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From: Chuck Taylor (no email)
Date: Thu Aug 05 2004 - 00:04:26 EDT
Robert Gainer asked about how a time sight works, and
how compass corrections fit in.
Robert,
The basic formula for a time sight, given in Bowditch
(1984 and other editions), is
cos(t) = ( sin(h) - sin(L)*sin(d)) / cos(L)*cos(d)
where you are finding
t = meridian angle
as a function of
h = observed altitude (corrected)
L = latitude, assumed or obtained by other means
d = declination
A time sight can be taken at any time you can see the
sun. The above formula is a derivative of the Law of
Cosines for a spherical triangle. You solve for
meridian angle (and hence longitude) by assuming that
the latitude is known.
Notice that if your assumed latitude is off a bit,
then your solution for the meridian angle (and hence
for longitude) will also be off by some amount. It so
happens that this error is minimized if you take the
sight at the time at which the sun crosses the prime
vertical circle (PVC). That is one reason why time
sights were traditionally taken at the time of PVC
crossing.
The time of PVC crossing is usually computed, either
by tables (such as Table 25 of Bowditch 1981 Vol 2),
or by formula. You don't need a magnetic compass for
that.
For an intuitive picture of PVC crossing, picture
yourself in the northern hemisphere (say at 48 deg
north latitude) between the spring and fall equinoxes.
Note that the sun rises somewhat to the north of
east, and sets somewhat to the north of west. Yet,
when it crosses your meridian at noon, it is to the
south. If it starts the day to the north of the
east-west line, yet is to the south of the east-west
line at noon, it follows that it must cross that
east-west line on its way south sometime between
sunrise and LAN. At that time it will bear due east.
That is the morning PVC crossing. Similarly, sometime
between noon and sunset, it must again cross the
east-west line going north. That is the afternoon PVC
crossing, at which time the sun will bear due west.
Others have pointed out that you don't always have the
benefit of a PVC crossing. For example, at latitude
48 north (where I live), there are no visible PVC
crossing between mid-September and mid-March (the
equinoxes). As I mentioned earlier, a time sight of
the sun can be taken at any time that the sun is
visible.
To understand the other major reason why time sights
were traditionally taken at the time of PVC crossing,
try to put yourself in the mindset of an 18th century
navigator, before the time of Sumner and St. Hilaire.
The concept of a line of position had not yet been
introduced. Finding your position meant finding your
latitude and then finding your longitude as a separate
and distinct process. First, you found your latitude
by observing Polaris or by observing the sun at LAN.
Note that finding your latitude is equivalent to
measuring the distance between you and the equator (or
pole) in a north-south direction. Naturally, your
observation is of a body that is due north or due
south of you. Similarly, finding your longitude was
thought of as equivalent to finding the distance
between you and the prime meridian (Greenwich for the
English-speaking world). What better way to measure
an east-west direction than by observing the sun when
it bore due east or due west? Given the historical
context, this made perfect sense to navigators of the
time.
After the introduction of the line of position methods
developed by Sumner and St. Hilaire in the 19th
century (although adoption and acceptance of these
methods lagged well into the 20th century), navigators
recognized that it was no longer necessary to take
separate sights for latitude and for longitude, and
that it wasn't necessary to shoot a sight at the time
of PVC crossing (or at LAN for that matter). Two LOPs
obtained at any time are at least as useful as
separate latitude and longitude sights. That is a
natural concept for us today, but it would have been
incomprehensible to most (if not to all) 18th century
navigators.
I hope this helps.
Best regards,
Chuck Taylor
North of Seattle
--- Robert Gainer <> wrote:
> Chuck said,
> >The more traditional way of determining longitude
> was
> >to use a time sight at the time the sun crosses the
> >Prime Vertical (i.e., the time at which the sun is
> due
> >east or due west of you). ...
>
> Chuck,
> I don’t understand how that will work. The magnetic
> variation and the
> latitude must be problems in that method. If you are
> at 23 degrees north
> latitude or greater the sun is never due east or
> west. If you do not know
> the magnetite variation with some degree of accuracy
> wont that have a very
> large effect on the method? Is this practical at
> all?
> All the best,
> Robert Gainer
>
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