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Kent wrote, about the refraction corrections-

"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."

He and I disagree here. In practice, in the case we are considering, the 
difference in refraction between the Moon's centre and upper limb is pretty 
infinitesimal: no more than 0.4 arc-seconds. Presumably, Kent is keen on 
such details because he wants to be completely rigorous, and get the 
principles right. And it seems to me that he hasn't got the principle right 
here. I will try to convince him.

The light-ray, coming down through the atmosphere, that is being observed 
with the sextant, is the light that is from the Moon's upper limb. And it's 
the bend in that ray that constitutes the refraction. Refraction is the 
difference between the angle at which the light arrives, and the angle at 
which it would arrive if the Earth had no atmosphere. So when we correct for 
refraction the observed sextant angle (above the true horizon) of the Moon's 
upper limb, that then gives us the true elevation of that limb, above the 
true horizon, as would be seen by that observer if the Earth had no 
atmosphere. Then the Moon's semidiameter allows for the difference between 
that true altitude and the Moon's centre; the true difference between them, 
refraction no longer playing a part.

Of course, if working with tables or formula that work with zenith distance 
rather than altitude, you have to use the complement (to 90�) of the 
altitude, but the reasoning doesn't change.

He continues [with my comments in square brackets]-

"Consequently in my way to do the reduction I start with refraction on the 
UL ..."  [all right so far...  ] "and then I calculate the difference in 
refraction for the UL - SD for the moon." [that's the bit that's 
unnecessary, and indeed wrong].
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 [no, 
I'm not] that
is fine but then there is, as I see it, a difficulty coming up later when
treating the distance."

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

Kent continues, about another correction which could be made to the lunar 
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 thiscorrection when reducing the 
distance."

[I agree that in principle, this is a correction that might be considered. 
In Raper, (1864) it's table 53, "correction of the lunar distance for the 
compression of the vertical semidiameter". And I agree that if one were 
forced into observing at very low altitudes, such as Kent quotes in his 
examples, then it might indeed be worth making. But most navigators would do 
the damnedest to avoid measuring lunars below 15� altitudes, in which case 
this correction never exceeds 3". In the examples we are considering, it's 
less than 1".]


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

[It depends on the precision one is hoping to work to, which is limited by 
the observing instrument, the observer's skills, and the environmental 
conditions he has to work under. I would ignore such corrections, for 
observations made in a marine environment. ]

"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."

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

About another correction, for the contribution to Moon's parallax due to the 
Earth's oblateness.

"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.

[The correction  I made is from Chauvenet, "Spherical and Practical 
Astronomy" (1863 and many later reprints), table XIII "Correction of the 
Moon's eq. parallax" (actually, a reduction)and in text Vol I, page 104 to 
126. Kent should take a look at Chauvenet, who provides all the rigour he 
could ever ask for! He also seems to get everything right.

It's also in a table in a modern Norie's "Reduction of the Moon's horizontal 
parallax", from which I took, in [5530] the value -0.0002�, to slightly 
tweak the parallax of 0.4622.

I agree with Kent that this correction can be worth applying at higher 
latitudes, but al lat = 14, it was hardly worth bothering with.

This corresponds to one of Kent's corrections to parallax due to oblateness, 
but leaves his second oblateness term unaccounted for.

In [5701], Kent describes another correction to parallax for oblateness, 
which depends on Moon azimuth, and I haven't even considered that one, 
described by Kent as follows-

"- 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."

By my rough estimate, that might shift parallax either way by up to 5 
arc-sec, so when striving for high accuracy, it could be well worth making. 
In the present case, it seems to increase parallax by 2". Do I have that 
right?

However, I haven't made any such correction, that depends on Moon's azimuth, 
and haven't found a reference to one, yet, in text in English. Still 
looking, though.

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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