NavList:

Frank:

What formula (or class of formula) do you think we should be using? Let me remind you and everyone else reading along that I designed this USNO clone to be a nearly exact duplicate of the features of the original USNO app, including some small flaws.

Well, personnally I can only refer to expert opinion. I understand it is a complex subject with many factors influencing the result, and that the formulas to calculate refraction are empirical. Like any model, they are either wrong or very wrong. That said their accuracy can be backed up by observation. I note that the Bennett formula is used in the Nautical Almanac, and that it is universally recognised to be of suitable accuracy for navigational purposes for hs values from 0 to 90°:

In addition, the two parameters 7.31 and 4.4 as they appear in the original formula were revised around the early 2000 version of the NA to 7.32 and 4.32 respectively, suggesting new observations were made and a revised fitting of the formula was performed, refining or actualising the calculation.

A question for you and anyone else following along: if the Moon's HP is 56.5' (as an example), and we have calculated its altitude (Hc) as 45°00' exactly, then what is its parallax in altitude? Are there two options? Do they both make sense?

From what I studied about sextant correction, and expressed in previous messages in this messaging board, this depends of course on the precision you want to consider. If a precision of 1' is good enough, then this has not a lot of importance, one can calculate Parallax in altitude (~ HP * cos(H)) using hs as the H in the formula. Now, if one wants to consider more refinements in the order of 0.1', then one should take into consideration not only further corrections, but also the order of operations - as the "H" in the P-in-A now becomes (hs - refraction - DIP + SD + augmentation). Looking at historical books, there are plenty of corrections that have been considered until the 80s that are not mentioned anymore. For instance, I found this in the Bowditch vol 2 ("Useful Tables") from 1981:

But reading the NA from 2002 and beyond, the notes on calculating the moon correction tables do not even consider augmentation or oblateness of the Earth and that a precision of much lower than 0.1' can be achieved by ignoring augmentation and simply calculating P-in-A before adding SD (instead of P-in-A after adding SD plus augmentation).

So I agree with you there are several ways to get a "good enough" result, but that "good enough" may be different depending on the needs of the navigator. I suppose that many refinements were dropped because of two reasons: one some of them were too unpredictable or difficult to estimate (such as illumination or planet phases) and that the emergence of radio and satellite positioning systems made the sight reduction process less critical as it used to be.