Author: VoijaRisa

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Almagest Book IV: The Lunar Anomaly and Eclipses

In the last post, we covered how the sun’s anomaly impacts things, but

we must pay no less attention to the moon’s [varying] speed. For if this is not taken into account, it will be possible for the moon, in many situations, to cover equal arcs in longitude in equal times which do not at all represent a return in lunar anomaly as well.

I’ll preface this section by saying this is, to date, by far the hardest section I’ve grappled with. I believe a large part of the difficulty came from the fact that Ptolemy is exceptionally unclear about what his goal is with this section. My initial belief was that it was to find the full period in which a the position of the sun and moon would “reset” as discussed in the last post. However, that’s something we’re going to have to work up to.

For now, we’re going to concentrate on just one of the various types of months. Namely, the “return in lunar anomaly” which is another way of saying the anomalistic month. Continue reading “Almagest Book IV: The Lunar Anomaly and Eclipses”

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Data: Stellar Quadrant Observations – 4/21/20

I managed to get out and do a bit of observing last night. It’s been 4 months to do the day since my last opportunity to do so and I started off by making a rather large blunder. The location I usually observe at has several pads for observing and I have one I usually use. It’s straight down a path that points generally north which makes finding the north star fairly easy, even without looking for the big dipper to verify which star it is.

Well, last night I looked for the star over where the path met the parking lot, but because I was on a pad a bit to the west of where I normally am, I accidentally aligned on Kochab, the second brightest star in Ursa Minor, and ended up making three observations over the course of a half hour before I realized the error. Fortunately, we still got in 11 observations, and all of them were new stars! Continue reading “Data: Stellar Quadrant Observations – 4/21/20”

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Almagest Book IV: The Solar Anomaly and Lunar Periods

In the last post, we explored various lunar cycles from astronomers predating Ptolemy in which the moon reset its ecliptic longitude and anomalistic motion to define a full lunar period. These ancient astronomers did this by studying pairs of lunar eclipses1but Ptolemy notes that this method

is not simple or easy to carry out, but demands a great deal of extraordinary care

The reason for this difficulty is that, without careful consideration there can essentially be false positives of eclipses separated equally in time, but do not in fact, result in the moon returning to the same ecliptic longitude or same speed.

One of the reasons is that the conditions necessary to produce a lunar eclipse are also dependent on the sun, which has anomalistic motion. As such, it could be entirely possible that the moon could not have yet returned to the same ecliptic longitude as a previous eclipse, but the sun’s anomaly could cause an eclipse anyway.  Thus, a pair of eclipses may be equally separated in time, but

this is no use to us unless the sun too exhibits no effect due to anomaly, or exhibits the same [anomaly] over both intervals: for if this is not the case, but instead, as I have said, the equation of anomaly has some effect, the sun will not have travelled equal distances over [the two] equal time intervals, nor, obviously, will the moon.

To illustrate this, Ptolemy starts with an example. Continue reading “Almagest Book IV: The Solar Anomaly and Lunar Periods”

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Data: Stellar Quadrant Observations – 12/21/19

Happy Winter Solstice to all. Despite wanting to get out at solar noon today to get an observation of the sun, I was asleep at that time as I have a cold that’s sucking all my energy despite not making me feel all that bad.

However, tonight was also a late moon phase so doing stellar observations was on the schedule. Quieteria helped me with these observations, but as it was very cold tonight, we called it quits fairly early. And because we’re in a rather blank patch of sky with mostly stars well past 4th magnitude, that only led to a fistfull of observations. Continue reading “Data: Stellar Quadrant Observations – 12/21/19”

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Almagest Book IV: Observations Necessary to Examine Lunar Phenomena

So far, Books I & II covered the motions of the sky and how to find the rising times of various points along the ecliptic. This was a good start because, in Book III, we explored the motion of the Sun which is confined to that ecliptic. So while the sun was somewhat complex because of its anomaly, it was still relatively simple. In Book IV, we’ll work on deriving a model for the motion of the moon.

Unfortunately, this is going to be a more complex model. Initially we could be concerned about the complexity of the model because the moon is not confined to the ecliptic – it bobbles above and below it by about 5º, but aside from discussing this briefly, we’ll safely ignore this for now and instead only worry about the moon’s motion in ecliptic longitude, that is to say, its projection onto the ecliptic.

However, what will complicate things is that one of the main things we consider regarding the moon, its phase, is also dependent on the sun. Thus, to consider the moon’s phases, we’ll need to be taking into consideration the sun’s anomalies at the same time we consider those of the moon. In addition, the points at which the moon is at apogee and perigee is not consistent as it was for the sun2.

The good news is that we’ve already explored the two models that Ptolemy uses to explain anomalies from the mean motion. As such, there will be far less exposition in this book and we’ll be able to dive in much more quickly. Continue reading “Almagest Book IV: Observations Necessary to Examine Lunar Phenomena”

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Almagest Book III: On the Inequality of Solar Days

Finally, we’ve arrived at the end of Book III where we’ve arrived at a well developed model of the solar motion. But before closing out, Ptolemy has one last chapter to discuss the inequality of the solar day. Ptolemy states the problem as follows:

the mean motions which we tabulate for each body are all arranged on the simple system of equal increments, as if all solar days were of equal length. However, it can be seen that this is not so.

What Ptolemy is really getting at here is that the term “day” is somewhat ambiguous. As such, the different ways by which we might measure a “day” are explored in this chapter. Continue reading “Almagest Book III: On the Inequality of Solar Days”

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Almagest Book III: On the Calculation of the Solar Position

This chapter is easily the shortest one in Book III. It literally consists of a paragraph (which I’ll quote in its entirely but break into two for ease of reading) that gives a very quick description of how one calculates the position of the sun at any given time from the epoch derived in the last chapter.

So whenever we want to know the sun’s position for any required time, we take the time from epoch to the given moment (reckoned with respect to the local time at Alexandria), and enter with it into the table of mean motion. We add up the degrees [and their subdivisions] corresponding to the various arguments [18-year periods, years, months, etc.], add to this the elongation [from apogee at epoch], 265;15º, subtract the complete revolutions from the total, and count the result forward from Gemini 5;30º rearwards through [i.e. in the order of] the signs. The point we come to will be the mean position of the sun.

Next we enter the same number, that is the distance from apogee to the sun’s mean position, into the table of anomaly, and take the corresponding amount in the third column. If the argument falls in the first column, that is if it is less than 180º, we subtract the [equation] from the mean position; but if the argument falls in the second column, i.e. is greater than 180º, we add it to the mean position. Thus we obtain the true or apparent [position of] the sun. Continue reading “Almagest Book III: On the Calculation of the Solar Position”

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Almagest Book III: On the Epoch of the Sun’s Mean Motion

We’ve come a long way in this book establishing a working model for solar motion. In fact, we’ve explored two models and derived a table that shows how far the sun would be away from its mean motion based on the mean position. However, at this point, everything has been done in terms of apogee and perigee.

In this chapter, we’ll be defining the “epoch“. What that means is that Ptolemy is going to pick a point in time, and define where the sun was on that date. Then, applying the tools we have developed in this book, we’ll be able to determine where the sun is at any other given date using the epoch as the starting point. For those that like things in a bit more mathy terms, it’s the location of the sun on the eccentre at time = 0, wherein Ptolemy will decide what that date is. To get there, we’ll first establish precise point on the eccentre at a known point in time, and then use the methods from this chapter to go backwards until we get to the chosen epoch date. Continue reading “Almagest Book III: On the Epoch of the Sun’s Mean Motion”