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Almagest Book V: Components of Parallax – A First Approximation

So far in this chapter, we’ve explored how to use the table of parallax to calculate the parallax of the moon and sun by knowing its distance from the zenith. But this in and of itself isn’t particularly useful. For example, when we did a sample lunar parallax calculation, we determined in that situation, the moon was about $1;10º$ off from its true position. But $1;10º$ in which direction?

Thus, the next step will be to break that down into its components, determining how far off in both ecliptic latitude and longitude the parallax makes the moon appear. Ptolemy again gives steps, but no example, so we’ll continue the previous example we did for the moon, following Neugebauer1. Continue reading “Almagest Book V: Components of Parallax – A First Approximation”

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Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith

Having computed the lunar parallax,

the sun’s parallax for a similar situation [i.e., as measured along an altitude circle] is immediately determined, in a simple fashion (for solar eclipses) from the number in the second column corresponding to the size of the arc from the zenith [to the sun].

Well that sure sounds easy. Let’s look at a quick example. Continue reading “Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith”

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Almagest Book V: Calculating Lunar Parallax Along Great Circle Through Zenith

Now that we’ve completed the parallax table, we should discuss how to use it. As usual, Ptolemy will now walk us through the steps, but does not provide an explicit example so I’ll follow an example from Neugebauer2, to help illustrate the procedure.

As our goal, Ptolemy states that our goal is to find

the moon’s parallax at any given position, (first) with respect to the great circle drawn through the moon and zenith

Continue reading “Almagest Book V: Calculating Lunar Parallax Along Great Circle Through Zenith”

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Data: Stellar Quadrant Observations – 3/6/2021

As usual, the weather in January and February of this year hasn’t cooperated. Nights were consistently cloudy during good moon phases or the temperatures well below freezing.

But as March has  rolled around, temperatures have begun to warm, and the forecast called for a clear night with a good moon phase so Yseult and I headed out to Danville Conservation Area to do some observing. Continue reading “Data: Stellar Quadrant Observations – 3/6/2021”

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Almagest Book V: Parallax Table

In the previous chapter, we worked out the parallax for the sun and the moon at 4 different positions. Next, we determined how we might make some corrections for the moon if it were between the positions we described, breaking the corrections into ones for the epicycle and for the eccentre. From that work, Chapter $18$ presents the results in tabular format. Since this one is pretty wide, I’ve again made the decision to put it into a Google Sheet.

Continue reading “Almagest Book V: Parallax Table”

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Almagest Book V: Lunar Distance Adjustments for Eccentre

So far in this chapter, we’ve reviewed how to calculate the lunar parallax for certain limits of the lunar position and looked at what’s necessary to estimate the effects for lunar positions away from those limits due to the epicycle. Now, we need to discuss the impact of the eccentre and how that we can estimate the effect on parallax due to it bringing the moon closer and further.

So let’s set up a generic diagram of our eccentric model, ignoring the epicycle and only concerning ourselves with the mean moon:

Continue reading “Almagest Book V: Lunar Distance Adjustments for Eccentre”

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Almagest Book V: Lunar Distance Adjustments for Epicycle

In the last post, we explored how to calculate parallax if the distance to an object is known and its distance from the zenith. This was done for the sun and the moon at four different distances. However, because the moon varies so widely in distance in Ptolemy’s model, we need a way to estimate between those positions and we’ll begin by looking at the effect the epicycle has on distance for various points throughout its cycle. To help us, we’ll start with a new diagram:

Continue reading “Almagest Book V: Lunar Distance Adjustments for Epicycle”

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Almagest Book V: Parallaxes of the Sun and Moon

Now that Ptolemy has worked out his model for the sun and moon in earth radii, we can use this result to calculate parallax for any position in our model. To begin, we will calculate

the parallaxes with respect to the great circle drawn through the zenith and body.

This is essentially the reverse calculation of what we did in this post, so we can reuse the same diagram:

Continue reading “Almagest Book V: Parallaxes of the Sun and Moon”