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Thanks fo Peter Fogg for his good-natured response. I think he and I are
converging, to some extent. He raises some points which we can draw on
further.

>Know of two methods to determine the precise moment of the Sun's meridian
>passage
>by observation.

Well, in a previous mailing I was careful to define a situation when
neither the vessel nor the Sun had any North-South component of motion.
Otherwise, an important correction must be made when finding the
moment-of-noon by the methods Peter describes to determine longitude: see
below.

>Firstly, to plot any number of sights up to and after LAN, then draw a
>curve that
>fits them. Apart from the difficulty of drawing smooth curves at any time, and
>especially on a moving boat, the resulting parabolic (?) shape has a
>fairly flat
>top, defying the choosing of an exact moment. As George says.

This plotting is a good technique, but Peter doesn't give a hint at the
length of the time-scale, before and after LAN, that he thinks the plot
should cover, which is an important matter. I have come many navigators who
think the job can be done over a few minutes around noon, Where I differ is
in maintaining that to get any reasonable precision in determining the
moment of noon, the plot should cover several hours, not several minutes.

Having made such a plot, a good way to find the moment of symmetry, for
noon, is to fold the paper, with a bright light behind it, so that the two
halves of the curve coincide. But it's not easy to plot precisely enough to
get that time to a fraction of a minute, when the plot covers several
hours: this is the weakness of the method.

>Secondly, making a timed observation of altitude before LAN, then noting
>the time
>when the Sun's altitude again reaches this level. The next step is to
>divide this
>time equally, giving, in theory, the exact moment of LAN. I don't understand
>George's objection that using this method has 'already conceded the principle'.

I have no objection at all to using this method: it's fine. The point I
wished to make was that it should measure time of noon, not AT or near
noon, but on either side of noon. Within limits, the further from noon the
measurement is made, the more accurate it will be. My comment was for the
benefit of those longitude-at-noon diehards, to point out that this is NOT
a longitude-at-noon technique. Perhaps I didn't make that clear.

Another method, not mentioned by Peter, is to use a noon sight to measure
your noon latitude (not the longitude) and use dead-reckoning to estimate
changes in the latitude until, much later in the day, when the Sun's
altitude has fallen considerably, the Sun altitude is measured again. At
that moment, then, at the new location L, you know the exact moment of that
second measurement according to your timepiece, the latitude (by DR), the
Sun's (corrected) altitude, and from the Almanac, the Sun's declination.
Take the complement of these three angles by subtracting each from 90�
(taking South as negative, which means that Southerly angles get added to
90�). You then get a spherical triangle with sides colat, codec (or polar
distance), and coalt (or zenith distance). Add these three, and halve the
sum, to give the half-sum s.

Then calculate an angle P from
P = 2*arccos(sqr(sin s*sin (s-coalt)/(sin colat*sin codec)))

P is the angle through which the world has turned from the moment of local
apparent noon at position L to the moment of the second altitude
measurement, also at position L, at 15 degrees per hour. So dividing by 15
gives the time interval, in hours, between the moment of local apparent
noon at position L and the moment of the second altitude measurement of the
Sun, also at L.

This is a precise method of discovering exactly when local apparent noon at
L occurred on the timepiece, depending on the DR for the change in latitude
over the period from noon, and on the clock for an accurate measure of that
period. It is accurate because it relies for its timing on a Sun altitude
late in the day, when the altitude is changing quickly. No correction for
North-South velocity of the ship or Sun, as discussed below, is required
when this method is used

Of course, a morning observation can be made in just the same way.

This was a common method of finding the moment of local apparent noon, to
determine longitudes in the days of lunar distances and also in the days of
longitude by chronometer.

>Celestial navigation in general uses a series of convoluted steps to
>arrive at a
>LOP, compared to coastal nav. where you take a bearing on an object on
>shore and
>there is your LOP. This process seems fairly straightforward and accurate
>enough.
>George has, if I understand correctly, pointed out that the Sun's seasonal
>north/south movement and/or the vessel's own north/south motion may conspire to
>rob this moment of its validity, although it seems the difference is too
>small to
>worry about. So much for robustness.

No! It's true that the difference between a LATITUDE determined from
maximum altitude, and a latitude determined from meridian altitude, is too
small to worry about, for the vessel speeds that most list members are
accustomed to. But a North or South component of velocity, even at just a
few knots, makes a marked shift to the TIME of the maximum altitude, and
seriously upsets a measurement of longitude that relies on the moment of
symmetry of the Sun's curve of altitude, or the mid-time moment between
equal altitudes.. For such a measurement, it is essential to take into
account the North-South motion of the vessel, as follows-.

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

Error between time of Sun's maximum altitude and time of meridian passage.
(adapted from Cotter, page 265-266)

To allow for the correction that is required by the North-South component
of ship's speed. When travelling towards the Sun, (for example heading to
some extent Southwards when the Sun is more South than you are) the effect
will be to delay the moment of greatest altitude until AFTER the moment
that the Sun has passed the meridian. The meridian moment is what you need,
with the equation of time, to get your long from GMT. So you must subtract
from the moment of maximum altitude (i.e. the mid-point between your equal
altitudes) a time in seconds calculated as shown below.

A table is provided below in which you enter your latitude and Sun
declination. You can easily interpolate by eye (or extrapolate a bit when
dec exceeds 20 deg). Precision is NOT needed. The table gives you a
multiplier for your North-South component of speed in knots to arrive at a
time correction in seconds.

Take the North-South component of your speed in knots (which you can treat
as a positive number, even if it's Southerly), and multiply it by the
number taken out of the table below. This is the correction in seconds of
time, always a positive number. Subtract it from your equal-altitude time
if your North-South component of speed is toward the Sun, in which case the
Sun at Noon the sun will generally be forward of abeam. Add the correction
to the equal-altitude time if you are now receding from the Sun, in which
case the Sun at Noon will be somewhere aft. You will see from the table
that when the noon sun is overhead (lat = dec) the correction is always
zero.

Time correction in seconds for each knot of speed in the North-South direction.


  SunDecN20     N10       0     S10     S20 SunDec
Lat
N60      21      24      26      29      32
N50      13      16      18      21      24
N40       7      10      13      16      18
N30       3       6       9      12      14
N20       0       3       6       8      11
N10       3       0       3       5       8
  0       6       3       0       3       6
S10       8       5       3       0       3
S20      11       8       6       3       0
S30      14      12       9       6       3
S40      18      16      13      10       7
S50      24      21      18      16      13
S60      32      29      26      24      32
Lat

You can see that at a North-South speed of 10 knots, the error for noon can
reach more than 5 minutes of time.

This approach neglects accounting for the North-South motion of the Sun,
which is no more than 1 knot at the equinoxes, at it's greatest. You can
allow for this speed of the Sun by looking up in the Almanac its change in
declination, in minutes, over an hour's interval. If the Sun is moving
toward the vessel, add that speed to the vessel's: otherwise, subtract it.

Any such error in timing has little effect on the measured LATITUDE at noon.

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

>There is a small advantage to using noon (or close to) as a time for Fix,
>in that
>calculations for an ongoing DR are simplified so that they can often be done in
>one's head in the cockpit, without going below, compared to using some other
>complicated starting point for time. Remember reading in a cruising guide to a
>part of the Queensland coast (Great Barrier Reef) words to the effect of
>that one
>loses half of one's brain while on a boat. So true. Perhaps its the constant
>motion, the loss of normal sleep patterns, but while on board I like to
>have a set
>routine to follow for nav, using pre-printed forms and as little thinking as
>possible.
>
>'Has that helped?' Yes, greatly. Am still puzzling over just how knowing in
>advance the exact moment of the Sun's meridian passage (from my nav.
>calculator)
>can be of practical benefit. Its turning out to be the most detailed and
>enlightening lesson on meridian sights, and the Time/Longitude problem,
>that I've
>come across.
>
>
>
>
>Harking back to taking sights of the Sun at 90� and 270�, how can one be
>sure of
>measuring the altitude of the Sun at the precise moment of this Azimuth?

Why bother? Why not simply treat it as an ordinary position line and work
out its azimuth, whether or not it's at 90� or 270�. After all, the Sun can
only reach those azimuths in summer. Astronav has to be applicable all year
round.

>  Since
>the Sun is not hovering about at these moments, but zooming across the sky.
>(For the moment I'll pass over how to know in advance when this will
>happen - but
>am happy to expand on this if anybody is wondering)
>The answer is to start taking sights 2/3 minutes before this moment, and to
>continue for 2/3 minutes after. Then plot these sights graphically, and compare
>them to a calculated line representing the change in altitude of the body
>over 5
>minutes of time (this technique was explained in detail in an earlier posting).
>Then any point along this line which best fits the multiple sights can be
>used, in
>this case the moment of exact azimuth, leading to a precise north/south
>LOP, which
>is the line of Longitude.
>
>As with any single LOP, the disadvantage is that one has no check on the
>accuracy
>of this line. This is where I disagree (provisionally!) with George who says:
>
>'In practice, only two bodies are required, not three. Two bodies will give
>two position circles, which cross at two places. One of these crossings
>will be the position of the observer, and the other can be ignored as
>irrelevant. As long as the two bodies are well separated in the sky, it is
>obvious which of the two is the relevant crossing because they are so far
>apart on the Earth's surface.'
>
>The problem is that we are assuming that we don't have a DR - lost amidst the
>watery wastes! - so how will we know which of the intersections is the
>right one?

Well, you can choose two bodies (or two successive positions of the same
body) that are sufficiently far apart in the sky that there is little doubt
about it. The unwanted position may then be on land, or in a different
ocean. Surely, any navigator will know which ocean he is in! But in
general, it can only help to have more than two position lines, I agree.

>Three circles lead to a single (?) triangle, traditionally known as a 'cocked
>hat', the centre of which is our Fix position. An important exception to the
>centre (as in 'the doctrine of least squares' - more info available on request)
>being the Fix is where there is a known danger - lee shore, reef, whatever - in
>which case the Fix position becomes the closest point to the danger. This
>will be
>along one of the LOPs, so could well be the real position.

Well, the meaningfulness of a "cocked hat", whether from land bearings or
astro sights, is frequently misunderstood. It's a surprising fact that no
matter how good the navigator, only one time in four will his cocked hat
embrace his actual position, which is three times more likely to lie
outside it. This is a universal truth, relying in no more than this
proposition: that each position line, being the best estimate that can be
made, is just as likely to lie to the left of the true position as to the
right. So it's a big fallacy to imagine that the true position must be
within the cocked hat. Instead of a probability of 100%, it will be just
25%.
>
>It seems to me that three sights, with well separated azimuths, are always
>preferable to two for this reason - two give an exact point of
>intersection, but
>no indication about how accurate it may be. In general terms (and there are
>specific qualifications) the size of the 'cocked hat' triangle is a good
>indication of the accuracy of the observations: the smaller it is, the
>better - if
>the three LOPs met at a single point then that would be perfection - and
>about as
>unlikely as any perfection in this life.

If a whole series of cocked hats are consistently small, that indicates
accurate navigation. But a single small cocked hat may be no more than the
operation of chance.

Thanks to Peter for providing some arguments to chew over.

George.

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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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