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This is part 4 of a series "About Lunars".

It will deal with-
Bruce Stark's Lunar Tables
The serious effects of lunar parallax.
The possibility of calculating altitudes from predictions, rather than
measuring them.
The possibility of calculating "lunars" from measured altitudes rather than
observed lunar distances.
Lunar measurements on-land.


4.1  BRUCE STARK'S LUNAR TABLES.

Thanks to the kindness of list member Bruce Stark, I am now the owner of
his "Tables for clearing the Lunar Distance, and finding GMT by Sextant
Observation". A long title, perhaps, but it defines exactly what these
tables do. You can get a copy via info@celestaire.com at $37 + carriage.

With a current Nautical Almanac, to give the coordinates (dec and GHA) of
the Moon and other-body, the tables allow the prediction of lunar distances
at two times, one hour apart, on-the-hour, which you hope will bracket your
GMT. The tables then allow you to correct ("clear") a sextant observation
of lunar distance between the Moon and other-body for the effects of
parallax and refraction, and calculate GMT from the way that this
observation fits between the two predicted lunar distances. No other tables
are required; everything is supplied in Bruce's book. It uses logarithms
throughout, but in a way that the user is hardly aware of it.

I have been very impressed by the ingenuity with which Bruce has modified
the trig and the logs to make everything as simple as possible for the
navigator, and avoid pitfalls that might otherwise lurk in his path.
Bruce's system avoids the need for any interpolations, yet maintains a
precision to 0.1 minutes, as far as I can judge by limited spot-checks. He
has allowed for small effects that are often neglected. Blank calculation
forms (for photocopying) are supplied, and these are essential. Examples
are given for the user to check his working.

Captain Cook would have been delighted to have a copy of Bruce's lunar
tables on board. He was provided with precomputed lunar distances in the
Nautical Almanac of his day, which made his task a bit simpler than ours.

I have one criticism, which Bruce and I have discussed, as follows. The
user has to go through a series of steps, which are well defined, but
nowhere are the underlying principles and equations EXPLAINED. This might
be acceptable to an old-style navigator who is prepared to follow a rote;
modern individuals with that mindset will nowadays all be using GPS. Anyone
who is measuring lunars today will be doing it (to some extent, at least)
out of intellectual curiosity, as has become clear from the correspondence
with list members. For them, Bruce's tables would, in my own opinion, be
rather more satisfying if background explanation were added.

4.2 FINDING THE LONGITUDE

The user of Bruce's tables (or of any of the other lunar techniques
discussed so far), ends up with a measure of GMT. He is put into the same
position as is the owner of a chronometer. If he does carry a chronometer,
then he is enabled to check its going. But even after all that he does not
yet have a longitude.

Using his now-known GMT, the modern navigator can establish position lines
for objects he can observe in the sky, from their measured and predicted
altitudes. By choosing suitable objects he can cross position lines to
provide his latitude and longitude. Those objects might be (but don't have
to be) the same ones that were used for the lunar distance.

In earlier times he would have to establish his longitude, by determining
his Apparent Local Time from a Sun (or star) observation, and compare that
with the known GMT to determine longitude. This process would call for a
known latitude, readily measured at noon, and followed-up since by the
"reckoning".

These matters will be considered in part 5.

4.3 A SURPRISING OBSERVATION. (Well, it surprised me!)

With some friends, I was examining a series of land-based Sun-lunars that
they measured last summer, over a period of about 1 hour, at a time when
the Moon was quite high in the sky, and drawing a plot of the observed
(uncleared) lunar distance against time. It turned out to be a rather
straight line, as expected. But instead of the expected slope of about 30
arc-minutes per hour, the observed lunar distance was changing at just over
20 arc-minutes per hour. Why so much slower? The answer lies in the way the
Moon's correction for parallax changes so rapidly with time.

4.4 EVIL EFFECTS OF MOON'S PARALLAX

As long as altitudes are above 10 degrees, refractions are no more than a
few minutes, and for the purpose of this argument we can neglect them. In
contrast, parallax makes an enormous difference to the apparent position of
the Moon, up to 60 arc-minutes.

As explained in an earlier part, all nautical tables have been calculated
from the point-of-view of an imaginary observer at the centre of a
transparent Earth. A real observer on the surface of the Earth sees the
Moon, against its background of stars, from a different perspective,
depending on just where he is. The difference in apparent angle is the
parallax. For the Moon, the parallax is so large because it is much closer
to the Earth than any other object in the sky. When the Moon is overhead,
its parallax is zero. When the Moon is on the horizon, it has its maximum
value (the Horizontal Parallax). In between, it varies as cos(altitude).
Parallax always makes the Moon appear lower in the sky.

For simplicity, imagine a navigator near the equator, and the Moon with
near-zero declination. This is the worst-case scenario, when parallax has
its greatest effect, and it is also simple to visualise. The Moon will rise
in the East, pass over his head six hours later, and set in the West.
Parallax will depress the apparent Moon at Moonrise by about 1 deg, then as
the Moon altitude increases, the parallax will decrease until at the moment
it is directly overhead, the parallax is zero. After the Moon passes "over
the top", parallax starts to increase again, but now it is in the opposite
direction, still depressing the Moon's position, more and more, but now
pushing it down to the Westward, until finally the Moon sets in the West,
with parallax at its maximum value of about 1 degree again. The Sun follows
a similar path, but a few hours apart.

So it's reasonably clear, if you think about it and draw a diagram, that
parallax increases the apparent speed of the Moon in its path across the
sky, from East to West, because the Moon appears to travel an extra 2
degrees all told. The rate at which the Moon is pushed by parallax becomes
greatest when it's at it's highest, overhead, at about 15 arc-minutes per
hour, at a time when the parallax itself is zero.

But what about the movement of the Moon with respect to the other bodies,
Sun, stars, planets? Because they are all so much further away than the
Moon, the effect of parallax on the position of those bodies is quite
negligible (for this argument). Parallax doesn't move them.

So as the Moon and the other bodies pass across the sky, more-or-less
together, parallax shifts the apparent Moon westwards, by up to 15
arc-minutes per hour if it's at its zenith, with respect to everything
else.

Now remember what we are trying to measure to obtain GMT. It's the position
of the Moon with respect to the background of Sun and stars. And with
respect to that background, the Moon is moving, always Easterly, at about
30 arc-minutes an hour, give or take a bit. However, we have just worked
out above that parallax causes the APPARENT Moon, as seen by an observer on
the Earth's equator, to be shifted Westerly at a rate of up to 15
arc-minutes per hour. So with respect to the star background, the Moon,
when directly above the observer, has lost half of its apparent velocity,
because of the changing parallax. And that apparent motion of the Moon is
what a lunar observer uses to determine GMT from the Moon's position!

We might call this effect "parallactic retardation" of the apparent Moon. I
have never seen it discussed in an English-language text. If anyone else
has, I would be interested to learn. Steven Wepster has found a discussion
in a German text by W. Jordan, Berlin 1885.

I have described this effect in simplified terms, by ignoring the tilt of
the Earth's equator to the ecliptic - quite an approximation!.

This effect has some serious implications for lunar observations when the
Moon is high in the sky. Not that one will get an erroneous answer, the
corrections for parallax see to that. But one will get a less precise
answer. If there's an error of 1 arc-minute in the lunar distance, then
with a high Moon, the resulting error in GMT becomes not 2 minutes of time
(as we presumed before), but 4. And so the resulting error in longitude
doubles, from 30 to 60 arc-minutes.

I have chosen a worst-case geometry of Moon and other-body as my example,
and often, the adverse effects of lunar parallax will be rather less than
described above. But the relation of the bodies in the sky is so variable,
and so hard to assess, that it's yery difficult to predict the effects of
parallax on the precision of lunars.

One recommendation could clearly be made as follows- If lunars were always
limited to times when the Moon altitude was less than 30 degrees, the
adverse effects of Moon parallax would be less that half the values
referred to above, because the rate-of-change of parallax varies with
sin(alt). And yet, Moon altitudes should be kept above 10 deg, to minimise
uncertainties in refraction, so it doesn't leave a big allowable range.

Has any reader come across any such recommendation in a text on lunars, I
wonder? Or any discussion at all of the evil-effect of parallax as
described above? It seems to have been universally ignored in
English-laguage publications.

4.5  LUNARS WITHOUT ALTITUDES?

A seductive argument.

Looking back at the methods we have used to clear the lunar distance, they
all require a measurement of the altitudes of the Moon and the Sun (or
other body) together with the lunar distance. The altitudes, however, are
only necessary to work out the corrections to the parallax and the
refraction, so they are not required to the same precision as the lunar
distance itself.

An argument can be made (and it has been made, and I accepted it myself at
one time) as follows-

Why go to the trouble of measuring the altitudes of the two bodies?
Instead, why not simply look up their sky-position in the almanac and
deduce their altitudes (or calculate them on a computer)?

The objection, of course, is in the counter-question "what GMT do you use
to look them up?" Because in the end, the GMT is what the whole exercise is
trying to discover. At the start, you don't know the GMT. So how can you
look up those altitudes to make the correction?

Well, the seductive argument goes like this. You are only using those
altitudes for corrections, so they don't need to be known that accurately.
So why not simply make your best guess at an approximate GMT? Look up the
altitudes, correct the lunar distance, work out a more accurate GMT. Then
if your initial guess turns out to have been a bad one, modify it to the
new GMT value, and go through it all again.

Although this procedure does work, to some extent, it doesn't work very
well. Not for the altitude of the Moon, anyway. Let's examine why.

It comes back to the effect of Moon parallax, which varies so rapidly
through the day. Imagine that our guess of GMT happened to be in error by
16 minutes of time. In the circumstances we are considering, of the Moon
near the observer's zenith, the lunar parallax is changing by about 15
arc-minutes in each hour. So our inaccurate guess of 16 minutes of time
will result in an error in Moon altitude of 4 arc-minutes. Because the Moon
and the Sun share the same track (roughly speaking, at least) from the
horizon to the zenith, An error in Moon altitude of 4 arc-minutes will
result in an error in corrected lunar distance of 4 arc-minutes also. And
because the corrected lunar distance changes with time at about 30
arc-minutes in each hour, this will result in a calculated GMT that is in
error by 8 minutes of time.

So now we can put this new value for GMT in as a better guess, and repeat
the calculation as before, which will provide a result that is only 4
minutes out. Next time, 2 minutes out. And so on. In the end, an accurate
result will appear, but it's a very slow and tedious convergence. That's
why I DON'T recommend calculating the altitude of the Moon from the
almanac.

What has been discussed was a "worst-case" situation, and with a different
geometry of Moon and other-body, convergence will happen faster.

In contrast, if the Moon altitude is measured at the same time as the lunar
distance is taken, rather than calculated, then we know that the altitude
is always appropriate to the moment of the lunar, and the parallax
correction will be correct, no matter what GMT the result turns out to be.

Calculating altitudes, instead of measuring them, was proposed as far back
as 1762 in Maskelyne's "British Mariner's Guide" and in his 1767 Almanac,
and by many others later. But I haven't seen a discussion of the weakness
of the method, as described above, in any text in English. Surely somebody,
somewhere, has noticed the problem of that slow convergence.

4.6 LONGITUDES WITHOUT LUNAR DISTANCES?

Why is it necessary to measure lunar distabces at all? It's an unfamiliar
and awkward process to many navigators. Can the same information be
obtained just by the familiar process of measuring altitudes? This question
arose recently on the Nav-L mailing list, and I will tackle it here by
quoting (edited) versions of Chuck Griffith's question and my own reply.
Chuck asked-

>Consider an alternative approach to finding GMT. Why can't we
>observe the altitude of the moon and one other body and, using our assumed
>latitude, solve for  the meridian angle of both bodies. The difference between
>the two angles should change by the rate at which the moon moves through
>the sky
>faster than another body. If that's true, can't we find the meridian angle
>between the two bodies for the even hours, say on either side of what time we
>think it is, and use the same inverse linear interpolation approach to find the
>time of our sight?
>
>Of course, I can think of a couple issues with this approach worth discussion.
>First, this only works when the altitude of the moon and the other body change
>reasonably with time, i.e., we can't do it when either body is close to being a
>meridian sight. Second, we need both altitudes simultaneously. I think this
>could be solved by alternately observing one body then the other several times
>and graphing the sights so that we could derive an averaged simultaneous
>altitude from the graph.

My response is as follows-

The question is a very fair one. It has been asked before, however;
starting in 1674. Francis Chichester, the famous single-handed
circumnavigator, proposed such a method in 1966, and a spate of
publications followed, on similar lines. These were answered in an
authoritative article by David Sadler, then director of HM Nautical Almanac
Office, in the RIN's "Journal of Navigation", 31, 2 May 1978, page 244,
entitled "Lunar Methods for 'Longitude Without Time' ".

From my point of view, the drawback of Sadler's article is that it is
illustrated by a diagram of such devilish cunning and complexity that I am
quite unable to make head or tail of it. If any reader manages to penetrate
its mysteries, I would be grateful for an explanation.

It's important to bear in mind that in any measurement that uses the Moon's
motion to provide time and hence longitude, accuracy in determining the
Moon's position is all. This follows from the fact that each minute-of-arc
error results in an error in the final position of the vessel of 30 minutes
of arc (which near the equator corresponds to 30 miles) or sometimes more.
It is FAR more demanding that the normal run of astronavigation.

The main virtue of a lunar is that the all-important measurement in which
so much accuracy is required, the angle-in-the-sky between the Moon and the
Sun (or other body) does not involve the horizon AT ALL. True, the
altitudes of Sun and Moon do have to be measured up from the horizon as an
auxiliary measurement, but this is only to get a correction to a
correction, and an imprecise value for those altitudes will be perfectly
adequate.

Why is the accuracy so degraded whenever the horizon is involved?

First, if there's any haziness in the air, the first thing to become
indistinct is the line of the horizon.

Second, even if the horizon is really sharp, it isn't exactly a
well-defined straight line (except in millpond conditions), especially from
a small craft. The horizon-line is made up from the peaks of overlapping
waves and swell, and the vessel, too, is riding on those waves. The
observer does what he can by timing his shots when he judges his vessel to
be on the top of its "heave", but it is inevitably a compromise.

Third, even if the horizon is both sharp and straight, its angle  can be
affected by anomalous refraction, which causes the dip to vary from its
predicted value. Air layers at different temperatures near the horizon can
cause the sun's image to be distorted as it rises and sets, and can in
extreme cases cause mirage effects when a distant vessel is observed as
floating well above the horizon, sometimes even inverted. Where none of
these objects is there to give a clue to the odd behaviour of light on its
path from skimming the horizon to the observer's eye, anomalous dip may
nevertheless be present, quite unsuspected and undetectable. An error in
dip of 1 minute may be quite usual, and 2 or 3 minute errors can also occur
occasionally. There is no way for the observer to correct for it. (Special
instruments to measure the dip-of-the-moment have been devised but are very
uncommon).

These errors may present no real problems in normal astronavigation. After
all, what significance has an error of 2 or 3 miles in an astro position?
However, to the lunar observer, where any such errors are multiplied 30
times or more in calculating his longitude, they are intolerable.

When objects lie in opposite parts of the sky, the Moon to the East, say,
and the Sun to the West, such horizon errors would actually add when
comparing the positions of the two objects.

This was well-known to eighteenth-century navigators, who accepted the
practical and arithmetical difficulties of measuring lunar distance up in
the sky, rather than altitudes up from the horizon, to cling on to all the
precision that they possibly could.

4.7 LUNARS IN YOUR BACKYARD.

It isn't necessary to go to sea to observe lunar distances. Because the
angle between Moon and other-body doesn't require a horizon, it can be
measured from on-land, and that is indeed a good way to practice, when the
observer is not being distracted and shaken-up by the motion of a vessel.

However, the required auxiliary measurements, the altitudes of the Moon and
the other-body, above the horizontal, do need to be made. If you don't live
with a view of the sea (or a large lake) then you have to create your own
horizon. This is easy to do.

What's needed is a pool of reflecting liquid in a tray, placed on the
ground. In the days of explorers, they would carry a few pounds weight of
liquid mercury for this purpose. Nowadays, old black engine-oil, or a
dollop of black treacle (molasses in American, I understand) will do the
job. The viscosity helps, in limiting ruffling of the surface by the wind.
If wind is a problem, surround the tray with a wind-break, or even cover it
over with a sort-of garden 'cloche', in an inverted vee, made of glass
which you have checked for flatness beforehand.
A good method for making that check is to look with the sextant at the two
views of a distant object with the index set to zero, as if checking
index-error. Then interpose the glass plate into one view-line from the
sextant (but not both), move the plate about, and check that there is no
shift between the images. If shift is observed, all is not lost: just
average two measurements between which the glass has been turned through
180 in its plane (not overturned).

Look down with your sextant (the straight-through view via the
half-silvered mirror) at the reflected image of the object in the pool, and
in the index mirror view the object directly, up in the sky. The angle
shown on the sextant will be angle between the object and its reflected
image, or twice its altitude above the true horizon. Because a normal
sextant can measure only to 120 degrees (sometimes a few degrees more),
then the maximum altitude that can be measured in this way is limited to 60
degrees (or slightly over).

The sextant angle, as measured, should be corrected for any sextant error
shown in the box, and for index error, and then halved. No dip correction
should be made.

After noting the observed altitudes in this way for both bodies,
corrections for refraction and for parallax must be made (unless Letcher's
method is being used which includes these corrections automatically). If
your altitude is significantly above sea level, make the appropriate
corrections to refraction caused by the reduced pressure, and also, if
necessary, any abnormal temperature. Note the corrected altitudes, and then
proceed as for any other lunar distance calculation.

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

This is the end of part 4. Part 5 will cover the final step of obtaining a
ship's longitude from a knowledge of GMT, whether this was derived from a
lunar or from a chronometer. That will conclude this series, unless further
questions arise that need answering.

George Huxtable.

------------------------------

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