24 November 2022 at 4:29 pm #613930
Wow, that’s great. It is nice to see that you could get a measurement for the magnitude and the obscuration which so nicely agrees with the prediction. Very satisfying. As regards my formula I am pleased that you could verify it. I too slogged through some trig but I spotted this simplified form of it at the end. I must work out what roughly the errors would be on these values given that you could measure things to within a pixel or two. I haven’t actually checked the assumption that the moon and the sun were approximately the same apparent diameter for this particular eclipse. Oh, actually, on that diagram of the eclipse in the handbook it has the apparent semi-diameters of the sun and moon at greatest eclipse (S.D.=…). Taking these to be the numbers we need it would imply that the moon was slightly smaller than the sun?
Thanks also for explaining about the eclipse diagrams in the handbook. Yes, that link you gave to that NASA page does definitely help! I will keep a note of it.
I also liked your two videos of the eclipse, especially the second one which shows that the path of the eclipse is a straight line. Very clever.
Duncan.24 November 2022 at 11:01 pm #613931
Happy to hear you liked my video animations. Given the “rather variable” (!) conditions I experienced during the eclipse; the need to take several shots on different exposures each time to combat the variable cloud cover, as well as experimenting with two different density filters for the same reason, and the problem of taking shots at the required 5 minute interval if clouds decided to intrude meant that the individual frames needed A LOT of processing to get them consistent enough to be assembled into the animations. I’m thus glad to know that my efforts have been appreciated! The second one is particularly intriguing, as you say – I haven’t seen anything similar on the Internet so I was rather pleased it came out so well.
I also found equations of the form you derived on the Internet but wanted to explicitly avoid that approach so I could get my results via an entirely different route and thus be confident in the comparison between the two methods. Doing it “the hard way” also enabled me to sort of work backwards from the answer and derive an again multi-step process for converting from magnitude to obscuration, which I’m sure will prove useful.
In terms of the measurement and calculation errors which might be expected, the fact that I was able to use a much larger image gave me advantages in both precision and accuracy. More precise as I was working at the 1 in 1000 level whereas you were at 1 in 250, and potentially more accurate as, because of the greater precision, I could determine my measurements at a smaller “step length”. That is, in my case a 1 pixel step hardly moved the “marquee” lines I was using to measure distances, and so the lines could be more closely aligned with the Sun, whereas your 1 pixel change will have moved the measurement point by four times as much so you might not have been able to line things up exactly. I still had the problem of estimating the correct points to measure from of course, given that my image was not sharp (because of the limitations of the home-made filters I was using), but even so I believe that my distances were probably correct to +/- 1 pixel. Translated to the difference this would make to the final result, as compared to the nominal 1200 by 846 figures, the maximum difference (in the cases 1201 / 845 and 1199 / 847) was just less than 0.5% which is still pretty acceptable.
The same website I used to determine the theoretical magnitude and obscuration also gave the ratio of the diameters at my maximum eclipse, which was 0.99253. The assumption of equal sizes is thus highly reasonable, involving potential errors of only the same order of magnitude as those involved in taking the measurements and probably less as the radii are used as part of more complex equations rather than as direct parameters. The SD values on the NASA chart are of course those relating to the point of absolute greatest magnitude (over eastern Siberia) so, while indicative of the situation, will not be applicable to the UK as the Moon will have moved along its orbit during the eclipse.
Steve28 November 2022 at 6:41 pm #613970
Hi again Steve,
I completely agree with you with what you said about the errors. The bit I meant to calculate was the error on the estimate of the obscuration and magnitude. If we take your +/- 1 pixel example, the largest value of the angle subtended by the chord w is derived from the ratio 847/1199 which corresponds to w=1.568860462 radians. The smallest angle is derived from the ratio 845/1201 which corresponds to w=1.56084678 radians. So the upper and lower limits to the obscuration percentage are 18.11% and 17.85% respectively, which roughly means that your observed value is 17.98 +/- 0.13 (compared to the predicted value of 17.96%). With respect to the magnitude the upper and lower limits are 0.2922 and 0.2894 respectively, which means that your observed value is 0.2908 +/- 0.0014 (compared to the predicted value of 0.2910). So it seems we would have to work a bit harder in accuracy before we could start to determine if the observed value really differs from the prediction (and this is what would be of interest).
Duncan.29 November 2022 at 12:44 am #613981
Oddly, although before I made this reply the front page of the website said the latest reply was from Duncan, and the total number of replies before this one was given as 23 with the last made by Duncan 5 hrs 57 mins ago (at time of typing), the number of actual posts was shown as only 22 with nothing visible from Duncan. An aborted reply, or a glitch somewhere?29 November 2022 at 9:20 pm #614014
Hi Steve, there was a problem with the system. I created a post and then made a couple of small edits and then it disappeared! I was hoping it could be resurrected but I think it has gone into the ether.
The short summary of what I said was this. I agree about the errors. Taking your results, the ratio 847/1199 gives an upper limit for the angle w and the ratio 845/1201 gives a lower limit. Taking these to be limits of the error we get for your measurement of the obscuration and magnitude 17.98 +/- 0.13 and 0.2908 +/- 0.0014 respectively. My conclusion is that we would have to work a bit harder to get these errors down if we were going to see any deviation from the predicted values of these quantities (which would be interesting).
Duncan.30 November 2022 at 12:58 am #614015
Interesting! I experienced exactly the same problem when I tried to edit a reply. Like yours, it just vanished. Maybe something to report to “the powers that be”?
Anyway – I agree with your “error bars”. Probably the only practical way to improve the accuracy and precision of the observations would be to increase the resolution of the image (mine was 1920×1920 pixels, cropped from 3264×2448 to centre it) and to use a better filter so the image was sharper. Mine was just a home-made item, constructed by inserting a small piece of Mylar sheet – formerly a lens from a pair of eclipse specs! – into the top of a closed cardboard cylinder sized to fit over my camera barrel. Cheap, but not exactly high quality! Then of course one would have to re-do the calculations to take account of the possibly differing apparent sizes of Sun & Moon. I’ve seen the required analysis on the Internet, but the result is by no means as simple as your “one-liner” formula! One must also consider the accuracy of the predictions themselves – do they assume a spherical Earth and Moon, for example, or use a “true geoid” and reliable limb profiles? At the level we are considering, such things become important.
I shall certainly be repeating these calculations the next time round though, on 29th March 2025!
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