A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Pike
Date: 2019 Dec 29, 12:15 -0800
Frank you wrote:
The equations for MOO and MOB are differential quantities. They're really only accurate for distances (changes in altitudes) that are "negligibly small".
An analogy: if we know the sine and cosine of some angle, we can calculate the sine for a slightly larger angle from
sin(x) = sin(x₀) + dx · cos(x₀),
where dx here is the small angular increment to get from x₀ to x (note that the angle dx should be a pure ratio (a.k.a. radians) so if you have the angle in minutes of arc, divide it by 3438). This is a differential relationship. You can either derive it with a little calculus, or you can derive it from the formula for the sum of two angles:
sin(x) = sin(x₀ + dx) = sin(x₀)·cos(dx) + sin(dx)·cos(x₀),
and then use the fact that cos(dx) is nearly 1 and sin(dx) is nearly dx for very small angles. This relationship,
sin(x) = sin(x₀) + dx · cos(x₀),
is exceptionally accurate for angular increments of a few minutes of arc, and it's not bad up to a couple of degrees.
I’m not sure that it needs to be as complicated as that. If you look at tables 1 & 2 in AP3270. All they are are plane trigometry tables. Although they appear in celestial tables and have units recorded in minutes of arc, they’re not particularly connected with the height of the star. It would be better in some ways if their units were recorded in nm rather than minutes of arc, because all they do is change your intercept values to have the same effect as using a protractor, dividers, and S=VT from the back of the Dalton computer to move your position line along track to use in a fix or E-W to allow for an observation early or late on pre-computation.
Their original usefulness is clear from the format of the table. If observing three stars with a bubble sextant a four minute gap between mid-times is a convenient work rate allowing the sextant to run down for one minute, the Hs to be recorded, the sextant to be rewound, the next star located, and the sextant to be restarted one minute before the next mid time. It also means that you can minimise on the time spent using the tables by having one assumed position and LHAs either one degree apart or a single LHA if you’re really into using MOB. Because things rarely go according to plan, there must be a set of factors to allow for the case of observations not being taken on time, and that’s why the bottom halves of tables 1 &2 are there.
I don’t believe the MOO tables were ever put there to allow for a much greater gap than about 8 minutes, but if you did so use them, I can’t see that they would be any less accurate than using a protractor, dividers, and the back of a Dalton computer, because all the MOO tables do is solve a geometrical problem using trigonometry.
Saving the makings of an LOP for plotting an hour or two later can’t be good practice. Two or three crossing LOPs is good, but a single LOP is better than none. Get the first LOP on the chart at the first opportunity, because in an hour’s time, the cloud might have come in and the first would be all you’ve got (that’s all Capt Sumner had). Then, if you get a second and third LOP later, move LOPs or assumed positions up using protractor, dividers, and S=VT to produce a fix. DaveP