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Data: Stellar Quadrant Observations – 8/21/2020

This week a thin crescent moon set early along with clear skies and pleasant evening temperatures set up another another promising night of observing. I was joined again by Their Excellency Yseult and we headed back out to the Danville Conservation Area. While a bit more of a drive, the impressively dark skies are certainly worth it as it opens up opportunities to add stars that we couldn’t see elsewhere to the catalog. Surprisingly, the nice night didn’t attract many other observers, but an astrophotographer named Will Day stopped by with his wife and child who had never been to the site before. He asked if we minded being included in pictures and we happily agreed. I think the results turned out well.

Overall, we took observations on 36 stars and the three visible planets that night, most of which we took turns observing as we could then use each of our results as a quick reality check against one another and hopefully produce better results. Continue reading “Data: Stellar Quadrant Observations – 8/21/2020”

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Almagest Book IV: Correcting the Lunar Mean Motion in Latitude

Previously, we looked at how Ptolemy made corrections to the anomalistic motion for his lunar model. In this post, we’ll be doing something similar for the mean motion of lunar latitude.

Ptolemy explained that the value we originally noted was in error,

because we too adopted Hipparchus’ assumptions that [the diameter of] the moon goes approximately 650 times into its own orbit, and $2 \frac{1}{2}$ times into [the diameter of] the Earth’s shadow, when it is at mean distance in the syzygies.

In short, Hipparchus’ figures were a good starting point but now we can do better by

using more elegant methods which do not require any of the previous assumptions for the solution of the problem.

Continue reading “Almagest Book IV: Correcting the Lunar Mean Motion in Latitude”

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Data: Stellar Quadrant Observations – 8/14/2020

This week’s weather was supposed to be dismal. And for many of my friends, it was. The derecho that ripped across the midwest left many people in the kingdom without power for several days. However, for me, it pulled the clouds that were supposed to last through the weekend out early, leaving me with a cloudless friday night with a waning crescent moon. About as good as I can ask for for observing. However, the humidity did linger with humidity readings upwards of 70%. While it didn’t feel uncomfortable due to the reasonable temperatures (low 70’s), it did mean that my glasses fogged quite badly and all the paper I take notes on was soggy by the end of the night. Despite that, last night was easily the most productive night I’ve yet had. I totalled 71 stellar observations and 3 planets.

Continue reading “Data: Stellar Quadrant Observations – 8/14/2020”

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Almagest Book IV: The Epoch of the Mean Motions of the Moon in Longitude and Anomaly

As we saw for the sun, to be able to use the tables of mean motion to predict the position of the moon, we’ll need to know where the moon was at a specific point in time. Ptolemy chooses as that point in time as the beginning of the first year of the Nabonassar reign. To determine the position of the moon on this date, Ptolemy starts with the second eclipse of the Babylonian triple we discussed in the last chapter and then calculates backwards. Continue reading “Almagest Book IV: The Epoch of the Mean Motions of the Moon in Longitude and Anomaly”

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Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly

Towards the beginning of Book IV, Ptolemy went through the methodology by which the various motions in the lunar mean motion table could be calculated. But if you were paying extra close attention, you may have noticed that the values that ended up in the mean motion table didn’t actually match what we derived. Specifically, for the daily increment in anomaly, we derived $13;3,53,56,29,38,38^{\frac{º}{day}}$. But in the table, we magically ended up with $13;3,53,56,17,51,59^{\frac{º}{day}}$. Identical until the 4th division.

So what gives? Why did Ptolemy derive one value and report another?

In Chapter 7, he gives the explanation: He found the value needed to be corrected and did so before he put it in the table. But before he could explain to us how, we needed to cover the eclipse triples we did in Chapter 6. So how do we apply them to check the mean motions? Continue reading “Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly”

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Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon

We’re almost finished with chapter 6. All that’s left is to determine the position of the mean moon during one of the eclipses which will tell us the equation of anomaly at that point. To do so, we’ll add a few more points to the image we ended the last post with:

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

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Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle

Continuing on with Ptolemy’s check on the radius of the epicycle, we’ll produce a new diagram based on the positions of the Alexandrian eclipses. However, instead of doing it piece-by-piece as I did when we explored the Babylonian eclipses, I’ll drop everything into a single diagram since we already have some experience and the configuration for this triple is a bit more for forgiving on the spacing:

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle”

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Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly

Modern commentary on Ptolemy often downplays the Almagest because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the Almagest was truly novel1. But I would hasten to remind that, while Ptolemy stood on the shoulders of those who came before, he certainly climbed there on his own, not simply accepting their results, but doing his best to validate them.

And we’re about to get a big dosing of that, because all the work we’ve done in the past three posts, we’ll be redoing with a new set of eclipses observed by Ptolemy himself, allowing for an independent check on the important value of the radius of the epicycle.

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly”

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Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon

In the last post, we were able to determine the radius of the epicycle when the radius of the deferent is $60^p$. It took a lot of switching between demi-degrees contexts, but in the wake of all that math, we’re left with a mess of lines and arcs that we’ve already determined. So in this post, we’ll use that starting point to go just a little further and determine the position of the mean moon, specifically for the second eclipse. To do so, we’ll need to add a bit more to the configuration we ended with last time:

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

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Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle

In the last post, we introduced three eclipses from Babylonian times which we used to build a couple intervals: The first eclipse to the second, and the second to the third. Using those, we used lunar and solar mean motion tables to figure out the solar position, as well as its change. Since the moon must be opposite the sun in ecliptic longitude for an eclipse to occur, we used the change in solar position to determine the true change in lunar position in these intervals. From that, we could compare that to the mean motion to determine how much of it must be caused by the lunar anomaly. But while we’ve determined this component, we haven’t done anything with them yet.

So in this post, we’ll start using these to answer several questions that will build out the details of the model. Specifically, we want answers to questions like what is the radius of the epicycle? Where, in relation to the ecliptic was the mean moon during these eclipses and what was the equation of anomaly? That’s a lot of information to extract so I’m going to try to break it up a bit and in this post, we’ll only tackle the radius of the epicycle2

To begin, let’s sketch out the epicyclic lunar model3 with the three eclipses drawn on it.

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle”