Author: VoijaRisa

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Almagest Book V: Calculation of the Lunar Position

It’s been awhile since I’ve done an Almagest post so quick recap. In Book IV, we worked out a first model for the moon, which was a simple epicycle model inclined to the plane of the sun’s sphere. In this book, Ptolemy showed us that this model was insufficient as the moon’s speed varies more than should predict and so we added an eccentric as well as having the center of the eccentre rotate around the observer. Finally, we introduced the concept of a “mean apogee” which is the position on the epicycle we’ll need to measure from in order to do calculations.

That’s been a lot, but with all of this completed, we should now be able to use it to calculate lunar positions which Ptolemy walks us through in this chapter. Unfortunately, he does this in the form of a generic prescription of steps instead of a concrete example. Fortunately, Toomer provides an example1 that I’ll use to supplement Ptolemy’s narration. Continue reading “Almagest Book V: Calculation of the Lunar Position”

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Data: Stellar Quadrant Observations – 11/6/2020

This is normally the part of the year where it gets difficult to go observing due to extremely cold temperatures at night. However, this week has seen daily highs in the mid 70’s and overnight lows in the mid 50’s. So with clear skies, I knew I needed to get out and observe.

Unfortunately, we’re at roughly a third quarter moon so it rose around 9:30 and was high enough to start being problematic by a little after 10:00. As such, I didn’t feel like making the long drive to Danville and just headed out to Broemmelsiek. The skies there continue to grow more and more light polluted so I was limited to pretty bright stars as it was. Continue reading “Data: Stellar Quadrant Observations – 11/6/2020”

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Almagest Book V: Constructing the Lunar Anomaly Table

In our last post, we showed how it is possible to determine the equation of anomaly by knowing the motion around the epicycle and the double elongation. This, combined with the position of the mean moon2 gives the true position of the moon. As usual, Ptolemy is going to give us a new table to make this relatively easy to look up. But before doing so, Ptolemy wants to explain what this table is going to look like. Continue reading “Almagest Book V: Constructing the Lunar Anomaly Table”

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Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions

Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.

As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.

Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle3 and need to factor in the double elongation of the moon from the sun.

To see how to do so, let’s get started on a new diagram:

Continue reading “Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions”

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Almagest Book V: Second Determination of Direction of Epicycle

In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.

Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one

in order to show that we get the same result at the opposite sides of the eccentre and epicycle.

Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”

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Almagest Book V: The Direction of the Moon’s Epicycle

When we built the first lunar model, it was done using observations only at opposition, which is to say, during eclipses which only happen during the full phase. In the last few chapters, we looked at quadrature, which is to say, during first and third quarter moon and derived a second anomaly. But what happens if we consider the moon when it’s somewhere between those phases?

Ptolemy gives the answer:

[W]e find that the moon has a peculiar characteristic associated with the direction in which the epicycle points.

So what does that mean? Continue reading “Almagest Book V: The Direction of the Moon’s Epicycle”

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Data: Stellar Quadrant Observations – 10/9/2020 & 10/13/2020

NOTE: This post is actually being posted in December since, as I was writing my year end summary, I realized I never posted these observations!

One of the Big Goals of this project is eventually follow Kepler’s methods to be able to derive the orbit of the planets. While I’m still nowhere near being ready for that as I’m still working on the Almagest, I know enough that I know the key observations are those taken when the planets are at opposition. Thus, the quadrant was originally built in time for the 2018 Mars opposition. But Mars only comes to opposition a little over once every two years.

And it’s now time for another opposition. Since I’ve been having some issues with the recently added azimuth ring, I wanted to make sure they were resolved before the opposition. Thus, I went out a few nights before opposition to see if I’d successfully resolved the issues. That night (10/9), I observed $29$ stars plus the three visible planets (Jupiter, Saturn, and Mars). As usual, a few of these observations got tossed, but the overall data looked pretty good. The average right ascension averaged $0.20º$ low and the declination came out $0.16º$ low which is fairly average. Unfortunately, I didn’t catch any new stars as this portion of the sky is pretty well mapped.

Then came the $13^{th}$ which was the important date of opposition. While waiting for Mars to get to a decent altitude, I was able to take observations of $20$ stars that I kept as well as Jupiter and Saturn.

Coming to Mars, I ended up taking $10$ readings in hopes it would average out well. Ultimately, the average for Mars came out with the right ascension being low by $0.11º$ but $0.44º$ low on the declination.

The overall data for the night wasn’t much better, but actually went the other way: The average right ascension was $0.41º$ high and the declination averaged $0.14º$ low.

As usual, the data is available in the Google Sheet.

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Almagest Book V: Second Anomaly Eccentricity

So far, we’ve stated that the effect of the second anomaly is to magnify the first anomaly. In the last chapter, we worked out how much larger. Since this second model works by bringing the moon physically closer and further by offsetting the center of the lunar orbit with an eccentric and having that eccentre orbit the Earth, we can determine how far that center must be. In other words, the eccentricity of this second anomaly. Continue reading “Almagest Book V: Second Anomaly Eccentricity”