Here is an extreme example that may make you less confused.
No need because I'm not confused now.
But you are correct in that alex314's initial premise: When you observe the sun (or the moon), you can see that the apparent angular movement is always the same. It is about 15 degrees per hour. It means, the position of the sun and the moon changes by 15 degrees per hour.
is incorrect except at the poles.
No, your statement is incorrect. but I can see why you are confused.
I will address the nominal, geometric case without regards to atmospheric refraction, and references to the position of the sun are to the center of the sun. At the pole,
at equinox, the sun will nominally trace the horizon. In this case the apparent angular motion of the sun is 360/degrees per solar day = 15.00 degrees/hour. (Easy to imagine, because the path of the sun is perpendicular to the Earth's axis of rotation.)
At the equator, the sun will trace a line from east to zenith to west in 12 hours = 15.00 degrees/hour. Again, easy to imagine and do the calculation in your head.
At any other latitude, while perhaps not as obvious, the path traced is at the same rate. You can use some simple geometry to convince yourself of this.
This situation changes when the sun is not at 90 degrees to the axis of the Earth's rotation. At the summer solstice, at the north pole, the sun traces a path in the sky at a constant elevation of ~23.5 degrees.
An object tracing a path in the sky at 23.5 degrees elevation in 24 hours is
not moving at an angular rate of 15.00 degrees/hour. This should be obvious from the analogy to Sigma Octantis. Sigma Octantis at an elevation of 89 degrees is not moving at 15.00 degrees/hour; there must be a monotonic function that goes from object on the horizon moving at 15.00 degrees/hour to object at pole moving at 0.00 degrees/hour.
Or, if you're mathematically inclined, you can calculate the angular separation yourself from
https://books.google.com/books?id=MTGYxQyW998C&pg=PA66&lpg=PA66&dq=%22the+angle+between+two+celestial+objects%22&source=bl&ots=UMhTOyKjZG&sig=X3M3S3h7M-EHDsF6BNBJHNBIvZ4&hl=en&sa=X&ved=0ahUKEwi90-GVmtLUAhVs04MKHb0gBOYQ6AEIKzAB#v=onepage&q=%22the%20angle%20between%20two%20celestial%20objects%22&f=falseusing either observed data or calculated solar position
https://www.esrl.noaa.gov/gmd/grad/solcalc/azel.htmlYou'll find that the solar rate of angular motion changes significantly. By no means is it always 15 degrees/hour.
View Sigma Octantis, about a degree away from the celestial pole. Engage the equatorial mount drive on your telescope to keep Sigma Octantis centered in the telescope. This drive will rotate the telescope around the equatorial mount axis at a rate of approximately 15.0 degrees per hour.
But Sigma Octantis did not move angularly with respect to you at a rate of 15.0 degrees per hour. The pointing of the telescope traced a portion of a cone with an apex angle of about 2 degrees. The angular rate at which Sigma Octantis moved with respect to the observer is a few tenths of a degree per hour, not 15 degrees per hour.
I have never mentioned "with respect to the observer". Everywhere I've been referring to rotation about the axis (gnomon) of a "properly aligned" sundial.
No. You butted in with that irrelevant observation, and have continued with it, even though you were told:
While this is true, it has nothing to do with the basic geometry that the apparent angular rate the sun and moon move is not always the same.
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Whether or not a sundial has equally spaced tick marks in a particular orientation or unequally spaced tick marks in a different orientation, [snip] does not change the actual observed apparent position of the sun in the sky or the change in position versus time of the sun in the sky.
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You're confusing uniform angular motion or measurement of a specific measuring device with the actual angular motion of an object in the sky.
Again, you're confusing uniform angular motion of a specific measuring device with the actual angular motion of an object in the sky.
It should be obvious that this is the case because Sigma Octantis is not viewed by the observer at 90 degrees from the celestial pole.
Why is that even relevant?
For the geometrical reasons stated above.
Similarly, because of the orientation of the earth's axis with respect to its plane of orbit, the sun is not always viewed by the observer at 90 degrees to the celestial pole, and changes throughout the year. Therefore the apparent angular motion rate changes as well, and is not constant.
The apparent angular rate of the sun about the earth's axis only varies slightly due to the earth's orbital ellipticity not due to the seasons.
What? Nothing about this discussion has anything to do with orbital ellipticity! Where did you get that? It has to do with the geometry of angles!
All the motion I have referred to about about the earth's axis of rotation and that is very nearly constant (unless you are bothered about a millisecond per day up or down). In earlier posts I tried to avoid this detail as it might lead to a flat ~ Globe slanging match.
No, I'm not concerned with a few milliseconds. I am concerned with an apparent angular rate of motion of a celestial object that is as slow as 13.8 degrees/hour rather than 15.0 degrees per hour. And I do try to take care to use significant digits in values I present.
The original post included
When you observe the sun (or the moon), you can see that the apparent angular movement is always the same. It is about 15 degrees per hour. It means, the position of the sun and the moon changes by 15 degrees per hour.
In the sphere model it is trivial: The earth rotated once full (360 degrees) in 24 hours (i.e. a day). That makes 15 degrees per hour. Simple.
No. It's not. The observations and conclusions are wrong. It's a bad argument when you present "here's a simple fact" that is wrong. Which is why I asked for:
"To refute this, provide observational information about the azimuth and elevation of the sun or the moon at pairs of times one hour apart"
and
"Show that for each pair of observations an hour apart, the angular separation is the same, and is about 15 degrees."
Which was not done.