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There seems to be an interesting argument brewing up between Herbert Prinz
and Bruce Stark, and I wish to wade in, in support of Bruce.

Truth to tell, I am unable to put my finger on the exact point of the
disagreement, but that isn't going to deter me.

The subject is as follows-

In measuring a lunar distance between the Moon and another body, to
determine Greenwich Time, the altitudes of both bodies are required in
order to make certain corrections to the lunar distance. If these altitudes
cannot be measured (because the horizon cannot be seen, perhaps) then they
must instead be calculated, using the Almanac predictions. However, to look
up the position of these bodies from the Almanac requires a knowledge of
Greenwich Time, and to compute the altitude requires a knowledge of the
observer's position. However, both the Greenwich Time and the observer's
longitude are unknown at this stage, and it's the object of the exercise to
determine them. So how can the necessary altitudes be calculated with
sufficient accuracy?

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

If initial values can be guessed for the Greenwich time then this allows
(interim) Almanac sky-positions of the two bodies to be calculated. Then
guessing an observer's longitude allows (interim) altitudes to be
calculated, so that (interim) corrections can be made to be made to the
lunar distance. Then an (interim) value for the Greenwich Time and hence
the longitude can be obtained, and as long as these are closer to the true
value than the initial guess was, the guessed values can be replaced by the
new ones, and the whole operation repeated. This process (iteration) can be
repeated indefinitely until eventually each cycle of iteration makes no
significant difference to the result.

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

It seems to me that the argument boils down into whether the iteration
converges slowly (in which case many iterations are required) or quickly
(in which a single iteration is likely to suffice). Originally, I believed
that the iteration was unavoidably slow (and said so, in postings to this
list). But more recently, Bruce Stark has convinced me that if a navigator
goes about it the right way, he can achieve very fast convergence. Bruce
has spelled it out well, and this posting will not add much that is new,
but may provide a somewhat different perspective.

Let me explain. What I am going to do is to simplify everything as far as
possible, neglecting any unimportant factors, but while disposing of this
bathwater, try to preserve the baby. The following arguments will be
roughly right, but not precisely so.

Although clearing the lunar distance is a complex matter involving the
altitudes, parallax, and refraction for both bodies, we will for now assume
that the refractions will be unimportant for our argument (we will assume
that altitudes are always above 20 degrees to ensure this), and assume that
parallax for anything but the Moon is negligible. Then the one dominant
term remaining is the parallax of the Moon, and we can forget about the
other-body, and its altitude, altogether. Next, we can assume the
worst-case (and simplest) geometry in which the Moon is at, or close to,
the observer's zenith. In that case the geometry is such that every minute
of parallax gives rise to a one-minute displacement of the Moon, in the
East-West direction. Any other configuration would displace the Moon less
than that.

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

Method 1.HERE'S THE BAD WAY TO CORRECT THE LUNAR DISTANCE..

Consider first this naive method, that ends up with a very slow
convergence. Don't commit it to memory, though, as it's such a bad way to
do the job. First, assume that an accurate measurement has been made of
latitude, from a Sun meridian altitude, and that later a lunar distance has
been taken, but without measuring the Moon's altitude, so that altitude
needs to be calculated. We don't know yet the Greenwich Time, but we will
make a guess at it. Just imagine that guess is an hour out: a big error, I
know.

Next step is to work out the position of the Moon in the sky, in
declination (dec) and Greenwich Hour Angle (GHA), as seen from the centre
of the Earth.. The Moon's dec can change by up to 15 arc-minutes in an
hour, increasing or decreasing, so our hour error might contribute an error
of up to 15 arc-minutes due to declination change. Now consider the Moon's
GHA, which  changes by nearly 15� in an hour. So our one-hour error in
assumed time may give rise to a colossal error of 15� in our estimated Moon
position in the sky, in an East-West direction. We can ignore the changes
in declination, in comparison.

With a high Moon, a 15� error in GHA can give rise to a 15� error in Moon
altitude. Now we have to obtain the Moon parallax correction from that
altitude. Leaving out some minor corrections, the parallax is, roughly
speaking

parallax in degrees = 1� x cos altitude

so when the altitude is near 90�, an error of 15� can result in an error in
parallax of up to 1� x cos 75�, or 0.26�

With our assumed worst-case geometry, this will produce an error in lunar
distance of the same amount, 0.26�. And an error in lunar distance of 0.26�
will give rise to an error of nearly half-an-hour in Greenwich time.
because the lunar distance changes by about half a degree per hour.

So there you have it. An error in the guessed Greenwich time of an hour
gave rise to an error in the calculated time of half-an-hour. That was one
"iteration". We could go round that loop once again, starting with a better
guessed time, to reduce the error further, to about 15 minutes, and so on.
It's a very slow convergence, and rather an impractical one. A bad way to
do it.

Note also, that we haven't even considered any errors caused by an initial
guess at the observer's longitude.

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

Method 2. HERE'S A MUCH BETTER WAY TO CORRECT THE LUNAR DISTANCE.

Just as last time, assume that an accurate measurement has been made of
latitude, from a Sun meridian altitude, and that later (or earlier) a lunar
distance has been taken, but without measuring the Moon's altitude.

This time, however, the navigator also measures an altitude of the Sun,
ideally that same evening or morning, at a time well away (a few hours)
from Noon. This is what was called an "observation for time", and taken
with the known latitude, allows an accurate calculation of the Local Hour
Angle of the Sun at that moment, within a very few minutes of arc. This
requires no knowledge of Greenwich time.

The Local Hour Angle of the Sun, expressed in degrees East or West of the
observer, is exactly the same as the Ship's Apparent Time, in hours before
or after noon, converted at 15� per hour. This allows the ship's clock to
be set, rather accurately, to the ship's apparent time, at that moment.
From then on, for as long as the clock keeps adequately good time, and the
ship's travel East or West is well logged, the local hour-angle of the Sun
can easily be obtained, at any moment, from the reading of that clock.

The eventual intention of the navigator is, presumably, to determine his
longitude from the Greenwich time he will obtains from a lunar distance, so
the ship's apparent time is needed anyway. It's no extra burden, then, to
measure it for the purpose of correcting the Moon's altitude.

We now have the LHA of the Sun, but to calculate the altitude of the Moon
we need the Moon's declination and LHA. Together with the observer's
well-known latitude these will provide a navigation triangle that will
allow the Moon's altitude to be calculated as seen from the ship's
position.

The Moon's dec presents few problems: we can make a guess at the Greenwich
time, as before, and even if that guess is an hour in error (just as we
assumed before) the dec interpolated from an Almanac won't be more than
about 15 arc-minutes away from its true value.

What about the LHA of the Moon? Well, now we know, accurately, the LHA of
the Sun, and we know that the LHA of the Moon will differ from it by just
the difference in their GHAs at that same moment.

That is, LHA Moon = LHA Sun + (GHA Moon - GHA Sun)

Again, we make the same guess at Greenwich time as before, and interpolate
from the Almanac the GHAs for Moon and Sun. But now there is a big
difference from the previous case. Although the GHAs of moon and Sun are
both increasing at that very high rate of 15� each hour, those rates are
closely matched, and their DIFFERENCE, the term in brackets in the equation
above, is changing by only 0.5 degrees in each hour, roughly speaking. This
is where Bruce's factor-of-30 improvement derives from. It's the nub of the
matter.

So if our estimate of Greenwich time was an hour in error, the resulting
error in the Moon's calculated LHA would be no more than about 0.5�, or 30
arc-minutes. This might combine with the possible error in dec of 15
arc-minutes to result in a displacement of the Moon's position in the sky
(with respect to the Sun) of no more than about 34 arc-minutes in the worst
case, as a result of the error of one hour in the assumed Greenwich time.

That displacement would give rise to a worst-case error of 34 arc-minutes
in the Moon altitude, which could alter the parallax correction by no more
than 0.6 arc-minutes. In turn, that could give rise to a worst-case error
in the lunar distance of 0.6 arc-minutes.

At the Moon's rate of motion of about 30 arc-minutes in an hour, that would
end up with an error in Greenwich time of no more than 72 seconds of time.
It has reduced our initial gross error of 1 hour in presumed Greenwich
time, to a rather trivial 72 seconds: a factor of 50 in one iteration! So
in most cases a second iteration will not be called for.

Although I have concentrated on the Moon's parallax, because it is by far
the major contribution, the altitudes of both bodies involved in a lunar
have to be calculated in a similar way, to provide both refraction and
parallax.

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

So you can see how working from a well-measured LHA of the Sun has
completely transformed the situation. Instead of requiring many iterations,
a lunar distance, using calculated altitudes, is likely to generate a
good-enough Greenwich time, and therefore longitude, in one go, as
navigators will usually be able to guess a value for Greenwich time that's
much less in error than 1 hour. Two iterations will always suffice.

I found it hard, over several months, to accept the advantages of method 2,
but now I am fully convinced that it's really the only way to do the job.

I ask Herbert Prinz which of these methods (or perhaps some other method)
he uses when the altitudes of the bodies have to be calculated.

Herbert says-

>in your latest post you offer a variant of the classical iterative
>approach, as it is described, for instance in Chauvenet's Manual, or in
>>Cotter's History.

I don't have ready access to Chauvenet, but I do to Cotter's "History of
Nautical Astronomy". What Cotter has to say on the topic is very condensed,
and not (to me) particularly clear. It's on page 206, as follows-

" The altitudes of the Moon and the second object are required in order to
ascertain the exact values of refraction and parallax-in-altitude for the
Moon, and refraction for the second body. Moreover, the time of the
observation must be known with tolerable accuract in order to ascertain the
Declination and Right Ascension of the Moon (and the Sun if he is the other
body). In circumstances when it is not possible to measure the altitudes of
the Moon and second object at the time the lunar distance is measured,
these angles must be computed by solving the appropriate PZX triangles".

In that way Cotter has rather skated round the difficulties of this topic,
and has carefully avoided committing himself to stating how, exactly, the
job is to be done. Who can blame him?

Cotter complicates matters by introducing the Right Ascension of the Moon,
a concept familiar to astronomers but which for navigators was replaced in
the Almanac by GHA, from 1952. Cotter was writing in 1968.

Method 2 appears to have a long history behind it. The father of the lunar
distance, Nevil Maskelyne, in the British Mariner's Guide (1763), addressed
this problem of computing the altitudes of the bodies if they were not
observed, and offered a solution which looks very similar to method 2,
though I have not analysed it in detail. Bruce Stark has inroduced me to a
manuscript by the American astronomer Robert M Patterson, providing
astronavigation directions for the 1805 Lewis and Clark expedition, which
followed method 2 to compute altitudes for the Moon.

It will be instructive to thrash out this interesting matter somewhat
further until all the interested parties manage to arrive at an agreed
answer to it.

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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