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Subject: Re: Errors in Cotter's book, updated
From: George Huxtable (george@XXX.XXX)
Date: Thu Jan 09 2003 - 10:23:50 EST
Fred Hebard said-
>In noon sights, would one ever correct for third-order effects such
>as acceleration or decceleration in the declination, say around the
>middle of May? These would be much smaller than your 15-second
>correction, mis-shaping your parabola around noon but not changing
>the point of inflexion unless you tried to measure it with only two
>readings. Compared to clock errors in the old days, the 15-second
>correction due to the change in the declination doesn't sound like a
>major one.
Taking his last point first, consider a pair of equal-altitudes of the Sun,
either side of noon, being used to establish longitude by chronometer. At
the rate the Earth turns, 15 degrees per hour, a time-error of 15 seconds
due to declination change would give rise to a longitude error of nearly 4
arc-minutes, or nearly four miles when near the equator. In the context of
chronometer navigation, quite a serious error, and one that's well worth
correcting for.For a vessel travelling North or South at 10 knots, the
correction becomes 37 miles, so gets very important indeed.
The error I am referring to above is the time-difference between the moment
when the Sun appears to be highest in the sky, and the moment of true noon
when it's on the observer's meridian.
The argument I put forward in my last mailing was only valid over a limited
range of times, either side of noon. It was put forward to counter Cotter's
assertion, which was as follows-
>page 264. Cotter says, about finding the moment of noon by equal Sun altitudes-
>
>"By taking the equal-altitude sights shortly before and after noon the
>necessity for >applying a correction for the change in the Sun's
>declination in the interval is >obviated, since any such change will be
>trifling."
>
>I disagree with Cotter's analysis here. It seems to me that the correction
>necessary for >a changing declination does not reduce as the interval
>chosen gets closer to noon.
Cotter was suggesting that the correction reduced to become "trifling", as
the pair of observations was made closer and closer to noon, and my
argument was that it remained unchanged, over a period near to noon.
My suggestion that a plot of altitude against time approximates to a
parabola certainly becomes invalid as one gets much further from noon, and
the argument for a constant time-correction may well become invalid then,
as Jan Kalivoda has pointed out to me privately.
Next, Fred asks me about the importance of second-differences in this
context, because the Sun declination is never changing at quite a steady
rate. It's a fair question.
The second-difference is the effect of curvature of the declination plot,
and would have its greatest effect near a solstice (not in May as Fred
suggests). There's an arithmetic error in the last of Fred's solstice
figures for rate-of change of slope, which should be 0.4 minutes per day
per day, not 0.6. This makes little difference, though. Without doing any
arithmetic, I would guess that the effects of any of the values in this
column are quite negligible.
Linearity (or rather non-linearity) might have a more serious effect when
interpolating in the old lunar distance tables to obtain Greenwich Time,
particularly because that interpolation had to be made over a 3-hour
interval. Nowadays, the almanac data (from which you have to make your own
prediction of lunar distance) are given at one-hour spacings, which reduces
such interpolation errors by a factor of 9.
On page 241, Cotter (referring to 3-hour lunar distance predictions)
states- "In extreme cases an error of 50' of longitude results when the
correction for second differences is ignored". Cotter does not state what
these extreme circumstances might be, and frankly, I find it hard to accept
that errors of 50' of long (which would correspond to a time error of 3 min
30 sec) are possible from this cause. Unless, perhaps, a very bad choice
had been made, of a star that was way out-of-line with the path of the
Moon.
My copy of Nories Tables, 1914, has table LIV, "To find the exact Greenwich
Time corresponding to the true Lunar Distance", which corrects for the
non-uniform motion of the Moon over the three-hour time period. The
worst-case correction that this table covers is no more than 31 seconds of
time, which would correspond to no more than 7 minutes of longitude. This
is a small correction compared with the other inaccuracies of a lunar, and
seems more real than Cotter's figure of 50'. I am inclined to reject
Cotter's worst-case estimate of 50' of longitude error, caused by
non-linearity of the Moon's motion, as fanciful. Can anyone offer other
thoughts on that matter?
George Huxtable.
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