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Latitude and Longitude by Sunline near LAN.

    This seems to be one of Frank�s pet sights and is outlined in the
more recent editions of Bowditch (1995 and newer).  Essentially we are
taking a range of altitudes over a period of time before, during, and
after the sun crosses the meridian of the observer (LAN).  The raw
sextant sights are then plotted on the Y-axis, and time (GMT) is
plotted on the x-axis.  Once plotted, the curve should be a parabola
and can be folded so that the two halves mirror each other.  Depending
on the number of sights and accuracy of those sights, the curve should
be fairly uniform.  The crease, when crossing the x-axis should show
you the exact time of LAN.  This time can be then entered into the
almanac to find the exact GHA of the sun at that moment.  GHA
converted to Longitude should give the Longitude of the observer at
the moment of LAN.  This curve will be skewed to a greater or larger
extent based on the change in declination of the body and the North/
South component of the ship�s motion.  Because of these factors, if
you use raw data, a prudent navigator would put only as much faith
into the position as can be reckoned knowing that errors may exist due
to these factors.
    To obtain Latitude, the nearest sight to the time of LAN (which
should be the maximum altitude, or very close) must be converted to Ho
then the z-distance and declination applied as a straight LAN problem.
    A second method to obtain Latitude and Longitude from these numbers
is to reduce the data as standard sunlines.  A great number of sun
lines reduced and plotted should give an accurate position if done
correctly.  This is best done with a computer reduction and plotter as
the sheer number of sights to reduce to obtain good accuracy will be a
burden to attempt using tables or even calculators.
    This method can contain several errors and can cause some
difficulties.  I think that the best time and place for this sort of
sight is near the solstice (when the change in declination is quite
small) and in the tropics (where the change in altitude is fairly
rapid over time.)  Fast moving ships, especially with Northerly or
Southerly courses, will also cause a skewing of the curve as the
altitude will change with the change in Latitude.
I performed this sight in late May in Latitude 15 deg North, while the
ship was drifting due to engine problems (crs 270, Spd 0.6 knots).
This allowed for a change in Altitude of about 35� of arc over a
period of 24 minutes.  During this time, I shot 31 sunlines and
graphed them.  One sight proved to be in error as it skewed the curve,
but not so much as to cause a problem.  The change of declination from
start to finish was 0.2� so the effects of time were minimal in this
case.  Near an equinox this error would have been much greater and
probably would have skewed the curve visibly.
    With the graph and a bit of eyeball interpolation, I was able to
determine the time of LAN within 2 seconds with a corresponding error
of Longitude of about 0.7� which is certainly an excellent result.  I
saw the first change in altitude of the sun 3 seconds after the time
of LAN computed from the GPS longitude, and used this for my latitude
line, providing a 0.2� error south is Latitude.  This gave a total
position error of 0.9 nm which is quite good for Celnav at sea.
    After the graphic reduction, I plugged 25 of the sights (the maximum
allowed) into the computer and did a standard running fix via
Celestial LOP�s (lots of sunlines).  I excluded the obviously flawed
sight and several of the 30-second interval sights near LAN, keeping
all of the other sights some distance from LAN.  The error in Latitude
from this method was 0.1� and the error in Longitude was 0.4� This
gave me an error in position of 0.4 nm at the time of LAN.  This
method under these circumstances provided a very good position.
    The attached spreadsheet (MS Excel) shows the raw data.  The first
column shows the number of the sight.  All shots were shot after 12
PM, and the second column shows the minutes of the noon hour, while
the third column shows seconds.  The fourth column �time� is decimal
minutes.  Next is the Hs measure, first in degrees, followed by
minutes of arc to the nearest tenth.  Next we have Hs in decimal
minutes.  The last two columns were time and Hs which were used for
the plot which is below and should be printable as a separate page for
people to try at a decent scale.  The box on the right is the data in
a more concise form including errors as computed by sailings.  I also
included a short form of my Lat by LAN computation.

The conclusion I find with this method is that if it is done at a time
and place to eliminate most of the variables, it can be utilized at
sea without too many calculations to determine both latitude and
longitude.  The real trouble begins if your latitude varies greatly
from your declinations, you are moving rapidly north to south, or you
don�t have the hour it takes to shoot and reduce this sight.  I still
believe that an AM sunline crossed with a LAN is your most convenient
method to get a noon fix and can be used without thinking of all of
the variables that will creep in using this method in various
locations and at times more distant to the Solstice.

I am attaching the MS Excel spreadsheet to my next post that I used
for the graphic display.  Unfortunately I cannot electronically
reproduce the sunline method of reduction, but it uses the same data
as on the spreadsheet.

Jeremy
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