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Perhaps I don't understand the full constraints you have in mind , but if
you have time and an almanac, you don't need an accurate DR position or
really any DR position at all. You can determine the circles of constant
altitude for three or more celestial bodies (assuming a spherical earth -
something quite reasonable for practical navigation) and find their best-fit
intersection numerically. A problem arises if you need to plot the circles
of constant altitude on a Mercator chart to find the intersection since they
are not simple circles or even ellipses, but rather complex figures (this
does simplify if the GP of the body is near your actual position, the "large
altitude" case, where circles are used as good approximations). Then you
really do want something like the Marcq St. Hilarie intercept method or even
the older noon sight along with the morning/evening "time sight".

The intercept method will work even with a rather bad DR because the
resulting LOP is not as sensitive to errors in DR position that preserve
azimuth. Errors in the azimuth direction are small when the distance of the
GP of the body is large compared with any DR error. This also remains true
even if the DR error is large but it doesn't affect the azimuth direction
significantly. But if the DR error is such that the calculated azimuth to
the body's GP is now in significant error, there would be a very significant
error in the LOP and final fix. The classic example is that of "large
altitudes" where the intercept method and its LOP are in fact replaced by an
altitude circle. In this case any azimuth errors ca n be large because the
GP and DR/actual position are way too close.

Hope this helps.

Rod Deyo

----- Original Message -----
From: "Peter Fogg" 
To: 
Sent: Tuesday, April 09, 2002 4:20 PM
Subject: Re: Timing Noon


> George Huxtable wrote:
> 'the Sun must be on the meridian IF AND ONLY IF he is at a (known)
longitude'
>
> The practical situation is being all at sea. The 'known' longitude is the
DR
> assumption. Unlike celestial navigation (CN) generally (more on this
below)
> this DR needs to be reasonably accurate for my postulated Noon Fix
position.
> Assuming morning star sights were taken, followed by a Longitude Sun sight
> when the Sun's azimuth was at 90� (more below), and the boat has been
heading
> in a constant direction at a known speed, without too much interference
from
> current, assuming all this (phew!) then the DR should be accurate enough
for a
> Fix - with the safeguard that if observations of meridian passage don't
> coincide with the exact, known, time then this is an indication that
perhaps
> the DR is out, or something else is awry. What I was trying to say in the
> earlier posting was that one takes observations of the sun over the few
> minutes the Sun appears to hang in the sky, while knowing the exact moment
it
> happens, providing the DR is good.
>
> Granted that I went a bit overboard about presumed accuracy while at sea.
>
> If the observer finds herself (or himself) at sea during summer in a wide
> range of Latitudes (reasonable enough assumption) then a Sun sight at 90�
of
> aximuth will yield a position line (LOP) running north/south - in other
words
> a meridian of Longitude. This can then be advanced to cross the Sun's
meridian
> passage sight, which gives an east/west LOP, or Latitude line. Then later
on
> in the afternoon the Sun's azimuth will be at 270�, leading to another
> Longitude LOP. And so we go on, constantly testing our DR assumption,
> constantly refining our calculated position.
>
> But what if we have no idea where we are to start with? I remember when
> studying CN (well, when starting this study) this bothered us all. What if
> there was this humungous storm? That went on for weeks? Followed by more
weeks
> of fog? And then a monsoon? Where would we be then? Although the
instructor
> was sceptical about our concerns he plotted a Fix using a DR some
thousands of
> miles away. Obviously the intercepts were long. THEN he took the Fix and
> re-used it as a DR and re-plotted, and lo and behold, the result was quite
> acceptable. Since then I've used this simple technique myself, when trying
to
> solve CN problems that deliberately don't give a DR position. With one, my
> first intercepts were 500 odd miles long, but the second calculation was
close
> enough not to encourage me to do the process over again - in practice one
> would go with that until the next round of sights.
>
> Still have a query. To what extent could one use a calculated Longitude,
such
> as the 90� and 270� azimuths, or perhaps meridian sights made on land,
with
> north and south sticks in the ground if necessary, to ultimately set one's
> clock? This is where this is all going, to try to find a reasonably simple
> (sorry, lunatics) way to establish, and regulate, the Longitude/Time
question.
>
> Bring back the Board of Longitude!
>
> 'presumably Thornleigh, near Sydney' Very close, its actually Westleigh,
about
> a mile to the north west, but as Westleigh is still, happily, mostly bush
> Thornleigh (with its railway station) is often the only suburb marked on
maps.
>
> Peter Fogg
>
>
> George Huxtable wrote:
>
> > Peter Fogg had some kind words to say and then-
> >
> > >The almanac contained within the electronic nav calculator I use gives
> > >me a precise moment for meridian passage of the Sun, according to the
> > >Lat/Long and date entered.
> > >
> > >For example, tomorrow Wednesday the 10 April, 2002
> >  at my position
> > >S33�44' E151�04, LAN occurs at 11h57m10s, the Zone Time is 10 hours
> > >ahead of GMT.
> > >I would be happy for anybody to check this data, any way you can, often
> > >wonder about just how accurate it is.
> >
> > I have checked this prediction for local apparent noon on that date and
at
> > Peter's location (presumably Thornleigh, near Sydney) on my own pocket
> > calculator. This was programmed using the data from Meeus "Astronomical
> > formulae for calculators", and hence Newcomb.
> >
> > It gives me the same answer, within a second, as Peter obtained, i.e.
> > 01:57:10 GMT.
> >
> > Where does this time derive from? At Greenwich, the Mean Sun passes the
> > meridian at 12 noon. Thornleigh is 151�04 further East in longitude, so
at
> > 15� per hour, the mean Sun would transit the meridian there 10h 04min 16
> > sec earlier or at 01h 55min 44sec GMT. However, the equation of time is
> > then 1min 26sec, in the sense that the real Sun lags 1min 26sec behind
the
> > mean Sun at that point of the year. The real Sun will then pass the
> > meridian of Thornleigh, and everywhere else on that same line of
longitude,
> > at 01h 57min 10sec GMT. This is the GMT of Local Apparent Noon at
> > Thornleigh on that day.
> >
> > >If correct, its a great asset, since it also gives me my precise GHA,
> > >and thus Longitude, at the moment of meridian passage, as confirmed by
> > >my (corrected for error) ship's clock.
> >
> > I'm puzzled about this. If Peter had set up a couple of posts in his
garden
> > on an exact North-South line, he would be able to tell when the Sun
passed
> > the meridian. But otherwise, how can he tell it from the clock, even if
> > that clock has been set exactly to read GMT? If the clock reads
01:57:10,
> > then he knows the Sun must be on the meridian IF AND ONLY IF he is at a
> > longitude of E151�04. But if he isn't at that longitude, the Sun won't
be
> > on the meridian. How can he tell if the Sun is on his meridian? He is
> > indulging in a circular argument, and assuming what he is trying to
> > measure, as I see it.
> >
> > All Peter is able to measure accurately at noon is the Sun's altitude,
> > which taken with its declination will give him his latitude. Finding the
> > longitude to any accuracy requires a further measurement of Sun
altitude,
> > earlier or later in the day (or both).
> >
> > If I have completely missed the point of what Peter Fogg is explaining,
I
> > hope he will forgive me and put me right.
> >
> > George Huxtable.
> >
> > ------------------------------
> >
> > 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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