That’s a great write-up Rick, and I’ll download it for reference. Major, major correction though: it’s Dave Johnston, with a t! Just kidding. You don’t have to credit me at all. I was intrigued by your question as I didn’t know anything about the orientation of the solar poles and it was worth it just to learn about that. So I thank you for bringing the topic to my attention!
Dave
You two are going to end up causing me to have to go buy an EQ mount in order to keep up with this discussion 
So am I right in thinking that the effect of the EQ mount is to ‘absorb’ the effect of the changing parallactic angle, such that a single alignment with the Position Angle is sufficient?
Dave, you said:
First, I don’t think what you are after is the parallactic angle. That seems to be related to the relationship between the apparent rotation of the disk of the sun as a result of using an alt/az tracking system rather than equatorial. It doesn’t relate to the angle of the poles of the sun as viewed from earth.
Here are some of the diagrams that Meeus uses in his book to explain the Parallactic Angle, q:
So, in my understanding it directly relates the angle of the Sun’s north pole relative to the observer’s local zenith, as viewed from Earth.
So, if you can ignore the Parallactic Angle and reference your camera to the Position Angle, I would guess that’s only because the EQ mount changes the frame of reference such that you don’t need to care about q changing over time?
(The other thing that confuses me frequently is the slightly loose way in which the term Position Angle gets used - it has a specific usage in the eclipse literature, a different one for the Sun, and yet another for lunar libration, where also the term Axis Angle is used by some authors!)
Stephen, as I understand it the Parallactic angle allows for correction of the difference between using an equatorial mount and an alt/az mount. with an equatorial mount, the angle of the camera follows the curve shown in Fig 3. So the bottom of the camera is tangential to the arc in the morning, and thus angled so the pole of the sun is straight up and down. At noon the camera is still tangential to the arc and is thus level, and the pole is again straight up and down. And in the afternoon, the same is all still true. So with an EQ mount there is no need to adjust for the “rotation” of the sun during the day, so long as you get the pole at the right angle to begin with, it will stay that way.
But with an alt/az mount, the camera is parallel with the ground, rather than the elliptic arc (unless you tilt it). So as shown in figure 3. the pole of the sun will change angle during the day. The difference in angle is the parallactic angle, I think. when using an alt/az mount, I guess you need to keep adjusting for that during the day. With an EQ mount you don’t, because the camera is always tangential to the arc.
At least that is what I think I understand.
The position angle (P) changes very slowly during a day, because it is related to the earths rotation around the sun, while the parallactic angle is related to the earth’s rotation around its axis. So as long as you find the correct P for the day and set your camera for that angle, and use and EQ mount, the north pole of the sun should remain “up” during the day. It also doesn’t see to matter when you are on earth (the P remains the same), while parallactic angle does.
I gathered a log of this by playing the app at that line (Solar Rotation). I experimented with changing the date and time, and also the location on earth, and seeing what changed in the results, and by how much. But it is possible that I read too much into it or reached a wrong conclusion, you should probably check anything I say!
Dave
I believe that this is correct, Stephen. Also agree with Dave’s comments. I’m pretty sure I have what I need now since we found that calculator. Sorry for the fire drill! Really appreciate the effort, though!
Thank you both - I think that all makes sense now. I realise that I already have the math implemented to calculate both parallactic angle (q) and position angle (P). I have them both shown in the embeddable simulator like this (top left):
The parallactic angle is calculating instantaneously as it varies constantly. The position angle is calculated once for the moment of max eclipse, as it hardly varies.
Rick - would this give you what you need?
I need to do some double checking of the values, but they’re already in use in the eclipse calculations, so I think they’re sound - I just need to check quadrants, complementary angles, edge cases etc.
Well I many not live long enough to verify it during that eclipse, but it looks great to me. I am amazed that you got this incorporate so quickly. It took me longer to even figure out what the issue was.
Now only .1% of people will even know what those are much less what to do with them. And a certain subset of the remaining 99.9% will ask you to explain it. But this certainly helps make TPE one of the most comprehensive eclipse resources for both experts and beginners.
I am doing dry run rehearsals this afternoon. I still need more practice!
Dave
I plugged in todays date and came up with a P of 334.4, which is the angle on the disk of the current south pole.
Meeus says that the position angle, P, is “the position angle of the northern extremity of the axis of rotation, measured eastwards from the North Point of the solar disk”. That first clause is a little hard to make sense of, but I think it implies this (green arrow is my addition) ?
I think there’s a more general use case for knowing the parallactic angle: to estimate the positions of sunspots after C3.
If you take a reference shot around C1 and note the rough positions of major sunspots - even make a rough map on paper, then you can use that - rotated by the change in parallactic angle to estimate when the same sunspot might be revealed as the Moon moves away.
I’m thinking it could be a neat partial phase project to make a video of major sunspot occultations, and then to catch them on the other side as they’re revealed once more.
But I think you need a sense of the change in parallactic angle in order to be able to estimate it (assuming you’re not using an EQ mount) - given how quickly 4K or 8K video fills a storage card, it’s one of those things you probably can’t just leave running.
That looks great, Stephen! you are amazing!! I can’t believe you did this so quickly. Wow!
We’re going to have to get you to come over to the dark side and get an EQ mount, Stephen!
A fun one to double check with would be 8/2/27 in Luxor, Egypt. The angle is on the other side of north…
How do you embed the simulator?? And where is the embeddable simulator. I think I missed something
Yeah, I think I had that garbled. I was trying to make sense of the P number, given that it is supposed to vary +/- 26.31 degrees. I was I know that 360 minus that number gave the correct P, but I was a little confused bout what the number was relative to, along the circumference. I was going counterclockwise around to the south pole, which was highlighted by the blue dot, which I thought was significant for the measurement. When I found that definition, it made sense.
Dave
If you give me an hour or so, it will be updated in the iOS app. I’ll post here when it’s live…
I see it! I see it! That is so cool! Thanks!!!
Details here
I wanted to double check my use of the term ‘supplementary’ for the value (360 - Position Angle). ChatGPT tells me this:
What do you think I should use? ‘Explementary’ or ‘conjugate’ ? I don’t think I’ve ever come across either term before (except for ‘conjugate’ in Life of Brian…)
Wow! Who knew. I guess I would pick Explementary if I had to pick one. The things you learn!
That is good to know. I didn’t think “supplement” was right when I used it, but I didn’t know what term was correct. I also think explementary would be preferable, as conjugate may suggest other relations. 
Dave
Explementary it is! I’ll update S(P) to E(P) in the simulator.
