When we developed the model for sun, one of the first things we did was to determine the rate of mean motion for the sun to which the anomalistic motion could be added. However, the moon is going to be more complicated for several reasons. As such, before going further, we’ll need to spend some time examining the various lunar periods.
The first reason the moon will be more complicated is that the moon has a motion the sun doesn’t. For the sun, we considered its motion along its own path: the ecliptic. This path is the fundamental plane for the ecliptic coordinate system. So the sun moves only in ecliptic longitude1, but because it’s path defines the ecliptic, it doesn’t have any motion in ecliptic latitude. This is not true for the moon which bobs up and down with respect to the ecliptic by about $5º$. Thus, we’ll also need to consider this motion which also has its own anomalistic motion. So now we don’t only need to worry about the mean motion in ecliptic longitude and its anomaly, we’ll also need to worry about the motion in ecliptic latitude which also has anomalistic motion.
Ptolemy states this by saying,
The moon’s motion appears anomalistic both in [ecliptic] longitude and latitude: the time it takes to traverse the ecliptic is not constant, and neither is the time it takes to return to the same latitude.
The second is that, for the sun, the period of the mean motion and the period of anomalistic motion were the same; They both returned to the same point in their cycle every year. For the moon we will again have both a mean motion and anomalistic motion, but it turns out they don’t have the same period.
The result of this is that the moon’s motion is relatively messy. The moon will not traverse the same distance around the celestial sphere even if you take the same interval of time starting and ending on the same date. This means that the moon’s path doesn’t make a closed circuit such that, when it returns to the same position in ecliptic longitude, it is not at the same latitude above or below the ecliptic2 nor does it have the same anomalistic speed. We’ll need to wait for the moon to make many cycles before all of these line up again.
An analogy of this would be cars lined up in a turn lane all with their turn signal on. They all have slightly different periods and if they all start at exactly the same time, they will quickly fall out of sync. But if you wait long enough, they will all converge again and a new cycle will start.
And that’s what this chapter is really about. What Ptolemy wants to look at is, if on a certain date, the moon has a certain ecliptic latitude and longitude with a certain anomalistic motion, how long will it take for all of those values to line up again? If we know that, we can divide that overall cycle by the number of smaller cycles from each individual motion to be able to determine the precise period of each of those different cycles.
Hence the ancient astronomers, with good reason, tried to find some period in which the moon’s motion in longitude would always be the same, on the grounds that only such a period could produce a return in anomaly.
So how did they find such a period? As you might expect from the last chapter, they looked to lunar eclipses. The Babylonians were prolific note-takers for astronomical events and recorded numerous lunar eclipses. So astronomers started looking through these records seeing if they could find a long standing pattern. This pattern would be one
consisting of an integer number of months, such that, between whatever points one took that interval of months, the length in time was always the same, and so the motion [of the moon] in longitude, [i.e.] either the same number of integer revolutions, or the same number of revolutions plus the same arc.
Ptolemy then cites “even more ancient [astronomers]3” who claimed to have found such a period of $6585 \frac{1}{3}$ days. This corresponded to
$223$ lunations, $239$ returns in anomaly, $242$ returns in latitude, and $241$ revolutions in longitude plus $10 \frac{2}{3}º$, which is the amount the sun travels beyond the $18$ revolutions which it performs in the above time (that is, when the motion of the sun and moon is measured with respect to the fixed stars).
This passage needs some explaining as Ptolemy seemingly pulls these numbers out of nowhere. First, a lunation is just a full lunar cycle: From full moon to full moon. So what Ptolemy is saying here is that in this period of $6585 \frac{1}{3}$ days, the moon went through $223$ cycles of the phases.
Next is returns in anomaly. This is a bit harder to understand as it’s not something that is quite so easily observed as the phase. However, the moon’s anomaly is actually quite pronounced. It’s daily motion in ecliptic longitude varies between about $10-14º$ per day. So while we don’t really think about such things, it actually is relatively easy to determine how many times the moon varied between these extremes.
Lastly, is the number of times the moon went through its cycle in latitude. Again, this isn’t something we think about often, but because the moon goes from about $5º$ below the ecliptic to $5º$ above it, that’s an easily observable $10º$ bounce. Thus, the numbers of this should be easy to determine too.
In truth, I’m certainly over simplifying this as I’m quite skeptical that astronomers really tracked such things in detail and “counted” them as I’m implying here. Rather, my suspicion is that from observing each of these periods we just discussed, the astronomers had a fairly good rough approximation on what the period was. So they probably approached this problem a bit backwards, taking this long period and dividing it by their first approximation individual period. But since the full period could only be an integer number of each of the individual periods, whatever whole number it ended up being close to would be the true number of cycles.
That’s a bit vague, so let’s walk through some examples starting with the lunations. As we stated, a lunation is the time time from one full moon to the next. The modern value of this is $29.53$ days. Dividing $6585.33$ days by this results in $223.01$ days which rounds quite nicely to Ptolemy’s $223$ days. So even if those astronomers were slightly off in their estimate in the mean period of a single lunation, the result would still get them close enough to $223$ days.
We can similarly do this for the up-down motion of the moon from the ecliptic which is the time taken to return to the same latitude. Another name for this is the “draconitic” or “draconic” month. This is slightly less than the longitude with the modern value being $27.21$ days. Again dividing this gives $242.02$ days which again rounds down to the $242$ days given by Ptolemy.
Next, we have the returns in anomaly. This is, predictably, the anomalistic month which has an average period of $27.55$ days for a modern value. Again, if we divide the total period by this, we get $239.03$ anomalistic months, which is indeed quite close to Ptolemy’s rounded off $239$.
Lastly, we can ignore the moon entirely and divide the period by the solar year ($365.25$ days) to get this to be a period of $18.03$ years. This is the $18$ years Ptolemy gives. However, that dangling $0.03$ years ($~11$ days), when converted to degrees is that $10 \frac{2}{3}º$ Ptolemy mentions.
But wait… I skipped one: the ecliptic longitude! Let’s do this one real quickly: The modern value for a return in ecliptic longitude (known as the sidereal month) is about $27.3$ days. So again dividing $6585$ by that gives $241.22$ cycles. That’s really not close to a nice integer number. What gives?
The answer is that The moon actually didn’t complete an integer number of this type of cycle in that period. Recall that this large period based based on lunar eclipses, which only happens when the moon is opposite the sun. Thus, the moon actually had to move a little extra to get back into position for the eclipse. which is why Ptolemy added the extra $10 \frac{2}{3}º$ to it.
So before moving on, let’s make sure we have the terminology straight here because Ptolemy flips back and forth frequently, and I may well too:
Lunation = Synodic month
Return in longitude = Sidereal month
Return in latitude = Draconitic/Draconic month
Return in anomaly = Anomalistic month
Now, let’s briefly look back at that value of $6585 \frac{1}{3}$ days. This is sort of a funky number since it contains that dangling $\frac{1}{3}$. As such, these ancient4 astronomers multiplied everything by three to get a total period of $19,756$ days. They called this period the “exeligmos”5.
However having established this, Ptolemy immediately tosses it out and cites Hipparchus who disputed this number, and instead used a period of $126,007$ days plus an equinoctial hour. Pedersen refers to this period as the Hipparchian period.
Again, this period was determined by Hipparchus by comparing his observations of eclipses from Babylonian records of eclipses hundreds of years earlier as compared to ones he may have observed. The precise method we’ll explore in a few posts.
Ptolemy does the same thing we just did above, stating how many cycles of each type of motion there would be in that period:
$4,267$ months, $4,573$ complete returns in anomaly, and $4,612$ revolutions on the ecliptic, less about $7 \frac{1}{2}º$6, which is the amount by which the sun’s motion falls short of $345$ revolutions.
Dividing the number of days by the number of months gives us the length of a synodic month. Ptolemy gives this as $29;31,50,8,20$ days7.
Ptolemy then notes
the corresponding interval between two lunar eclipses is always precisely the same when they are taken over the above period [$126,007$ days $1$ hour]. So it is obvious that it is a period of return in anomaly since [from whatever eclipse it begins], it always contains the same number [$4,267$] of months, and $4,611$ revolutions in longitude plus $352 \frac{1}{2}º$
Essentially, Ptolemy restates the original premise on which we founded this determination of periods and cycles.
While the Hipparchian cycle is built on a repeating cycle of eclipses, Ptolemy also expounds on what would happen if we looked for a period based on simple syzygys8. In other words, if we consider a full moon, is there a period of full moon cycles in which there is a return in longitude and anomaly?
This is easy to arrive at from what we have above as any period based on eclipses will require a return in ecliptic longitude and anomaly. The number of returns in latitude in the Hipparchian period was $4,267$ synodic months, and the number of returns in anomaly in that period was $4,573$. If we find the largest common factor between then and divide both by that, then that would represent the number of months (synodic and anomalistic) that will represent a full cycle. It turns out that $17$ is the only common factor. So if we divide both by $17$, we get that there is a common return in ecliptic longitude and anomaly every $251$ synodic months and every $269$ anomalistic months.
However, that’s based on that Hipparchian period which still isn’t perfect because it
did not contain an integer number of returns in latitude.
In other words, it gets the moon back to the same longitude, with the right anomaly, but the latitude is still off9. It’s very close to a return in latitude as well which should be obvious since it was based on eclipses and an eclipse could not occur if the latitude were not very close to 0º since then the moon would not fall in the Earth’s shadow. But rather, it’s off by just enough that it results in eclipses with different “size and type of obscuration10.”
To try to smooth things over, Ptolemy tells a story about how Hipparchus deduced a relationship between the number of lunations (synodic months) and returns in latitude (draconitic months)11. Regardless of how it was derived, Ptolemy gives the relationship that $5,458$ lunations12 = $5,923$ returns in latitude13.
That gives us a pretty good understanding of the basic lunar periods, but there’s still some wrinkles. The first, which we’ll start exploring in the next post, is that, as we noted above, eclipses aren’t solely caused by the moon returning to the same ecliptic longitude and latitude with the same anomaly, but are also dependent on the position of the sun. The result of this is that the ancient astronomers Ptolemy cited here, needed to be extra careful to ensure that the durations they cited weren’t being impacted by the anomaly of the sun. We’ve glossed over that here and simply presented figures which already took this into consideration, but Ptolemy still wants to make sure we understand a bit more about this, which we’ll explore in the next post.
In addition, we’ve also talked about the moon’s anomalistic motion but need to understand more about how it will impact when eclipses occur. With both of those added to our knowledge, we’ll then put these pieces together to determine what sorts of lunar eclipses are best to minimize the impact these can have on things.
Happy New Year to everyone!
- Toomer’s translation generally drops the “ecliptic” part when describing ecliptic latitude and longitude. However, I prefer to keep it because we will later be discussing terrestrial latitudes and longitudes. To help distinguish, I will try to leave the note the “ecliptic” in even if it is obvious from context.
- i.e., in ecliptic latitude.
- Likely Babylonian.
- Most likely Greek.
- The previous figures for lunations, returns, etc… can also be tripled and Ptolemy does this.
- There’s a mathematical error here. This should be calculated by multiplying the number of days by the mean motion of the sun (and then subtracting out the multiples of $360º$). If that is done, Pedersen notes that it only gives about $3 \frac{1}{2}º$ short of another full revolution. While this may be due to a simple mathematical error, Pedersen also notes that if we divide the number of synodic months ($4,267$) by the length of the year as calculated by the Babylonians ($12;22,8$ synodic months), you get $344;58,42$ years which is $345 – 0;1,17$ years where $0;1,17$ years is $7;42º$ which is pretty close to what Ptolemy cites. Hence, this may be due to a poor choice of calculation methods.
- Pedersen notes that if you actually perform this calculation yourself, you’d get $29;31,50,8,9$ days. He notes that this is only a discrepancy of about $\frac{1}{12}$ of a second. This went unnoticed in Ptolemy’s time, but is in fact commented upon by Copernicus in the mid $1500$’s.
- This is a term I’ve not used previously in this blog to my recollection. It just means when an object is in a straight line with the Earth and sun, i.e., a conjunction or opposition.
- Hence why he did not include it in the figures above.
- Toomer notes that by “type” Ptolemy means whether the obscuration begins from the north or south of the lunar disk.
- I say a story here because both Toomer and Pedersen call it out as false as the relationship was well known to Babylonian astronomers before Hipparchus. They both site Kugler’s Babylonische Mondrechnung if you speak German.
- I should note that Ptolemy frequently refers to synodic months or lunations as just “months” here which threw me for a bit.
- This can easily be confirmed by multiplying the number of synodic months by the number of days in each one, then dividing by the number of days in the draconitic month. Really, we could have chosen any number of synodic months to start with, but this was likely chosen because the result is that both are expressed in an integer number of months.