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Eclipse Limits for Solar Eclipses – Latitudinal Parallax: Alternate Method

When writing the post on finding the latitudinal parallax as part of determining the limits for eclipses, I commented in a footnote that I’d developed a different method for determining this. While Ptolemy’s methods are reasonably accurate, I figured I should go ahead and share the one I came up with using the first case (when the sun is at the summer solstice and the moon is south of the ecliptic from Meroe) as an example. To do so, let’s consider again the configuration of the sun and moon in that instance.

Continue reading “Eclipse Limits for Solar Eclipses – Latitudinal Parallax: Alternate Method”

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Data: Stellar Quadrant Observations – 12/3/21

The weather so far the past few weeks has been unseasonably warm. While there have certainly been days where the temperature has barely gotten above $40º$, we’ve also had quite a few where it’s gotten back into the $70$’s prompting me to have my windows open… in December.

One of these nice weather nights ended up happening on a Friday with a near new moon so I headed out to Danville Conservation Area to do some observing. This time I brought Mielikkä Koiviston along who is new to observing. As a result, a lot of our time was spent discussing the night sky and how things work and we only got in $30$ measurements in the time we were out.

The night started off pretty well, doing some observing in Aquarius and Pegasus, picking up five stars for which I previously didn’t have any observations. But as the evening wore on, I wanted to jump over to Orion because it’s a fun constellation to go through as there’s so much to do. Unfortunately, something happened around that time (likely the azimuth ring got bumped and out of alignment) because every measurement was suddenly well over a degree off. I ended up tossing a good chunk of the data as a result, but kept $19$ observations.

As usual, the data can be found in the Google Sheet.

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Almagest Book VI: Eclipse Limits for Solar Eclipses – Solar & Lunar Anomalies

So far, when considering the distance the sun/moon can be from one of the nodes, we’ve worked out how much the longitudinal and latitudinal parallax impact things and all that’s left now is the fact that the sun and moon aren’t always at their mean position. They both have anomalies which we’ll need to consider. This is because the big goal of this book, so far, is to reduce the amount of math we have to do when checking for an eclipse. While we could go through all the effort of calculating the true position, that’s extra steps. Wouldn’t it be nicer if we could just stop at the mean position if it’s not in the window in which an eclipse can occur?

To that end, our final step in this series of posts exploring the limits for solar eclipses is to translate the true positions to the mean positions.

Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Solar & Lunar Anomalies”

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Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax

Now that we’ve determined how much further from the nodes parallax can cause solar eclipses to occur due to the latitudinal parallax, we need to consider the longitudinal effect. As with the last post, Ptolemy is absolutely no help in this. He simply tosses out some values with no explanation or work stating

When [the latitudinal] parallax is $0;08º$ northwards1, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards2, it has a maximum longitudinal parallax of about $0;15º$…

Seeking some assistance, I again refer to Neugebauer and Pappus, but immediately run into an issue. Neugebauer minces no words and states

Ptolemy is wrong in stating that $p_\lambda = 0;30º$ and $p_\lambda = 0;15º$ are the greatest longitudinal components of the parallax for locations between Meroe and the Borysthenes. It is difficult to explain how he arrived at this result.

Well… this will be interesting to try to untangle then. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax”

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Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax

Now that we’ve determined how far away from the nodes a lunar eclipse can occur, we’ll work on doing the same for a solar eclipse3. But before diving in, I want to say that this has been one of the most, if not the most challenging section of the the Almagest so far. One of the primary reasons is that Ptolemy shows no work and gives almost no explanation on how he did this. When such things happen, I often turn to Neugebauer’s History of Ancient Mathematical Astronomy which I did in this case. There, Neugebauer refers to Pappus of Alexandria, a fourth century mathematician who did commentary on the Almagest and walks through a process that arrives at the same values as Ptolemy.

However, there was a very large amount to unpack in just a few pages there and, unlike most cases where I can simply work along with it and see where things are going, this time I had to really understand the whole process before the first steps made any sense. This led me to agonize over what was going on with those first steps, amounting to several days of effort and rewriting this post from scratch several times. The result is twofold. First because I feel this section can only be approached by understanding the methodology before diving into the math, there’s going to be far more exposition than normal and, as a result, this is likely to be one of my longer posts. Second, the struggles I had with trying to understand the method and rewriting this post so many times has left me with a lot of fragments of thoughts in my brain and in the blog editor. I’ve done my best to clean it up, and maybe it’s just those thoughts swirling around in my brain, but this post just doesn’t feel as coherent as I like. Apologies in advance if you struggle to follow. Know I did as well.

Anyway, moving on to the topic at hand.

Normally, I like to start with a quote from Ptolemy to give us some direction, but I think Ptolemy did such a poor job of laying this section out, I’m going to avoid doing so for the majority of the post. Instead, let’s try to understand the process by recalling what we did with the moon and discussing how things will change. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax”

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Mean to True Conjunction Alternate Method

In our last post, we walked through Ptolemy’s method for finding the angular distance between the mean and true conjunction.

When I initially wrote the post, I followed Neugebauer’s explanation as Ptolemy’s was quite difficult to parse and although they contain a lot of the same key elements, one important piece is left out of Neugebauer’s solution. Specifically, the part where Ptolemy’s iterative method leads naturally to the increase over the anomaly of $\frac{1}{12}$. Writing a later post, I realized that this was an important piece of information since it pops up later and thus made the effort to more completely understand Ptolemy’s method and rewrote the post to explain it. However, I didn’t want to lose the original work, following Neugebauer since readers may appreciate some explanation of Neugebauer’s work as it too is quite dense. Thus, I’ve included that original text beneath the fold as a separate post. Continue reading “Mean to True Conjunction Alternate Method”

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Almagest Book VI – Lunar Eclipse Limits

Way back in Book V we determined the angular diameter of the moon as well as earth’s shadow at apogee. In the last post, we repeated the procedure for perigee. In the Almagest, Ptolemy doesn’t actually say what those calculations are for and instead, starts working out some figures for the sun. However, to try to keep things in a more reasonable flow (in my opinion), I’m going to skip to the end of this chapter and discuss why we care about the moon’s diameter and earth’s shadow.

In short, lunar eclipses can only happen near the lunar nodes. But, it doesn’t have to be exactly at a node. First off, the earth’s shadow has some width to it. In addition, the anomalies of the sun and moon play a role, which means the actual range the eclipse could occur in is surprisingly wide. So in this post, we’ll work on that. Continue reading “Almagest Book VI – Lunar Eclipse Limits”

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Data: Stellar Quadrant Observations – 11/6/21

It has been seven months since I have done any observing. Between unfortunate weather during good moon phases this summer, being exhausted from mundane life things, running an event… it just didn’t happen.

But now that a lot of that is behind me, we had a crisp fall night with daytime temps in the 60’s and lows in the 40’s. Chilly, but entirely manageable. Especially since I purchased a heated vest last winter. I’ve been looking for a pair of heated pants to go with it, but haven’t found any with sufficiently good reviews at a decent price just yet.

Yseult joined me again, and we got through 60 observations. Continue reading “Data: Stellar Quadrant Observations – 11/6/21”

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Almagest Book VI: Lunar Diameter and Earth’s Shadow at Perigee During Syzygy

In the last post, we explored how to make use of the table of mean syzygies to calculate the true syzygies. However, that chapter focused mostly on finding the time when the moon and sun would have either the same or exactly opposite ecliptic latitude. But what got left by the wayside was the lunar ecliptic latitude. We did a bit of work on calculating the argument of it but, aside from my mention of it in the afterword of the post, we never really completed that calculation. And eclipses of either type cannot truly occur unless the lunar ecliptic latitude is reasonably close to zero.

So we could calculate the ecliptic latitude of the moon for every conjunction and opposition but instead, Ptolemy decides we should first do a bit of a sanity check before getting any more involved. To do so, Ptolemy wants to examine how far from a node is it even possible for the ecliptic longitude of the syzygy to occur and still have an eclipse. If it’s outside of these limits, then no further calculation is necessary. To do this, Ptolemy is going to need to know some additional values. In this post, we’ll explore the angular diameter when the moon is at the perigee of its epicycle at syzygy4 as well as determining the width of Earth’s shadow at that distance. Continue reading “Almagest Book VI: Lunar Diameter and Earth’s Shadow at Perigee During Syzygy”