of parallax as a newer development in astronomy only seriously studied since the 1700s. Parallax was a concept in Ptolemaeus's works by which he assessed that the stars were indefinitely far away. But parallax for the Moon, amounting to a full degree of displacement between the horizon and zenith, was known and recognized in classical Greek days. It was thru this parallax that the Moon was fathomed to be quite 60 Earth radii away. I haven't found that the Babylonians specificly knew about parallax for the Moon but they recognized that the Moon was ahead of her geometric position when rising and behind that position when setting. I omit the ancient treatment of parallax and apply here, given the mid northern latitude of Babylon and New York, an allowance for it. It is four minutes, the time needed to move the extra one degree of parallax-induced depression, times a latitude leverage of 1.5. These six minutes are subtracted from the moonrise timings and six minutes are added to the moonset timings. That is, without parallax (Moon at infinity) she should have risen six minutes earlier but didn't because she was depressed by parallax below the horizon. Corresponding logic applies to moonset; parallax makes the Moon set too soon. This is not exact, I know, but I do want to keep things simple, as if they are otherwise so. Bear in mind that if this magilla with the Moon gives you a headache, consider that you're merely revisiting what iron and bronze age cultures doped out three thousand years ago.
Latitude ------ The Moon does not run in the ecliptic; we would have eclipses every full and new Moon if she did. Our predecessor starwatchers knew this. The lunar orbit is inclined 5-1/2 degrees. As the Moon runs thru this inclined orbit she deviates north and then south of the ecliptic. She crosses the ecliptic twice, on the northbound arm and on the southbound arm, each month. At maximum excursion from the ecliptic, the Moon is 5-1/2 degrees away, north or south, from the ecliptic. The north-south distance of the Moon from the ecliptic is her latitude. The downrange location along the ecliptic, counting from the vernal equinox, is her longitude. I consider here only the longitude component. Moreover, the orbit is not fixed in the stars but swashes around the ecliptic once every 18ish years. Or, if it's any easier to visualize, the nodes of the orbit migrate thru the zodiac. The Babylonians figured all this out and laid out a cycle of lunar phases and positions, along with a theory of lunar and solar eclipses, based on this inclination and nodal migration.
Motion ---- The Moon's motion along her orbit is smooth but not steady. By the behavior of orbits, she runs faster round the perigee and slower round the apogee. Our Babylonian starwatcher knew nothing about this cause of the variable lunar speed but he did exploit this feature. He learned that within a month the Moon went thru a full cycle of speed from fastest to slowest. I'm going to make up Babylonian terms for the point in the Moon's path for the extrema of speed. The place where the Moon runs fastest is 'celerilune' and that where the Moon runs slowest is 'tardilune'. These are Latin-based words; I don't speak Babylonian. The speed of the Moon was measured by the length of the arc run by the Moon in one day, or as we would say, the daily motion. The usual units are degree/day. The need for the daily motion was downright critical. With it the Babylonian was able to predict, to within minutes!, the moment of new or full Moon by taking a couple simple timings.
Timing the interval ----------------- A cardinal reason for using the interval or difference between two events was the lack of good timekeeping devices. What our ancestors did was note the diurnal arc thru which the Sun or Moon travels during the span between the two events. This arc, by the known rate of diurnal rotation, was converted into time. Cunning? Like a fox. In many inscriptions, this conversion was not explicitly done. The tables give the actual arc itself in [modern] degrees of angle. We may read that the Moon set four degrees after the Sun. We would transform this, at the rate of four minutes per degrees of diurnal motion, to sixteen minutes of time. By this trick the Babylonian stargazer could pin his timings to within a half degree of arc or two minutes of time.
Boxing in new Moon ---------------- How did our ancestral stargazers determine the moment of new Moon? Or of full Moon? By an amazing and deadly accurate program of timing certain risings and settings. In all there were six combinations of these events, four near full Moon and two near new Moon. In archaeology they are called the 'Lunar Six'. There are six only because Babylonian math wasn't up to understanding negative numbers. Today if we were to use these timings we would say there are four of them. Consider the new Moon. As the Moon shrank her crescent toward the end of her lunation and edged toward the Sun in the predawn hours, timings were made of her rising. The Moon rose before the Sun, less and less so as new Moon approached. On the very last day before new Moon, when on the very next day there would be no Moon in sight, the final timing was made. And a timing was also made of the sunrise coming immediately after moonrise within a hour or so. Now we have the interval between the moonrise and sunrise on the day of last visibility of the Moon immediately before new Moon. We now wait one full day to let the Moon pass around the Sun and move into the evening sky. We inspect the sky near the sunset point and, lo!, we see the newly waxing Moon. The new month begins. We note when the Sun set and wait until moonset, coming within an hour or so. We now got the difference between sunset and moonset on the day of first visibility of the Moon immediately after new Moon. New Moon must have occurred sometime between these two timings.
Boxing in full Moon ----------------- Despite the fullness of the disc, the Moon is revoltingly hard to see on the horizon in less than optimum viewing conditions. But gamefully our ancestors collected masses of full Moon observations. Consider a Moon which is not quite full. She is less than 180 degrees downrange from the Sun so at sunset she is already in the sky. We could have actually seen it rise some moments before sunset and indeed this is one of the two measurements the Babylonians made. So we have on the day before full Moon the interval from moonrise to sunset. Much later on that same night, we see the fullish Moon in the west before sunrise. Of course it plodded along the zodiac for these many hours but let's say it still is not quite full. It will set a little before the Sun and we have an other timing. This is the span between moonset and sunrise just before full Moon. When the Moon is a bit past full she is more than 180 degrees downrange and is not yet risen at sunset. We wait and then see the Moon rising. Our timing is now between sunset and moonrise on the day just after full Moon. Correspondingly near the next sunrise we still see the Moon in the west at sunrise; she sets shortly thereafter. The timing here is between sunrise and moonset just after full Moon.
Lunar six ------- Go back and verify that we accumulated six timings, set out here for reference. Scholars of Babylonian history call these te 'lunar six' but we'll see in a moment that modern astronomers would reduce these to a 'lunar four'.
just after new Moon ---- from sunset to moonset just before full Moon -- from moonrise to sunset just before full Moon -- from moonset to sunrise just after full Moon --- from sunset to moonrise just after full Moon --- from sunrise to moonset just before new Moon --- from moonrise to sunrise
Looking over this table you notice that the events around full Moon are really not distinct events. The timings at sunset before full Moon are the same in substance as those after full Moon, but in reverse order. In the former case the Moon rises before sunset; latter, after. We may in modern context combine these into a single 'full Moon at sunset' timing and define it as (moonrise) - (sunset). If the Moon rises first, this subtraction is negative and the minus sign is a flag for 'before full Moon'. The same subtraction taken after full Moon yields a positive result and the plus sign is the 'after full Moon' flag. A coalition of the timings at sunrise has the new one definition of (moonset) - (sunrise). A negative result signifies a Moon before full; positive, after. As it turns out, historians already preempted the term 'lunar four' for the four timings around full Moon and have expressed no inclination to alter the ancient definitions. We will stay with the set of six here. Just be sure to subtract the EARLIER event from the LATER one, OK?
Impediment against studies ------------------------ Until literally the final couple decades of the 20th century, scholars of ancient astronomy suffered appallingly from the lack of cheap simple accurate means to reconstruct the skies in these early eras. Most historians knew little about astronomy and had almost no entree' to astronomy references. In much of the older litterature on ancient astronomy we find crude graphical constructions of the skies and some conceptual mistakes. In fairness to these scholars, even astronomers had little to go by for far past aspects of the skies. They simply didn't need such information, not even for long ago solar eclipses. Precession, which most astronomers ignore for the brief span of their own careers, becomes hugely important for ancient astronomy. Without good knowledge of the precession rate, attempts to build an ancient sky were chancy. In fact, as late as 1960, astronomy litterature had annoyingly different precession rates! For shortterm applications, within a couple centuries, this was fine. For prolongation into the dawn of history you could slip the sky by many degrees. Some historians tried using mechanical planetaria, like that at Hayden Planetarium in New York. As wonderful as these machines were, they suffered from being too, well, mechanical. They do not incorporate the subtle but crucial effects of interplanetary gravity. A planetarium's planet runs on a 'trolley track' made of rigid gears and cams and rods. When the first home computers arrived, programs in BASIC were written for astronomy. Perfectly adequate for spans near to today (in the late 1970s) they fell apart for any longterm retrocycling. Part of the constraint was the low processing accuracy and small internal memory of the machines. The other was that nonastronomers didn't appreciate the nuances of solar system dynamics. They commonly wrote a 'trolley track' orbital simulator with results that were as calamitous as a, erm, trolley car collision.
Modern planetarium programs ------------------------- With the advent of large memory, accurate arithmetic processors, and a general maturation of software composing, computer programs for astronomy today, since the 1990s, are pretty much perfect for reproducing ancient skies. Many of these are free or cheap, obviating the nagging expense of collecting lunar ephemerides in books and tables. The computers themselfs are now so cheap, in the mid hundreds of dollars for an individual, or free as part of a college office's furniture. While I could start now to simulate an ancient Babylonian sky, which a good computer planetarium can reliably do, there's no point to doing so for this article. In the stead we'll trace out how to find the moment of full Moon in July 2001, taking data from a planetarium as if they were extracted from sky watching. July in New York is routinely murky and hazy and humid. On many days the Moon is obliterated by shmutz several degrees above the horizon.
Moonrise and moonset ------------------ According as the operation of your planetarium you may develop a table of moonrise, sunset, &c. Or you may have to manually set the sky to each event and read the date/hour field on the screen. Either way is acceptable, altho the latter is more fiddly. It is also more confident. In a table of Sun and Moon events, arranged in order of date like an ephemeris, there is a digusting ambiguity. Sunrise and sunset are firm. The Sun rises in the morning hours and sets in the afternoon hours. No problem. Moonrise and moonset are tricky. You just have to know how your planetarium handles them. The Moon can rise or set at any hour of the day because our time system is based on the Sun and not the Moon. A table of events for a given day will show 'moonrise' and 'moonset' as occurring at certain hours. So far so good. Usually the table has a fixed layout with moonrise and moonset in certain columns, regardless of the hour they occur at. You may at first assume that the Moon must rise first, cross the sky, and set. The event in the 'moonrise' column must precede the event in the 'moonset' column. Not. It can turn out that what 'should' be earlier is in fact the later event. Or maybe not; it really is earlier. Here's why. For our example, elaborated below, on 4 July 2001 I find moonrise at 19:01 EST and moonset at 03:26. (To Hell with daylight time!) We must ask some questions. Is the 03:26 moonset taking place LATER in the night FOLLOWING the moonrise of 19:01? Which is to say, is the table STARTING its calculation at moonrise on the 4th and CONTINUING it to moonset on 5th? Or is the planetarium starting AT 0h ON THE 4th and CONTINUING thru its OWN 24h? Which is to say, is the mooonset occurring first in time and some 16 hours LATER in the SAME day the Moon rises? There is often no clue to what the program is doing; the instructions and explanatory material are silent. You may in the end have to figure this out by brute force. Set the program to moonset on and read out what the hour is.
Full Moon of 5 July 2001 ---------------------- I deliberately picked this full Moon for a worked out example because there was a lunar eclipse with it. This means the Moon was close to the ecliptic. She was at her descending node so that assorted extra calculations which were part of the Babylonian lunar mechanism can be left out. With a planetarium program I pretended I'm watching the Moon approach her full moment. I used SkyGlobe and manually rotated the sky to touch the Sun and Moon to the horizon. I read the times from the screen display. At sunset on July 4th, the last time the Moon is already in the sky at sunset, I capture the rising time of the Moon and the setting time of the Sun, all in EST from Brooklyn. (Hey, some people rank Babylon right up there with Brooklyn.) At sunrise on the morning of 5 July I see that the Moon set for the last time before sunrise. I capture the hour of moonset and of sunrise. I do thee measurements for the following two sunsets and sunrises and build this table below. SkyGlobe probably can not give a true sunrise or moonrise against the upper limb. The symbols for the Sun and Moon are not of true angular size. I took these moments to be the instant the discs were bisected by the horizon. This does not alter the concept or the final results. And, yes, in the table, the rise and set times are entirely within the same 24h day; there is no rollover across midnight.
| 4 Jul | 5 Jul | 6 Jul | explanation --------+-------+-------+-------+-------------------- moonset | 03:26 | 04:14 | 05:07 | bisection of Moon symbol paralx | +0:06 | +0:06 | +0:06 | allowance for parallax corr MS | 03:32 | 04:20 | 05:13 | geocentric moonset | | | | sunrise | 04:32 | 04:33 | 04:33 | bisection of Sun symbol | | | | MS - SR | -1:00 | -0:13 | +0:40 | (moonset) minus (sunset) arc | --- | -5:53 |+18:07 | arc of 24h daily motion ========+=======+=======+=======+ moonrise| 19:01 | 19:52 : 20:37 | bisection of Moon symbol paralx | -0:06 | -0:06 | -0:06 | allowance for parallax corr MR | 18:55 | 19:46 | 20:31 | geocentric moonrise | | | | sunset | 19:24 | 19:24 | 19:23 | bisection of Sun symbol | | | | MR - SS | -0:29 | +0:22 | +1:08 | (moonrise) minus (sunset) arc |-13:39 |+10:21 | --- | arc of 24h daily motion
All this makes sense with a couple diagrams. The activity at sunrise has us looking toward the antisolar point on the western horizon; this is also the center line of the Earth's shadow for the lunar eclipse that happens at this full Moon. The Sun is at our back about to break the horizon.
O 6 July, FM+ \ western horizon \ -----------------------------O-FM--------------- \ \ O 5 July, FM-
The hour is sunrise itself, essentially the same on the two days of 5 and 6 July, 04:33. At sunrise on the 5th, the Moon already set; she's now down. We observed it, in playing like we are Babylonians, to set at 03:32. (I'm including the parallax correction from now on.) On the 6th, the very next sunrise, the Moon is still up and doesn't set until 05:13. Obviously sometime BETWEEN the two sunrises the Moon achieved her full phase. If we recognize that the arcs computed in the table above are those from the points in this diagram labeled (FM- to FM) and (FM to FM+), we see immediately that the entire arc (FM- to FM+) is nothing more than the sum of the two pieces. That is, the entire arc is (+0:40) - (-0:13) = 53 minutes. In angular arc this amounts to 13.25 degrees. Our Babylonian colleague would say that the Moon's speed at full Moon is 13.25 degree/day Now in the ancient records not only are the partial arcs written down but also the sum to make the full daily arc. There is a vital reason for this.
When was full Moon? ----------------- Without much error we can linearly interpolate between the endpoints of the daily arc by the portion of it cut by the two pieces. That is, the two pieces we measured on the 5th and 6th are in the ratio of the the moment of full Moon between the two sunrises. We have
FM = ((FM- to FM) / (FM- to FM+)) * (24h) + (sunrise) = ((0:13) / (0:53)) * (24:00) + (04:33) = (05:53) + (04:33) = 10:25
A thoroly analogous diagram can be drawn for the events at sunset with the nearly full Moon rising.
O 4 July, FM- / eastern horizon / ----------------------------O-FM------------------- / / 0 5 July, FM+
On the evening of the 4th of July the Moon is already up at sunset and we measured her rising at 18:55. Sunset came at 19:24. On the next evening, the 5th, the Moon is not yet up at sunset. We must wait until 19:46 to see her rise. From the table we see that the Moon rose on the 4th some 29 minutes before sunset and on the 5th she rose 22 minutes after sunset. The entire arc between the two sunsets is the sum of these two timings, (+0:22) - (-0:29) = 51 minutes. In angular measure this is 12.75 degree. The discrepancy between this and that derived from sunrise observations seems to be from the granulation of the timings, only to the nearest minute with a minute swing one way or an other. The same math applies to get the moment of full Moon
FM = ((FM- to FM) / (FM- to FM+)) * (24h) + (sunset) = ((0:29) / (0:51)) * (24h) + (19.23) = (13:39) + (19:23) = 34:02 -> 1 day + 10:02
The full Moon occurs at 10:02 on the day after the sunset, or on the 5th of July, not the 4th. The two moments are a bit mismatched and we do a simple arithmetic averaging.
full Moon = ((sunrise full Moon) + (sunset full Moon)) / 2 = ((10:25) + (10:02)) / 2 = (20:27) / 2 = 10:14
The full Moon occurred by a regular ephemeris at 10:05 on 5 July 2001. This method elaborated three millennia ago does work!
The problem with new Moon ----------------------- For the moment of new Moon we would like to have a situation like in this diagram at sunrise
C NM- / eastern horizon / -----------------------------------* NM ---------------------- / / C NM+
On the day before new Moon we see the vanishing crescent just before sunrise and can make a timing. We want to wait a day and see the Moon just after new as she rises following the Sun. We need the timing of the Moon at NM+ in the diagram. First the Sun rises and a bit later see the Moon come up. This is an impossible observation! No way in Hell could Babylonian or us today see the NM+ alignment. Thus, totally unlike for the full Moon we can not directly get the arc of daily motion MN- to NM+. By analogous fate we can not at sunset see the Moon setting before the Sun on the day before new Moon. Again we lose our grasp on the arc NM- to NM+. But we NEED this daily motion to perform our ratio of the timings to get the moment of new Moon!
The extraordinary mental ingenuity -------------------------------- An other diagram. S
. . M . . . . . CL. E .TL . . . . . . . . S'
In this diagram we got the Moon's orbit (deliberately exaggerated in excentricity; we'll need this shape later) with the alignment for a new Moon. The Moon M stands between us on Earth E and Sun S. Imagine the Sun to be some 400 times farther off than the Moon. Note that the Moon is in a part of her orbit where the daily motion is intermediate between the fastest and slowest. I labeled the fastest and slowest points as CL for celerilune and TL for tardilune. Please bear in mind that Babylonia never knew about the physical orbit of the Moon. They worked only with the path she traced in the sky. The line of apsides, joining the perigee and apogee, rotate gently but not by much within one year. Hence the Moon can be seen with a certain daily motion in the same direction in the zodiac, closely, within a given year. Of course, the phase will continually shift as the Sun apparently rounds the Earth in annual motion. In particular, six months ago the Sun was off toward S'. The Moon is now near her full Moon phase. Now pay attention. When the Moon is near full we have the situation I elaborated earlier. We can actually obtain the arc traversed by the Moon in one day. This is the method of the two timings made on consecutive days bracketing the full Moon moment. This arc is characteristic of the part of the orbit where the Moon stands during that full Moon period. But this part of the orbit is also where the new Moon is six months later when we can not by direct observation get the daily arc. So! The Babylonian astronomer figured out that if he used the daily motion of a full Moon six months BEFORE the new Moon he's now studying, he has nearly enough the correct daily motion of that very new Moon. It's the same as that of the full Moon! With the surrogate arc and the one end of it actually timed, he can figure out when the new Moon took place! I do not go thru a worked computation because the maths are quite similar to those for the full Moon. The trick once again: Use as the daily motion the arc for a full Moon six months EARLIER than the instant new Moon.
The saros cycle ------------- Ancient astronomers, like those of Babylonia, used the saros cycle to predict eclipses. This cycle is 6585-1/3 days. Well, yes and no. To appreciate the eclipse features of the saros, you need eclipse records from all over the world. For starts, each eclipse of a given saros occurs one-third of way farther around the globe, so that, at best, only one in three of the eclipses take place in the longitude zone of Babylonia. Of these one-third, Babylonia will miss some from a latitude too far north or south. On top of all this, there are many saros cycles operating at once, each with its own set of eclipses. Sorting all these out with scanty data was beyond hope for the Babylonians. Now I guess if a computation showed that the new Moon occurred when it was down and an eclipse may be in the making, the Babylonian may have assumed that the eclipse happened in some other part of the world. I don't know if Babylonia had such a global picture of the Earth back then. More likely they treated the unseen eclipse as a dud in their theory. Hence, while our ancestral stargazers knew of the saros they didn't make much use of it for eclipse predictions. They had a far far more important routine use of the saros.
Round and round the Moon goes --------------------------- Moon phase (new for a solar eclipse; full, lunar) and the nodes must line up to make an eclipse. When the line of nodes rotates around to the same point on the ecliptic and a new Moon occurs there, we got the next solar eclipse of that saros. This is NOT the very next eclipse in time! There are several cycles running concurrently, each with its own eclipses. Our colleagues of old found that also the line joining the celerilune and tardilune, what we now know as the line of apsides, also swings around the orbit in very nearly an integer number of laps along with the nodes and phases. So not only will an eclipse take place after one saros interval, the Moon will cross over the Sun at almost the same speed as for the previous saros's eclipse. Look again at the orbit diagram above. This orientation of the orbit repeats after one saros so that when the Moon is new at M, her motion is the same as for the previous saros, 6585-1/3 days before. With the extensive records of lunar behavior, the Babylonian assembled tables, called Goal-Year tables, with the lunar motion and position for the coming year, six months earlier in the previous year, and for one, or perhaps two, saros cycles earlier. The six-month section we now can explain as being the data for getting the daily arc of a new Moon based on a full Moon six months earlier. In the latter six months of the instant year, the data from the front six months are used.
Filling in the gaps ----------------- The Goal-Year table was used for anticipating the events of the instant year, as opposed to the diaries or logbooks which contained actual observations. But what happens, as it will, if bad weather intervenes against getting one or an other timing? In such a case, the ancient astronomer had to replicate the setup of the instant new or full Moon by using the alignment from one saros earlier! He knew that the daily arc of a full Moon 6585-1/3 days before would be the SAME as it is right now. If he loses one end of the arc from adversity of the weather, he can reconstruct it from a replicated value! It is this regular use, potentially twice in every month!, of the saros that makes it one of humankind's most amazing intellectual achievements, alongside the Newton theory of gravity and Einstein theory of relativity. The bit in the usual astronomy books about eclipses is really an extra goodie.
Summing up -------- The lunar motion is far from simple, yet a people rising out of the bronze age living off of the soil of fickle rivers had the will and spirit to maintain a vigil on the Moon for nearly a thousand years. And they got her inconstancies sussed out pretty nearly right on the money. So well did they succede that much of the astronomy of these people is bodily carried over to today's home astronomy.