In the last post, we determined the basic properties of the eccentric model to predict the motion of the sun. Now, we’ll use these properties in conjunction with the model itself to be able to predict “the greatest difference between mean and anomalistic motions” by referring back to our original model. Continue reading “Almagest Book III: On the Anomaly of the Sun – Difference Between Mean and Anomalistic Motion”
Almagest Book III: On the Anomaly of the Sun – Basic Parameters
Now that we’ve laid out how the two hypotheses work and explored how they sometimes function similarly, it’s time to use one of them for the sun. But which one? Or do we need both?
For the sun, Ptolemy states that the sun has
a single anomaly, of such a kind that the time taken from least speed to mean shall always be greater than the time from mean speed to greatest.
In other words, the sun fits both models, but only requires one. Ptolemy chooses the eccentric model due to its simplicity.
But now it’s time to take the hypothesis from a simple toy, in which we’ve just shown the basic properties, and to start attaching hard numbers to it to make it a predictive model. To that end, Ptolemy states,
Our first task is to find the ratio of the eccentricity of the sun’s circle, that is, the ratio of which the distance between the center of the eccentre and the center of the ecliptic (located at the observer) bears to the radius of the eccentre.
Continue reading “Almagest Book III: On the Anomaly of the Sun – Basic Parameters”
Almagest Book III: Hypotheses for Circular Motion – Similarities in Apparent Apogee and Perigee Distances
The last symmetry between the two models Ptolemy wants to point out is that,
where the apparent distance of the body from apogee [at one moment] equals its apparent distance from perigee [at another], the equation of anomaly will be the same at both positions.
Fortunately, the proofs for this are quite simple. Continue reading “Almagest Book III: Hypotheses for Circular Motion – Similarities in Apparent Apogee and Perigee Distances”
Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly
So far in this chapter, we’ve been looking at the two different hypotheses to explain the non-constant angular motion of objects in the sky. Ptolemy claimed that these were equivalent under certain circumstances and, in the last post, we showed how they do indeed produce the same results in the specific case of the object travelling 90º in apparent motion from apogee1 and that it always takes longer for the object to go from slowest motion to mean, than it does mean to fastest.
But that’s not really a full demonstration that they’re functionally the same. So in this post, we’ll show that their apparent angular position from the mean (known as the equation of the anomaly) is always the same, so long as there’s a few things that are consistent between models. Continue reading “Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly”
Almagest Book III: Hypotheses for Circular Motion – Similarities in Time Between Apogee and Mean Between Models
In the last post, we introduced two different models that could potentially explain the anomalous motion of the sun (or other objects). Specifically, the sun sometimes appears to move faster along the ecliptic than at other times2. The first model was the eccentric model in which the observer was placed off center. The second was the epicyclic in which the object would travel around the deferent on an epicycle.
Ptolemy stated that for the simple motion of the sun, either of these models would be sufficient. However, he wanted to demonstrate a key equivalence. Specifically that
for the eccentric hypothesis always, and for the epicyclic hypothesis when the motion at apogee is in advance, the time from least speed to mean is greater than the time from mean speed to greatest; for in both hypotheses the slower motion takes place at the apogee. But [for the epicyclic hypothesis] when the sense of revolution of the body is rearwards from the apogee on the epicycle, the reverse is true: the time from greatest speed to mean is greater than the time from mean to least, since in this case the greatest speed occurs at the apogee.
Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses
So far in Book III, we’ve been looking at the mean sun3. However, the sun’s motion along the ecliptic is not even. In other words, its speed along the ecliptic is not constant. This poses a problem because, to Ptolemy:
the rearward displacements of the planets4 with respect to the heavens are, in every case, just like the motion of the universe in advance, by nature uniform and circular. That is to say, if we imagine the bodies or their circles being carried around by straight lines, in absolutely every case the straight line in question describes equal angles at the center of its revolution in equal times.
Here, Ptolemy states that he accepts uniform circular motion as true. So the question becomes how can something moving at a constant speed appear to have not constant speed? Continue reading “Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses”
Data: Stellar Quadrant Observations – 8/24/19
Life has been eating me pretty heavily lately so I haven’t had any time to work on the Almagest, and barely any time to observe. But last weekend, I took the quadrant to Baroness Wars over in the Midrealm. The location wasn’t all that great. The dark sky map said it was like being in the suburbs of St Louis, and the fact that they left some lights on illuminating a racetrack or something next to the county fairgrounds we were in also sucked, but I was still able to get a few observations in over the two nights I was there.
Continue reading “Data: Stellar Quadrant Observations – 8/24/19”
Data: Stellar Quadrant Observations – 7/27/19
Last month, I came across this dark sky finder website. Unsurprisingly, dark skies in the Eastern half of the US are fairly hard to come by. However, Mark Twain National Forest in southern Missouri puts a significantly large dent in light pollution that much of the surrounding area is quite dark. Sadly, as the name suggests, the forest doesn’t have many clear sites suitable for astronomical observing, but I was able to find a campsite that’s part of the Rocky Creek Conservation Area that what looked to be a large field based on Google Maps that sat in the dark area.
This weekend I found someone to take over my herald duties to Their Majesties and made the 2.5 hour drive to check it out. Continue reading “Data: Stellar Quadrant Observations – 7/27/19”
Statistical Review of the Great Quadrant
Kingdom A&S is in just over a month and coincidentally falls on the one year anniversary of first light of the quadrant. Originally, I’d planned to do a deep dive statistical review of the quadrant, looking at sources of error, but this would be a modern mathematical review of an instrument that isn’t entirely period. Discussing it with friends, we decided it was a little too meta/degrees of separation.
Instead, I’ll do a blog post! So if you really want to get into the accuracy on the quadrant, hold on because I’m about to get mathy. Continue reading “Statistical Review of the Great Quadrant”
Data: Stellar Quadrant Observations – Lilies War
This past week was Lilies War, north of Kansas City. With a bit of packing tetris, I was able to fit the full quadrant along with most of my camping gear5 to attend. Despite Lilies being traditionally extremely hot6 and stormy, the weather was quite nice this year with highs in the 70’s most days and lows in the 50’s7. This unseasonably cool weather did bring some storms, but nothing like what we’ve experienced the past few years. Yet between storms I was able to get in a few nights of observing, getting in observations of 59 stellar targets, 26 of which are new to the catalog.
As always, data below the fold. Continue reading “Data: Stellar Quadrant Observations – Lilies War”