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George wrote: First, let's go back to the matter of correcting the sextant
reading of the observed lunar distance, for semidiameters, arriving at the 
value
d (which will later be "cleared" to provide the true distance D). I thought
we had resolved that between us, but it seems to be worrying Kent, still.

Kent Nordstom seems keen to understand the remaining differences between
his lunar calculation and my own, and I agree that its well worthwhile. 
Let's
goon, to the bitter-end!.....Yes and no. I agree with all that except
forthose last two terms . This
is not the place for refraction correction to come in.

I think George is not correct on this. There is a need (small) to correct 
for
refraction when reducing the LD as well. It might be easiest to explain by
starting with what has to be done when reducing altitudes. In Jeremy's case
the UL of the moon was measured. The refraction has to be done for the
geocentre of the moon and not on the altitide for the UL. Consequently in my
way to do the reduction I start with refraction on the UL and then
I calculate the difference in refraction for the UL - SD for the moon. This 
is then
a correction for achieveing a correct refraction value. In this case the
angle between the distance and the vertical is 90d (of course). If George is
already from the beginning calculating the refraction for the geocentre that
is fine but then there is, as I see it, a difficulty coming up later when 
treating the distance.

I all cases there is an angle between the distance line and the vertical
used for measuring the altitude. Refraction compresses the body - more on
the LL than on the UL - which means that there is a need to correct also
the distance for refraction. The body is not circular but compressed in
directions apart from th the compression in the vertical. The arguments for
this correction is altitude and angle between the distance and the vertical.
This correction should be done for the moon and the sun. As an example for
altitude 8d 30s and angle of 45d the correction is about 5 arcsec's, while
when the angle is 0d, as in altitude reduction, the correction is 10
arcsec's. If the other body is the sun then a similar correction should be 
used. So it is not negligible to avoid this
correction when reducing the distance.

If George takes a look in the English reference below and puts this into
graph I believe George will agree with me.

I have three Swedish references and one English: Jeans, Navigation and
Nautical Astronomy 1853, page 105, header: Refraction, oulines the need for
this correction. Note: I have not searched actively for English referencies.

George wrote: I'm not familiar with that correction term >which was cos 
aziumuth x (diff geographic and geocentric latitude>,
and perhaps Kent will explain it, or refer to a text that does. But as far 
as I can estimate,
its practical effect in our exercise is less than an arc-second, and I doubt
if it can ever work out to be much more than that, so it seems well worth
ignoring

Firstly I conclude that George does not compensate for the earth oblateness. 
Unfortunatle I do not have any
English reference for this way to correct for earth oblateness, but I have 
two Swedish. I guess that this way of
calculating has been described in the German "Lehrbuch der Navigation"
concerning "moondistanzen, strenger metode" from about 1860 (textbook
navigation...rigirous method). But I am sure that George os able to do a 
search amongst English litterature.

Kent N

----- Original Message ----- 
From: "George Huxtable" 
To: 
Sent: Sunday, July 06, 2008 12:46 AM
Subject: [NavList 5705] Re: Lunar trouble, need help







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

First, let's go back to the matter of correcting the sextant reading of
observed lunar distance, for semidiameters, arriving at the value d (which
will later be "cleared" to provide the true distance D). I thought we had
resolved that between us, but it seems to be worrying Kent, still.

He now writes-

"For finding the apparent distance (what I believe George defines as d) my
model does the following:
Obs. distance + corr. for index error +/- SD for the moon +/-SD for the sun
(if used) +/- corr. for augmentation of the moon +/- refraction correction
for the moon +/- refraction correction for the sun (if used)."

Yes and no. I agree with all that except for those last two terms. This is
not the place for refraction correction to come in.

"The correction for augmentation is based on the arguments moon's
semi-diameter and apparent altitude. Is this what George means... "but
neglected to do so in correcting the lunar distance for semidiameters (when
it's crucial)."??? Is the "augmentation factor "someting else?"

No, nothing else, that's exactly what I was referring to. As far as I could
tell, Kent took his Moon semidiameter from Henning Umland, but then omitted
to adjust it for augmentation, when correcting to obtain d. We are in
agreement about Moon SD (as it would have been measured from the Earth's
centre) as 15m 25s, but that valued should then be "augmented". I had done
that by calculating a factor to multply by, which was 1.0147, as explained
in [5615], but augmentation could just as well have been taken from a table
in Norie's, which instead gives it as an amount to be added, of about 13
arc-seconds. It appeared that Kent hadn't applied that correction to SD when
obtaining d from observed lunar distance, which seemed to explain the
difference between us rather well. If I've misunderstood, I hope Kent will
put me right.

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

Next, back to the first part of his mailing, about a discrepancy in our
values for Moon parallax. That seems to be simply explained by a slip (easy
to make) of a degree in the altitude.

But let's touch, now on what Kent's recent message states-

"What my model does is as per below. As can be seen the calculation includes
two corrections for earth flatness (maybe a better English term is
oblateness?):
- find the azimuth to the moon
- find the difference between the geographic and geocentric latitude
- multiply  this difference with cosine for the azimuth
The azimuth is approx. 111d and the diff. between the latitudes is 5m 45s.
The product is +2m 6,46s, which gives a "local altitide" of  60d 38m 57,73s
+ 2m 6,46s =  60d 41m 04,19s to be used for parallax calculation. Due to the
earth oblateness the value is added to the true local altitude if the
azimuth is greater than 90d (the moon is pointing away from the pole),
otherwise the value is negative."

I'm not familiar with that correction term, and perhaps Kent will explain
it, or refer to a text that does. But as far as I can estimate, its
practical effect in our exercise is less than an arc-second, and I doubt if
it can ever work out to be much more than that, so it seems well worth
ignoring.

He continues-

"Next is a small correction to the moon's HP with the arguments latitude and
HP. This gives a "HP" of 56m 36s - correction 0,78s = 56m 35,22s."

This seems to correspond with the table in Norie's headed "reduction of the
Moon's horizontal parallax", which I mentioned in my last post-

"Now, if we're bothered to, we can make the correction for the reduction in
the Moon's HP on account of the spheroidal shape of the Earth, described by
Kent as "Earth flattening. At such a low latitude of 15�, this amounts to
only .01' (taken from a table in a modern Norie's), so we end up with a
corrected HP of 56.60' [or 56m 36 sec]".

That value is within a second of Kent's result; which isn't surprising
really, because the only difference between our procedures was that I had
ignored a correction that turned out to be perfectly negligible.

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

I asked Kent for some other quantities he used in his calculation, and he
came up with-

"My input data were:
- moon's refraction -29,82s
- sun's refraction -1m 21s
- sun parallax 7,3s"

For two of those, the Sun parallax and Moon refraction, he and I agree
precisely.

For Sun refraction, however, we diverge a bit, as I had made it 1m 30s. That
was a bit sloppy, as I simply hadn't bothered to make the correction for
atmospheric temperature and pressure. That was a mistake, especially as
Jeremy had reported such a high temperature, of 98�F, which implies that the
correction is well worth making. I had simply taken the refraction
"straight" from the table in the Almanac, to be 1.5 arc-minutes. If I'd made
the necessary correction (which was on the borderline between 0.1 and 0.2),
it would have reduced the Sun refraction to 1.3 or 1.4 (1m 18 sec or 1m 24
sec), very much in line with Kent's value.

For working a lunar to highest sccuracy, the steps in refraction, of 0.1
arc-minutes, in the table in the Almanac, are rather coarse. At the
high(ish) altitudes we are looking at, refractions are very predictable, and
known to better precision than that table implies. Norie's has a table which
gives refraction, and its corrections for temperature and pressure, to an
extra decimal place, and that table predicts a Sun refraction of 1.37
minutes, or 1m 22 s, almost identical with Kent's figure.

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

So in the end, by a process of give-and-take, I think that between us, Kent
and I have eliminated all our differences, down to a very few seconds. If he
thinks that any remain, no doubt he will say so.

For me, it's been a rather informative exercise. I hope Kent feels the same.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.





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