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George Huxtable wrote [5580]: Where does �Tan. App. Altitude x 60.53� come 
from....



As Frank Reed correctly pointed out [5591] it shall of course be Tan 
(90-app.altitude) x 60,35�, i.e. tan Za x 60,35�. The factor 60,35� was used 
in Swedish navigation litterature around 1840. George mentions that this 
factor rather should be 58,2� representing todays knowledge. That is, as I 
see it, partly true. The factor 60,35� has been provided to me from the 
Observatory in Stockholm to be used with the true zenith altitudes for 
altitudes above approx. 10 degrees. The factor is taken from modern English 
litterature. In my modelling I use Tan (90-app.altitude) x 60,35� for 
altitudes above 20 degrees and in normal atmospheric conditions. A reduction 
factor is used for deviations in pressure and temperature. As probably noted 
the use of app. Altitude instead of the true altitude together with the 
factor 60,35� gives small deviations compared to the refraction table I 
have. I am fully aware of that.



George Huxtable wrote [5580]: Then the amended formula would provide 
predictions for altitudes greater than 20 degrees. But it breaks, down and 
badly, at lower altitudes.



Yes, this is correct. For altitudes below 20 degrees I use the formula 
defined by Smart. Also here I use a reduction factor for deviations in 
pressure and temperature. I could not at the time, when I did my 
programming, find a suitable formula from the old days. My way of writing 
about refraction for low altitudes was indeed bad and incomplete.



So again, what I have tried to achieve is a re-construction of how the LD�s 
were measured and reduced in the old days (by Swedish navigators) and not 
necessarily what will be achieved with todays know-how.



George Huxtable wrote [5582]: I did�nt understand all that (and still don�t) 
and asked for some relevant numbers to help me out. On the other hand, 
perhaps, it is better if I explain how I would do....



Frank Reed wrote [5591]: What he is describing is an ordinary time 
sight....Real navigators used the Sun for local time and carried it on a 
watch.....



The method with GHA and LHA for finding longitude provided by George is well 
known. But in the old days it was not done like that. George points out that 
I complicate things. Yes, I do because, again, my way of doing it is a 
re-construction of old methods. As I know there were two ways in the old 
days. The first way is what I outlined and this way can be used for any 
celestial body, incl. the sun and provides the local time (MT). The other 
method was to use the sun for generating the local time by finding the sun�s 
apparent time and correct it with the TE. This later method required 
(requires) less calculations and was therefore probably preferred by 
navigators.



Also keeping the local time with a watch was used but after what I have 
gathered this was not a common method likely because watches were not good 
enough. But this method, when working, was certainly much easier for the 
navigator than the first method outlined by me.



Kent N

----- Original Message ----- 
From: "George Huxtable" 
To: 
Sent: Tuesday, June 24, 2008 11:01 AM
Subject: [NavList 5580] Re: Lunar trouble, need help



Kent Nordstrom's lunar calculation is now starting to converge with my own.
However, I find much to disagree with in the following paragraph-

"My refraction values are calculated as Tan app. Altitude x 60,53� for
standard atmospheric conditions (760 mm Hg and +10 degr. C). Then a
correction for actual T and B is made for altitudes from 20 degrees and
upwards . For lower altitudes another way of correction for B and T is done.
You could always argue about the relevance of this approach, we all know
that refraction is difficult to model, in particular on low levels. Anyway,
with the approach taken my results coincide with refraction values taken
from tables quite well (Tables from 1845, based on Janet Taylor, Henry Raper
I assume)."

Where on Earth does "Tan app. Altitude x 60,53� " come from? For one thing,
it's quite the wrong way up, showing refraction increasing with altitude
rather than the other way round, and I suspect that must be the result of an
accidental slip in his email text, rather than in his calculation.. It would
be more correct (but for high altitudes only) if the tan were changed to
cot, or (which amounts to the same thing) altitude were changed to
zenith-angle, and if, at the same time, the constant was changed from 60.53"
to 58.293" (see Meeus, also Smart).

Then that amended formula would provide reasonable predictions, for
altitudes greater than 20�. But it breaks, down, and badly, at lower
altitudes, and then a more complicated expression is required. These errors
have NOTHING to do with any corrections for temperature and barometer, for
which refraction can ALWAYS be modelled, very accurately, as being simply
proportional to air density at the observer.

No easy analytic function exists for correcting refraction over the whole
range of altitudes, but a good empirical approach by Bennett (see Meeus)
gives, for refraction in arc-minutes at apparent altitude h0 degrees -

R = cot (h0 + 7.31 / (h0 + 4.4))

which I like to tinker with by changing it to-

R = cot ( h0 -.000861 h0 + 7.31 / (h0 +4.4)), which makes refraction zero at
90�, as it has to be.

I've no doubt that the high-altitude refraction figures that Kent actually
used, in correcting that lunar, were a reasonable shot at reality, and would
not much upset his lunar distance. It's his text-explanation that I find
unconvincing.

He invokes the names of Taylor and Raper, both of whom provide tables of
refraction, but neither explains the details of their basis, and I am pretty
sure they are NOT calculated according to Kent's precepts.

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

Kent wrote-

"I agree with George that I should have used 06-20-13 instead of 06-20-44
for
referring the moon�s altitude to the mean time of LD�s. This change has a
very small effect on my LD."

Yes, we agree about that, but now, does his recalcalculated Moon mean
observation-time agree with mine?

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


George wrote �But next, the bit that really puzzles me about Kent's
treatment is how he
manages to invoke sidereal time. That would come in only if he was getting
his astronomical positions from an Astronomer's almanac that gave
sky-positions in terms of right-ascension, rather than hour-angle�.

And Kent replied-

"My approach for calculating the MT is as I understand the very same as used
in the old days.

This is of course for finding the difference between the GMT and the MT,
i.e. the longitude. It is a separate calculation. So let me outline what I
do:

  1.. Find the LHA at the predicted GMT (we know (roughly) the latitide,
from the generation of true distances at say time 3, time 6 etc. we can
predict the declination at the GMT. True altitude is from the altitude
reduction. Aries GHA to the GMT can be predicted as well).
  2.. Find the Aries GHA for upper meridian passage in Greenwich, (normally
one day before).
  3.. Convert this GHA to time.
  4.. Correct for assumed longitude and then you get the time for Aries UMP
on your location.
  5.. Find the difference between Aries GHA and the body�s GHA at GMT. This
is the RA.
  6.. Find the sum of the LHA and the RA, which is the sideral time.
  7.. Subtract the value for Aries UMP at the location. Now you have the
difference in sideral time.
  8.. Convert this difference in sideral time and you get the MT at
observation.
And now you can easliy find the difference between the GMT and the MT. I
hope this explaination clarifies. I should also add that one must be very
observant when doing this calculation, it is easy to do wrong."

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

Well, I have struggled a bit with that, withouf fully understanding what
Kent is doing, which may well be my problem rather than his. It seems a very
roundabout procedure. It might help me if Kent would kindly provide the
actual numbers he has worked with in this case, including the almanac
predictions for Sun and Moon (and Aries?) positions, and from which almanac
they were
taken.

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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