Another year behind us. Sadly, this year was not everything I’d hoped for in terms of this astronomy project. Continue reading “2021 in Review”
Almagest Book VI: Solar Eclipses Separated by Five Months
Ptolemy next looks at whether or not it is possible for a solar eclipse to occur five months after a previous one. We’ve already done a fair bit of the heavy lifting for this topic as some of the math we did when considering lunar eclipses separated by five months will still apply. In that post, we determined that the moon would have moved on its inclined circle by $159;05º$ between true conjunctions. This does require we adopt the same assumptions of the sun moving its greatest distance and the moon moving its least.
What we’ll need to focus on for this post is redoing the eclipse limits for the situation in question. Continue reading “Almagest Book VI: Solar Eclipses Separated by Five Months”
Almagest Book VI: Lunar Eclipses Separated by Seven Months
So far in this chapter, we’ve demonstrated that it’s quite possible that, if either a solar or lunar eclipse occurs, there will be another one six months later. Then, in the last post we showed that, if a lunar eclipse happens, it’s possible (although unlikely) that there can be another one five months later. In this post, we’ll explore whether or not, for lunar eclipses, you can have two separated by seven months.
The procedure will be the same as in the last post. First we’ll determine the motion of the luminaries, taking into account the discrepancies due to the anomalies, as well as the differences between mean and true syzygy, and then compare those to the eclipse windows which we’ll again recalculate for this situation. Continue reading “Almagest Book VI: Lunar Eclipses Separated by Seven Months”
Almagest Book VI: Lunar Eclipses Separated by Five Months
In the previous post, we showed that , if a solar or lunar eclipse occurs, it is possible that another may occur six months later. Now, we’ll turn to ask whether or not another lunar eclipse can happen five months after a previous one. To answer this question, we’ll first work out how much the moon would have moved in that time period and then compare that to the eclipse window.. Continue reading “Almagest Book VI: Lunar Eclipses Separated by Five Months”
Almagest Book VI: Solar and Lunar Eclipses Separated by Six Months
Continuing in the theme of checking as few as possible syzygies for eclipses, Ptolemy now turns his attention towards
the problem of intervals at which, in general, it is possible for ecliptic syzygies to occur, so that, once we have determined a single example of of an ecliptic syzygy, we need not apply our examination to the [ecliptic] limits to every succeeding syzygy in turn, but only to those which are separated [from the first] by an interval of months at which it is possible for an eclipse to recur.
Continue reading “Almagest Book VI: Solar and Lunar Eclipses Separated by Six Months”
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”
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.
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”
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”
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”