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About Lunars, Part 1.

Part 1 explains what a Lunar Distance is, how it is measured and "cleared"
to obtain a Corrected Lunar Distance.

Part 2  explains how to obtain tabulated lunar distances, so that the GMT
can be deduced. Some examples are given.

Part 3 will come at a future date. It will discuss the possibility of
calculating the altitudes required with a lunar distance, instead of
measuring them: also the question of observing Moon altitudes in place of
lunar distances. It will show how to obtain local apparent time and use it
with GMT to obtain longitude, as early navigators had to do.

INTRODUCTION.

I am aware that this multi-part posting is over-wordy and rambling, and
really needs much tightening up. It's a try-out for NAV-L subscribers,
ahead of putting the thing out in another format. So I would welcome
readers' comments, particularly about any sections they have found
difficult or impenetrable; preferably via the NAV-L list.

It is, no doubt about it, a complex matter.

I am grateful for many discussions with list-member Steven Wepster, and
with Catherine Hohenkerk of H.M. Nautical Almanac Office. Any errors are
entirely my own, however.

It's rather too much to hope that what follows will be entirely free of
mistakes, and I would be grateful to have them pointed out.

I should make it clear that I don't claim to be an experienced observer of
lunars at sea; they are few and far between. I would be most interested to
hear from anyone that is.

USEFUL READING.

I recommend Charles H Cotter's, "The history of nautical astronomy",
(London, 1968) which has all the technical details anyone might wish for,
but is hard to find. For an easier read, try Derek Howes' contribution to
"the Quest for Longitude", ed. Andrewes, (Harvard,1996). This is a
beautiful but expensive volume, stuffed with interesting information.

I have recently become aware of "Self-Contained Celestial Navigation with
H.O.208" by John S. Letcher, Jr., 1977. This has a useful section on lunar
navigation, with some novel and interesting ideas, well explained. I would
not agree with every word he says, though.

WHAT ARE LUNAR DISTANCES?

The phrase "Lunar Distance" confuses many. It's nothing to do with the
distance of the Moon. It's the angle-in-the sky, to be measured with a
sextant, between the direction of the Moon, as seen by an observer on
Earth, and some other body visible in the sky. That measurement is often
abbreviated as "taking a lunar".

The purpose of a lunar is to obtain the Greenwich Mean Time, from anywhere
on Earth. I will stick to the old abbreviation GMT, but for those that
prefer to be more modern, it's indistinguishable in practice from UT.

A lunar uses the precise position of the Moon in the sky, in relation to
other bodies, to provide the Mean Time at Greenwich. With a really
trustworthy chronometer, or an accurate quartz watch, on board, there is no
need for a lunar. In these days of radio signals, against which the watch
or chronometer can be checked, and GPS, which gives a precise position in a
flash, lunars are an anachronism. Why bother, indeed?

You have to put yourself back into the 18th or 19th century, times when
chronometers didn't exist, or were ruinously expensive. If a tiny speck of
grit got into the works of your one chronometer, and its rate changed, then
it could lead you into danger. There were no radio time signals to put you
right: you were on your own. If you had a second chronometer, and they
started to disagree, how could you tell which was in error? Really three
chronometers were needed, to be safe. Few mariners could afford such an
expensive luxury.

Well into the 1800s, then, mariners would use lunars instead of a
chronometer (or if they had one, as a check on its going). The last
exponent of the lunar art was probably Joshua Slocum, sailing "Spray"
around the world single-handed in 1895-1898.

The user of lunars needed a pre-calculated table of lunar distances, which
Nautical Almanacs carried until 1906. Because lunars had by then fallen
into disuse, these tables were dropped. That was the end of lunars.

In those days of longhand calculation, the complex motion of the Moon made
accurate prediction of its position a very tedious matter. Nowadays, with a
pocket calculator, portable computer, or modern almanac, you can do in
seconds what once took almanac-calculators hours. It is now possible for
you to obtain lunar predictions for yourself. Lunars are back in business
again, for anyone who wishes to follow-up the early days of
astronavigation.

HOW DOES IT WORK?

If you go out on a clear night, when you can see the Moon against a
background of stars, take a good look and remember just where it is placed
against those stars. Now go out on the next night and look again, and you
will see that the Moon has shifted very noticeably against those same
stars, toward the East. The movement in 24 hours is about 12 degrees. Why?
Because the Moon travels right around the star background, 360 degrees, in
the course of its motion round the Earth in a month. It's the
fastest-moving object in the sky, with respect to the stars, by far.

Even over an hour, if you observe closely, you will see that the Moon has
moved by its own diameter, roughly speaking. And another observer,
elsewhere on Earth, who can see the Moon at the same time, will see it in
the same position against the star background (give or take some important
corrections). So observers, anywhere in the world, by observing where the
Moon is in the sky and making those corrections, can synchronise their
watches to an agreed time scale, that of Greenwich Mean Time, or GMT. They
are using the exact position of the Moon in the sky as a clock that anyone
in the world can read.

The position of the Moon can also, more conveniently, be measured against
the Sun, and also against the planets. All these move much more slowly
against the stars than the Moon does.

There are some snags, however-

1. The Moon is a clock that moves only very slowly. In an hour, it moves
only 1/2 degree or so. IF you can measure its position to 1 minute of arc,
this corresponds to 2 minutes of time. This is about the best accuracy that
you can realistically hope for. If you have some sort of timekeeper, you
could then set it, from the Moon, to that precision, two minutes of time,
and use it with an altitude to determine your longitude.

The trouble is, a timekeeper that's only been set within two minutes is not
a very good one. Because the whole sky appears to move around at 15 degrees
in an hour, in that 2-minute-of-time error interval it moves 30 minutes of
arc. As a result, your longitude has been determined to a precision of only
30 minutes of arc, or 30 miles at the equator. After experience with GPS,
you may wonder whether such a crude position is any use at all. It should
be clear, now, how important it is to achieve the utmost precision in the
lunar distance observation.

In the late 1700s, navigators were delighted when lunars became available.
Until then, they had only dead-reckoning, which could put them thousands of
miles out on a long voyage with unknown ocean currents. Cook thought that
the 30-mile accuracy of a lunar longitude was all that navigators would
ever require (how wrong he was!)

2. How do you measure, precisely, the position of the Moon in the sky as it
travels round its orbit? Unfortunately, the sky isn't marked out with
convenient lines of declination and right-ascension. At a fixed
observatory, a quadrant could be set precisely in the ground, and the Moon
position could be read off. But that was no use to a mariner.

This was resolved in the 1760s by the contribution made by Maskelyne, with
Mayer. He invented the first pre-computed tables, giving at 3-hour
intervals, the predicted angles, to be compared with angles measured in the
sky, between the Moon and other objects. For each day, he would select a
few convenient objects, some to the East of the Moon, some to the West,
which were near to the Moon's path across the sky, and at a convenient
angle from the Moon for measurement. The Sun was included when its angle to
the Moon was in the range 40 to 120 degrees. Around New Moon, the Moon was
invisible for a few days, so lunar distances were then useless, and those
days were omitted from the tables.

HOW IS A LUNAR DISTANCE MEASURED?

It is presumed that anyone reading this will be familiar with the use of a
sextant to measure the altitude of a body in the sky, make the necessary
corrections, and obtain a position line.

The difference with a lunar is that the observer is not measuring an
altitude, straight up from the horizon. Instead, he wants the angle between
the Moon and another body, up in the sky. The other body is often the Sun,
but it might be a planet or one of those stars.

For the method to work properly, the other body has to be somewhere near to
the path of the Moon around the sky. The Moon is always within 5 degrees or
so of the ecliptic (which is the path of the Sun). So the Sun always meets
that requirement, as do the bright planets. Only a limited number of stars
are acceptably close to the ecliptic, and lunars are limited to using a
selection from these stars, which are Altair, Fomalhaut, Hamal, Aldebaran,
Pollux, Regulus, Antares, Spica, and Markab.

For now, we will consider the Sun as the other body, but bear in mind that
it might be a planet or a star.

The horizon plays no part in the lunar distance observation itself, and the
view, straight through the sextant, is instead used to observe the fainter
of the bodies observed (usually the Moon). The brighter one (usually the
Sun) is viewed in the index mirror. To see them both, the sextant may have
to be tilted at what seems like a most unnatural angle, pointing up in the
air, and over on its side, even upside-down. This is something that just
has to be accepted. The index is then adjusted until both bodies can be
seen. This is a rather hit-and-miss affair, and is a test of the lunar
observer's skill, especialy in rough weather. It is much easier to get both
bodies in view if the angle between them can be estimated roughly
beforehand. Of course, the shades have to be used appropriately.

Once the two bodies are both in view, it's necessary to bring the images of
the limbs of Sun and Moon together so that they just (and only just)
"kiss". The observer rocks the sextant a bit so the two bodies just skim
past each other, in a similar way to taking an altitude against the
horizon. Their limbs just touch,as they pass, with no overlap. This part of
the operation demands the utmost possible precision. The two discs close,
or part, very slowly, so it's not practical to wait for the kiss to occur,
but to achieve it by turning the sextant knob. The moment when this occurs
is the moment for which GMT will be obtained, so that time should be
recorded using the best timepiece available. At the end, it's the error in
this timepiece that will be determined.

The sextant reading should be noted, to the utmost accuracy possible. On a
big ship in calm weather, or from land, it may be possible to observe this
lunar distance to a small fraction of a minute of arc. This is rather a
forlorn hope from a small craft, however.

Usually (depending on its phase) the Moon shows a fuzzy, shadowed limb, and
a crisp, rounded limb, which is always the limb closest in angle to the
Sun. The Moon limb that is measured to is always the crisp round limb.
Sometimes, with a star-or-planet lunar, that will be the further limb,
measured "across" the Moon disc from your object, not the nearer. It's
important to note which, because it affects whether the correction for Moon
semidiameter is to be added or subtracted.

To increase accuracy, it's recommended to repeat measurements of lunar
distance at very short intervals, and average the angles and the times. If
an odd number of measurements is taken at equal intervals, this becomes
easy. The index error of the sextant should be checked well, preferably
before and after.

Because the sextant may need to be pointing up in the air, it may be useful
for the observer to lean back on a seat, or even lie down. However, it's
necessary for his head and the sextant to remain free to move, to
compensate for the motion of the vessel. He has to find a position where
both bodies he measures are unobscured by sails, but it doesn't matter if a
sail comes across some of the angle between them.

There are many corrections required to be made to the measured lunar
distance angle, and they all need to be made as precisely as possible. For
this purpose, it is necessary to measure the altitudes, above the horizon,
of the Moon and the other body involved, at the same average time, or
nearly so, as the lunar distance was measured.

In the Navy, this could require three observers with three sextants
altogether, and some ceremony. However, others in more straitened
circumstances can make do with a single observer making all observations in
a carefully timed sequence: first Sun alt., then Moon alt., then several
lunar distances, then Moon alt., then Sun alt. In this way, the averaged
altitude observations can be at the same averaged time as the lunar
distances.

The corrections are such that the Sun altitude is not needed to great
accuracy, but the Moon altitude should be measured with precision.

For the Sun altitude, the lower limb can be aligned with the horizon, as
usual with any Sun altitude observation, and then semidiameter will later
be added. For the Moon altitude, the upper or lower limb has to be chosen,
whichever is the round limb, and the fuzzy shadowed limb avoided. The
observer must note whether upper or lower was chosen to align with the
horizon, because the Moon semidiameter will later be subtracted, or added,
respectively, to the observed altitude.

Now all the observations have been made and noted down, and the
calculations can start.

CALCULATIONS REQUIRED.

Get ready for quite a lot of mathematics now. It's an unavoidable part of
working lunars.

The problem is this. The compilers of the lunar distance tables do not know
where on Earth a navigator is going to be who is using them. The best they
can assume is that he is at the centre of a transparent Earth,
("geocentric"), in which case parallax and refraction would always be zero.
However, a real navigator will be somewhere on the Earth's surface, so he
will have to make allowance for the effects of parallax and refraction
before the tables can be any use to him.

Parallax and refraction shift only the altitude of the Moon and the Sun (or
other body), hardly at all the azimuth. However, the lunar distance is
generally in a direction that is skewed to those shifts. So the way it is
affected by those shifts is rather complex, and requires a knowledge of the
altitudes of the centres of the two bodies, the combined correction for
(refraction + parallax) for each, and the measured lunar distance angle.

Refraction corrections are in general not very large (a very few minutes at
most, provided altitudes do not go below 10 degrees). It's still important
to correct for refraction as exactly as possible. In comparison, the
parallax correction for the Moon can be immense (up to 1 degree) and
changes rapidly through the day: it's a dominant factor.

It's necessary to aim for the highest possible standard of precision. In
perfect conditions, by averaging many measurements, one might aim to get a
measured lunar distance to, say, 0.2 or 0.3 minutes. To retain this
precision, the many correction terms should be estimated, where possible,
to within 0.1 minutes, and the calculations carried out to that same
accuracy. The Nautical Almanac tabulates positions of Sun and Moon to 0.1
min

If printed tables of logs, trig functions, and log trig functions are to be
used, then 5-figure tables are required. Generally, scientific pocket
calculators will provide sufficient accuracy. If a computer program is
being used, which offers a choice between single- and double- precision,
double-precision should be chosen. If no such choice is offered, then it is
likely that the program is inherently precise enough.

It's up to you whether to do your calculations in degrees-and-minutes, or
in decimal degrees. It depends somewhat on whether you are using tables or
else a computer or calculator. Beware, although some calculators offer
conversions between the two, some get it wrong, especially for negative
angles.

INITIAL STEPS.

First, average the lunar distance measurements and the altitudes, in such a
way that the averages correspond to the same moment of time.

Correct all three sextant observations for scale calibration errors (from
the certificate in the sextant box) and index errors. Correct the Sun and
Moon altitudes for dip and semidiameter (subtracting if upper-limb).
Correct the observed lunar distance by adding the semidiameters of the Moon
and (if it's used) the Sun (Moon semidiameter may instead have to be
subtracted, in the case of a star or planet, if its distance was measured
"across" the Moon).

For the semidiameter of the Sun, you can take the daily figure (in minutes)
from the almanac, and convert to degrees as required. For the Moon
semidiameter, make a very rough guess at GMT (within a couple of hours will
do), look up Moon's Horizontal Parallax (HP), tabulated each hour in the
Nautical Almanac, convert to degrees, and multiply it by .2724. Then
multiply by (1 + Sin (alt)/55), which is the "augmentation of Moon's
semidiameter". This is better than simply using a tabulated value for the
Moon's semidiameter.

This work results in three quantities, d (apparent lunar distance between
Sun and Moon centres); also s (apparent Sun-centre altitude) and m
(apparent Moon-centre altitude), measured up from the true horizontal.
Altitude s may apply to a star or planet, instead of to the Sun.

The apparent lunar distance d is slightly affected by the effects of
refraction and greatly affected by the effect of  parallax (mainly that of
the Moon). However, because of the skewed angle that the lunar distance
makes in the sky, these effects combine in a rather complex manner.
Allowing for these combined effects is what's known as "clearing the lunar
distance". It provides the true lunar distance, which we denote D.

CLEARING THE LUNAR DISTANCE.

Cotter goes through many methods for this operation, and mentions many
more. He refers to a paper published in 1797 which described forty such
methods!  I propose to refer to Young's method only (J.R.Young, Practical
Astronomy, 1856).

Correct both observed altitudes, m and s, for refraction, using whatever
method suits you best. If atmospheric conditions are abnormal, correct
refraction for temperature and/or pressure beforehand. The refraction
correction to the apparent altitude is always a SUBTRACTION. The altitudes
must also be corrected for parallax. For the Moon, where getting its
parallax right is very important indeed, look up the current value of
Horizontal Parallax as before, apply "Reduction of the Moon's horizontal
parallax" by multiplying it by (1- (sin(lat))^2/300), multiply by cos(alt),
and that is the parallax correction, which always INCREASES the apparent
altitude (and always greatly exceeds the refraction). The symbol ^2 means
"to the power 2", or more simply "squared".

For the Moon, then, the corrections are such that M will always exceed m,
and for the Sun (or other body) S will always be less than s.

So now we have the apparent and true altitudes of the Sun (s and S) and the
Moon (m and M), and the apparent lunar distance d. We require the true
lunar distance D, in which the effects of refraction and parallax have been
allowed for.

For this, Young gives-

cos D = (cos d + cos (m+s))*(cos M*cos S)/(cos m*cos s) - cos (M+S)

or D = acs ((cos d + cos (m+s))*(cos M*cos S)/(cos m*cos s) - cos (M+S))

where acs means "arc cos" or "the angle whose cosine is"
The derivation is given in Cotter page 213.

Because it relies in finding an arc-cos to obtain D, this will be rather
imprecise around lunar distances of 0 deg and 180 deg, but that's not a
real drawback for our application.

Note that all angles should be in degrees unless your program will work
only in radians, in which case it's necessary to convert by multiplying or
dividing by 180/pi appropriately.

Also note that some programs such as Quickbasic do not provide functions
such as acs (arc cos) and asn (arc sine). However, atn (arc tan) is
provided. Instead these functions must be obtained indirectly using
 acs(x) = atn ((sqr(1-x*x))/x)
 asn(x) = atn (x/(sqr(1-x*x)))
where "sqr" means "square root of".

At last, this provides a corrected lunar distance, D, which can be compared
with tabulated lunar distances to obtain Greenwich Time.

This is the end of part 1. Part 2  explains how to obtain tabulated lunar
distances, so that the GMT can be deduced. Some examples are given.
Part 3 will come at a future date. It will discuss the possibility of
calculating the altitudes required with a lunar distance, instead of
measuring them: also the question of observing Moon altitudes in place of
lunar distances. It will show how to obtain local apparent time and use it
with GMT to obtain longitude, as early navigators had to do.


George Huxtable.

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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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