NavList:

John Karl, you wrote:
"I guess I still don’t quite understand the topic at hand."

Ok. I can see that... I will try again.

And you wrote:
"Like you say, the question can be cast as a multi-variable differential calculus problem."

It CAN be, but that doesn't mean that's the first thing you should do, and it also doesn't mean that this issue cannot be understood by a person who can't work it as a calculus problem. By the way, the exercises in your book are concerned with a very different issue so I will get to them in the "PS" below.

Let's look at this without calculus first...

In the problem of observing lunar distances, we need to correct for the effect of parallax and refraction, just as we need to correct an ordinary celestial line of position altitude for parallax and refraction. The difference with "lunars" is that the lunar arc, a piece of a great circle, is generally in a "funny" direction across the sky, skewed relative to the horizon (and the local vertical), so the corrections for refraction and parallax for the Moon and the Sun have to be factored in differently. Those altitude corrections, as well as the factors which determine what fraction of them lies along the lunar arc, depend on the altitudes of the Moon and Sun (I'm saying "Sun" here for simplicity, but the other body could be a planet or star --it doesn't affect the logic of the analysis). So the easiest way to work, and the one that was usually, though not exclusively, employed historically is to measure the altitudes of the Moon and Sun more or less simultaneously with the observation of the angle between the Sun and the Moon. The lunar itself, the angle between the Sun and Moon, has to be observed with care using a fine sextant. The altitudes, by contrast, are supporting players, and they can be measured rather casually using a lower quality instrument if desired. All of this is just background. No surprises, I hope.

Most of the historical resources on lunar distances, including the navigation manuals like Bowditch, generally confirm the above and say that the altitudes don't matter much in a lunar observation and that an error of a "few minutes of arc" will not significantly affect the outcome of the calculations for the cleared lunar distance. But that's not very specific, is it? We can do better. We can be specific about the allowable error in the observed altitudes, and that is exactly what I am talking about here. SUPPOSE I make an error of, let's say, TEN minutes of arc in the altitude of the Moon or in the altitude of the Sun; what is the actual error that results in the cleared lunar distance?? And further, are there circumstances where an error in the altitude of the Moon could be much larger --even SEVERAL DEGREES-- with no impact on the cleared lunar distance (to jump to the conclusion, the answer is yes, as I have said already in this thread, and as I have discussed at length in previous NavList posts).

How can we determine the specific error? It's easy today. We can calculate each and every case of lunars at some "mesh" size. Every single lunar distance geometry involves a certain difference in azimuth (that's your "RBA", John) between the bodies and some altitudes for the bodies. I can run all azimuth differences from 0 to 180 degrees, all Sun altitudes from 3 to 90, and all Moon altitudes from 3 to 90, each of these at a step size of 1 degree or 0.1 degree or whatever is convenient (dropping altitudes below 3 degrees because of uncertain refraction). And with a few hours of coding on a computer and an extremely short amount of time actually calculating, I can directly calculate the effect of an error of 10 minutes of arc in the Moon's altitude for all possible lunar geometries. Of course, historically, this would have been an enormous calculational project, and very likely, that's why the details of this were unknown historically. By doing this, you would quickly find that the altitude of the Moon hardly matters at all when the observed lunar distance (center to center) is close to 90 degrees. You would also discover that the altitude of the Sun does not have this property and has to be measured to within a few minutes of arc (typically +/-6' for 0.1' accuracy). You would also notice that the altitudes matter significantly less when either body is fairly close to the zenith, and you would notice that the required accuracy in the altitudes increases without limit as we get close to zero in the lunar distance. But the key "miraculous" factor here is that the altitude of the Moon can be in error by SEVERAL DEGREES (!) when the lunar distance is close to 90 degrees without any significant impact on clearing the lunar distance. And in the context of the discussion about "impossible triangles", it also makes no difference whether the zenith distances add up to be greater or less than the observed lunar distance, regardless of any supposed commandments of Euclid. You get the same results regardless. The process of clearing lunars is "robust" with respect to errors in the observed altitudes, especially the Moon's altitude for distances near 90 degrees.

I've described a brute force approach to determining the error in a lunar resulting from an error in an observed altitude of the Moon or the Sun. There is also that somewhat prettier approach involving differentiation. If you write down a good approximate expression for D-D' (the difference between the observed distance and the cleared distance), and then differentiate with respect to the Moon's altitude, dropping small terms, you will get

d(D-D')/dh_moon = HP*cos(h_moon)/tan(D).

With a little re-arranging and inserting the mean value for the Moon's HP, you'll get
err_D = (err_h_moon/60)*cos(h_moon)/tan(D)

where err_D is the resulting error in the lunar distance in minutes of arc for a given error in the Moon's altitude, err_h_moon, in minutes of arc. If I require a maximum error of no more than 0.1' in the cleared lunar distance, then the maximum allowed error in the Moon's altitude is

maxerr = 6'*tan(D)/cos(h_moon).

It's a nice short formula that tells us very quickly what the actual specific limits are on those 19th century recommendations of a few minutes of arc of acceptable error. Similarly, the maximum error for the Sun's altitude under the same conditions is

maxerr = 6'*sin(D)/cos(h_sun).

You'll note that these equations have the same limit for short distances (sin(D) and tan(D) are nearly equal for small angles) and also that the required accuracy has that "miraculous" feature at D=90 degrees for the Moon's altitude. These formulas ARE approximate to some degree and the "brute force" approach shows significant quantitative differences at very low altitudes but the qualitative behavior is the same.

This is not just math manipulation. There are significant practical issues at stake here. Suppose you're taking a lunar observation alone. Taking the altitudes of the Sun and Moon simultaneous with the lunar distance is impossible. But as long as you're within the limits above, you can take the observations for altitudes more casually, especially the altitude of the Moon. And in particular, if the lunar distance is close to 90 degrees (a convenient time for shooting lunars for many reasons), then you can take the altitude of the Moon at your leisure. There's no need for accuracy in the altitude of the Moon. Even altitudes which seem to yield "non-triangles," or "non-entities" in the mocking words of that letter-to-the-editor-writer "Nauticus" back in 1786, still yield fully valid solutions to the problem of clearing lunars within the specific constraints described above.

-FER
PS: You described some exercises in your book:
"When I do that (as in Exercises 4.8 and 4.9 in my book), I get results very similar to what I’ve seen you present. The one-body clearing correction to the lunar distance is (my Eq 13.9):
"- A dH + 0.5 cot LD (1 – A^2) dH^2 /3438"

This isn't connected with altitude accuracy. What you're doing here is trying to derive the SERIES EXPANSION for the problem of clearing lunars. This is equivalent to the direct triangle solution which you prefer, within certain reasonable limits. It has been known in detail since the late 18th century. Deriving this series expansion certainly has some interest as a mathematical exercise but it's not even remotely necessary from any sort of practical standpoint (would you derive h=(1/2)g*t^2+v0*t from a series expansion of a Keplerian orbit before you used it?). And you really missed out by not including the actual series expansion equation (as opposed to this "one-body" version) perhaps with an explanation of how important it was historically (ALL of the methods for clearing lunars in Bowditch, for example, were series solutions).

Regarding the equation as given in your exercise, you wrote:
"A is cos RBA (relative bearing angle)."

No, it isn't. I'm sure if you check again, you'll remember that A is the "corner cosine". That's the cosine of the angle between the vertical circle at the body and the lunar distance arc. It's the angle between the arcs of the triangle "at" the Moon, not the angle between the arcs at the zenith.

You wrote:
"The second term, the quadratic one, is what you’ve called "Q". "

FYI, for a quick "calculator" solution, I usually only give one term (the larger), but that "Q" should really be three terms from a strictly mathematical point of view. It's easy to see why. If you have a general series expansion of some quantity z with respect to variations in a single variable x, you get

z = z0 + A*dx + (1/2)B*dx^2 + ...

where A and B depend on the details of the problem. If you have two quantities, x and y, which can vary, then the general series expansion will be

z = z0 + (A*dx + B*dy) + (1/2)(C*dx^2 + 2*D*dx*dy + E*dy^2) + ...

where the A,B,C,D,E depend on the details of the problem (in the case of the lunars series expansion, they are simple geometric factors). Obviously at second-order, there are three terms in the series expansion: one in dx^2, one in dy^2, and the cross-term in dx*dy. For lunars analysis, the first quadratic term, proportional to the square of the Moon's altitude correction, was picked up in series solutions starting with Lyons's method in the late 18th century (though it wasn't always used even decades later since it, too, is insignificant for lunars near 90 degrees). The term proportional to the square of the Sun's altitude correction was never required in any practical case. The cross-term, proportional to the product of the Moon's altitude correction AND the Sun's altitude correction was added into various tables (e.g. in Bowditch and Thomson) starting around 1825. Its effect is quite small, but it helps a bit for lunars where the distance is rather short and either altitude is low.

(and again, to reiterate, I'm putting this in a "PS" because it's not related to the issue of altitude accuracy --this business about "Q" is just math for clearing lunars generally using a series expansion).

You added:
"Well almost -- your Q has the factor 0.55 in front, while I get 0.50."

The actual series expansion of course gives 1/2, as above. As I noted in a couple of posts (years ago now), increasing the factor by ten percent has the practical effect of reducing the error from neglecting the cross-term for no extra calculational cost. In other words, you get better results at short distances and low altitudes "for free" by introducing very small errors (typically a second of arc or smaller) for larger distances. The factor 0.55 simply works better.

You concluded:
"In your example you’re changing the moon’ altitude, while keeping LD constant, and letting the RBA change. But if LD has been kept constant, how can its change be calculated?"

I trust that the above (before the PS) already answered this question. If not, could you please elaborate.

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