Derivation of the tripod rule
Hi Fred,
I have attached a PDF to this post with the formula derivation. Looking back at the original post and recalling my derivation of the formula, I was deriving a formula that made allowance for the fact that these days, we look at a lot of downsized images on screen. Your formula makes no such allowance. Fixed tripod shots are never going to be perfect. Hence I introduced the single quantity N  number of pixels drift to allow for reduced image size viewing, and tolerance for nonround stars. Assuming a downsized image, effects like seeing and airy disc don't need to be taken into account.
Your approach is different to mine. You seem to be looking at using short exposures as a quasihigh precision guiding for stacking. My feeling is that if high precision guiding and 100% viewing is the target, the photographer should probably be using a tracking mount! I was looking at wide angle reduced resolution images where airy discs and seeing effects could be disregarded. We have two different but equally valid approaches. I must admit that I did not explain the quantity N and considerations around its value carefully enough in my original post. The value of N should never be set to 1. The star image is always going to be spread at the sensel over >1 pixel by the bayer matrix, seeing etc depending on lens focal lengths and f ratios.
Your statement on your web page that
"Cette règle date du temps de l'argentique et certains photographes l'on améliorée en tenant compte de la déclinaison minimale contenue sur l'image. D'autres formules plus compliquées ont été écrites mais aucune ne donne de résultat convaincant dans tous les cas. Elles sont toutes basées sur des méthodes empiriques."
translates roughly to a statement that your formula is the only formula that works and no others found on the internet are accurate or "convincing" in all cases. They are all empirically based. I think your statement is a bit of a stretch. Both your derivation and mine are based on wellknown celestial and optical theory. Neither of us have invented anything new. There is more than one way to arrive at the same result. I have included a plot of a comparison of the indicated exposure times from your formula and mine for lenses ranging from 5mm  500 mm. At f2.8,f4,f5.6 the differences are negligible.
At f22, there is a significant difference between our results. However, If we restrict the comparison to practical uses not theoretical extremes ( no exposures at >f8 for a static shot, no phone cameras with 1 micron pixels, no >500mm focal lengths ) then the airy discs and seeing effects are always less than pixel dimension of 5.5 µm and the results of the two formulae are very similar as can be seen in the illustration below. IIS does not allow excel files to be posted so I have taken a screen shot of the results.
kind regards
Joe
