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I had written-

" Given the chance, I would certainly try to get an example calculation from
him, even with fictitious numbers; and ask how a navigator at sea, without
internet access, would obtain sufficiently accurate predictions of the
Moon's position, other than by using the Nautical Almanac."

 and Ken Muldrew answered-

| Why not use the Nautical Almanac?

There are several problems with using the nautical almanac.  Rather than
explain them anew, I will quote (at some length-sorry) from an old posting
of mine, Navlist 1417 of 14 Oct 06, (with an error amended).

===================

Now, back to the matter in question, the precision of Frank's proposed
method of position-finding without a horizon, which he claimed to be
within 6 miles.

Let's return, first, to the precision in measuring a lunar distance,
the angle between Moon limb and star, for which he claims 0.1';
better, at a guess, than has ever been claimed by any navigator in the
history of navigation. This involves a subtraction of index error from
observed angle. To get the result to 0.1', what accuracy is required
for those two separate observations, and what accuracy does he
achieve? Where does irradiation, at the Moon's limb, come into this,
in determining those two quantities: does he claim immunity from its
affects, or that somehow it doesn't apply?

Let's investigate an error-budget in the overall determination. That
was one reason why I have asked for an explanation of the procedure,
with relevant numbers, because Frank hasn't stated where his celestial
predictions are to come from. Is his observer, from his jungle
clearing, privileged to obtain JPL predictions, beamed to him from a
satellite or a convenient mobile-phone antenna? Or will he have a
laptop with him, programmed up with JPL (or similar) algorithms? Or
will he rely, like the rest of us do, on the Nautical Almanac, as I am
going to assume from now on.

Using the almanac, how do you find what the predicted angle should be,
between Moon and star, at a particular moment of GMT? You will have to
look up many quanties in the almanac, each one stated to the nearest
0.1', a potential error of +/- 0.05'.

First, to find the star position, you need-
1. star dec
2. star HA
3. GHA Aries to nearest hour-point.
4. Aries change, for part-hour

and for the Moon-

5. dec to nearest hour-point
6. d correction to dec for part-hour
7. GHA to nearest hour-point
8. GHA change for part-hour
9. v correction to GHA for part-hour

these allow the lunar distance to be calculated, for a mythical
observer at the centre of the Earth.

9. Moon semidiameter must be allowed for

The following terms vary with the altitudes of the bodies-
10. HP cos alt (Moon parallax)
11. star refraction
12. temp and pressure correction to star refraction.
13. Moon refraction.
14. Temp and pressure correction to Moon refraction.

Depending on the details of the geometry, it's possible for any one of
those 14 terms to affect the result by up to .05', one way or another,
simply because of the way they are tabulated in the Nautical Almanac.
Of course, most are KNOWN to a much higher accuracy, but not from that
Almanac.

It is, of course, true that all 14 are highly unlikely to all add up
in the same direction, to their full extent. They are also highly
unlikely to cancel out to zero. Without making a full statistical
analysis, it seems reasonable, to me, for the standard deviation of
the resulting scatter to be taken as root-14 x the spread of each
component, or 3.7 x .05', or 0.19'. If anyone can suggest a fairer way
to combine those errors, I hope they will.

So, just the uncertainty in taking numbers from the Almanac, on its
own, has already contributed 11 miles (rms) either way to the scatter
in the result, even before any observation errors have been accounted
for. Above, we were analysing the error involved in one Moon-star
position line. There will be a similar error in the other position
line.

So much for Frank's overall uncertainty of 6 miles.

Does that explain why I was keen to get a full explanation of the
procedure, with numerical examples?

===============================

We still haven't got that numerical example, after several requests; but
then, that's Frank.

After that posting, I reassessed the matter, and later decided (and Lars
Bergman agreed) that a standard deviation of +/- 0.1' would be a fairer
assessment. Note that such a 0.1' error, for just one of the two
lunar-distances, already  takes up the whole error-budget, and leaves no
room for any observation error at all.

However, the matter is really much worse than that assessment, when using
the Almanac. Under "Accuracy", we are told that the largest error that can
occur in the GHA of the Moon is 0.3', though I am sure that most of the time
it's much less.

That sums up why an Almanac is unsuitably inaccurate for this application.

We can presume that subsequent reduction work will be done by calculator
rather than by tables, so will not add further inaccuracy.

Ken continued-

| Or a software almanac that has its
| database stored locally? These are common enough. Here is one that is
| somewhat quirky, but free, and the source code is included so that you can
| write your own to suit your own purposes:
|
| http://www.ucalgary.ca/~kmuldrew/celestial/almanac.html

Yes, that is indeed a good way to handle the problem. I have such a
home-brewed program myself, running very slowly on a pocket calculator. Some
quite elaborate sums are needed, to predict the Moon to a high enough
precision, but there are many programs around that will do it it, and no
doubt that applies to Ken's version also.

But if having such a program on-hand is an actual REQUIREMENT for using
Frank's procedure, then Frank should say so. He has been asked what his
assumptions are, about that matter, but has not responded. But then, that's
Frank.

Most likely, he presumes that our navigator will have ready access to his
on-line calculator, just as he would have on land. But he hasn't said so, as
far as I remember.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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