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Data: Stellar Quadrant Observations – 5/4/19

This year has been pretty rough for observing. With the quadrant being damaged and having to toss out a night of data in February, combined with cloudy weather in March and April, I haven’t been able to get much done.

But last night the weather cooperated and I was able to take the quadrant out, this time with the assistance of Megan doing the recording, and me doing the sighting. While I’m still not as good as Padraig, the results were fairly good and posted below the fold.

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Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships

We’re almost to the end of book II. There’s really 2 chapters left, but the next one is almost entirely a table laying out the values we’ve been looking at here recently, so this is the last chapter in which we’ll be working out anything new.

In this chapter, we’ll tackle the angle between the ecliptic and a “circle through the poles of the horizon”. If you imagine standing outside, the zenith is directly overhead which is the pole for your local horizon. Directly opposite that, beneath you, is the nadir. If these two points are connected with a great circle, that’s the great circle we want to find the angle of with respect to the ecliptic. Because we measure upwards, from the horizon, along an arc of these great circles, to measure the altitude of a star, these are often called altitude circles.

But while we’re at it, Ptolemy promises that we’ll also determine “the size of the arc…cut off between the zenith and…the ecliptic.” In other words, because the ecliptic is tilted with respect to the horizon, the arcs between the two will be different.

To get us started, Ptolemy begins with the following diagram.

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Almagest Book II: Angle Between Ecliptic And Horizon – Calculations

We’ll continue on with our goal of finding the angle the ecliptic makes with the horizon. Fortunately, this task is simplified by the symmetries we worked out in the last post meaning we’ll only need to work out the values from Aries to Libra. Unfortunately, this value will change based on latitude as well as the position on the ecliptic, but we’ll still only do this for one location. And for that location, Ptolemy again uses Rhodes.

First we’ll start with angles at the equinoxes:

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Almagest Book II: Angle Between Ecliptic and Meridian – Angle Calculations

Now that we’ve gotten a few symmetry rules developed, we can return to the main objective of calculating the angle between the ecliptic and meridian at different points along the ecliptic. Specifically, Ptolemy sets out to do this at the first point in every sign. But thanks to the previously derived symmetries, we’ll save ourselves a bit of work.

First Ptolemy does some very short proofs for these angles at the meridian and solstice, and then a slightly more complex one for the signs between them.

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Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries

In my last post, I mentioned that entered a paper based on the rising sign calculations presented in this post into an A&S competition. This was a very interesting piece to do because it showed how well woven the roots are, as doing so made use of almost every section we’ve gone through previously. As such, it felt like a good capstone for book II. But it doesn’t end there.

Rather, Ptolemy decides to go on for several more chapters as this book is focused on the great circles on the celestial sphere. While we’ve covered the ecliptic and celestial equator pretty extensively, we have done less with the horizon and meridian which is where Ptolemy seeks to go for the last few chapters in this book. Specifically, we’ll be covering:

  • The angles between the ecliptic and meridian
  • The ecliptic and horizon
  • The ecliptic and an arc from horizon to the zenith (an altitude circle)

All followed by another summary chapter at various latitudes. As the title of this post may have indicated, we’ll be covering the first of these in this post1. Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries”

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Queen’s Prize Winter 2019 – Computing Astronomical Rising Signs for Any Latitude & Excel Calculator

Although I only briefly mentioned it in this post, this past summer I’d taken the quadrant to Queen’s Prize which is Calontir’s novice level A&S competition held once during each reign. One of the challenges I faced was that the quadrant didn’t entirely fit in any of the categories. While it was designed to function like a period instrument, it was built using decidedly modern methods, and the goal of the quadrant wasn’t the instrument itself, but the measurements it could take. So the overall reaction from the judges was “super cool, but it’s hard to judge on its own merits.”

Thus, it was suggested that a more appropriate format for me might be a research paper. As the measurements from the quadrant2  is still a project in process, I knew I would need to do a paper on something else. While I was sorely tempted to simply print out all of my Almagest posts as one massive paper, I was inspired when writing this post on applications of the rising time tables. In particular, a member of my Barony, Padraigin, is interested in medieval astrology which makes use of rising signs; something that Ptolemy describes how to calculate. However, based on the work done thus far, it was only simple to do for latitudes given in the rising time tables. My goal was to try to condense the methodology for computing a rising time table at any latitude into a relatively short paper.

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Data: Stellar Quadrant Observations – 1/4/19

Last night was a nearly new moon. The temperature was uncommonly warm for this time of year3 but that didn’t end up disturbing the seeing too much as we had an exceptionally good night for observing. Combined with a rich field of stars in Taurus and Orion passing during the evening and Padraig and I had the most productive evening yet cataloging 59 stars, 16 of which were not in my original list of targets, but due to being able to see down to 5th magnitude, we were able to grab them.

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