Tag: moon

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Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee

In the last post, we looked at how much the total equation of anomaly would change during syzygy due to the eccentre we added to the lunar model in this book, when the moon was at its greatest base equation of anomaly. As Ptolemy told us, it wasn’t much. However, there was a second effect that can also change the equation of anomaly, which was based on where we measure the movement around the epicycle from. Namely, the mean apogee instead of the true apogee. This has its maximum effect when the moon is near apogee or perigee so in this post, we’ll again quantify how much.

Let’s start off by building our diagram:

Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”

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Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly

Syzygy is one of those words that has popped up very little in the Almagest so every time it does, I’m always thrown off a bit1. Especially when Ptolemy is going to spend an entire chapter discussing a topic that has scarcely even come up. But here we have Ptolemy spending the entirety of chapter $10$, to demonstrate that these modifications we’ve made to the lunar model have a negligible effect because he fears readers might think it does since

the centre of the epicycle does not always … stand exactly at the apogee at those times, but can be removed from the apogee by an arc [of the eccentre] of considerable size, because location precisely at the apogee occurs at the mean syzygies, whereas the determination of true conjunction and opposition requires taking the anomalies of both luminaries into account.

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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 example2 that I’ll use to supplement Ptolemy’s narration. Continue reading “Almagest Book V: Calculation of the Lunar Position”

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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 moon3 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 epicycle4 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.

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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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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”

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Almagest Book V: Size of the Second Anomaly

So far, what we know about Ptolemy’s second anomaly is that it doesn’t have an effect at conjunction or opposition. Its at its maximum at quadrature, which is to say, a $\frac{1}{4}$ and $\frac{3}{4}$ of the way through each synodic month5. Its effect is to re-enforce whatever anomaly was present from the first anomaly. Ptolemy laid out a conceptual model in Chapter 2, but to determine the parameters of the model, we’ll need to first explore how much this second anomaly impacts things.

Continue reading “Almagest Book V: Size of the Second Anomaly”