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George H, you wrote:
"Greg's postulated value of 6 minutes of arc was indeed unreasonable, and
based on a misunderstanding, as I have explained above."

His reasoning, which he explained later, was invalid, but nonetheless, the
range of variability he quoted is NOT "unreasonable" though we could argue
over a factor of two depending on whether we're talking about mean
variation, a standard deviation in variation, two s.d., etc.

And you wondered:
"And even if that assertion is valid, how on Earth does a navigator know,
when trying to use a sunset time to ascertain his position, whether Frank's
restrictions apply or not, within the "variation within the variability"
that he conjures up?"

Well, come on. There are many ways that this information can be available.
Consider the case of a navigation enthusiast practicing from land: I look
around and I see a snow-covered landscape in New England, and in front of me
I see the warming waters of the Atlantic in early Spring. That's a case
where you need to worry about temperature inversions and unusual refraction.
On the other hand, consider the case of a navigator at sea: I know I left
Bermuda two days ago, and I'm travelling east. The weather is moderate and I
am well south of the Gulf Stream. Those are conditions where I can expect
relatively little variability of refraction at the horizon. And of course,
there is one very useful clue: the appearance of the Sun itself. If the Sun
is strangely distorted just after it rises or before it sets, then that's a
darn good indicator that the timing will be off.

And:
"If we are talking about use of timing a sunset for real navigation,
requirements for position knowledge can be very relaxed in mid ocean. It is
only when a land mass is being approached that navigation becomes critical.
So observations made at coastal sites, with a view over the ocean, will be
particularly relevant in assessing the navigator's problem."

Yep, that's the irony about celestial navigation. It works best where you
need it least. Then again, these problems are not significant when you're
sailing towards a mid-ocean island. They're also not as significant in
warmer climates. It's true that there are areas and times of year where the
refraction variability will be a show-stopper --in the high Arctic, for
example, but it would be silly to worry about such issues when they don't
apply. In addition, if you were in mid-ocean, in reality, let's say this
summer, are you really saying that you wouldn't care about an accurate
position? Do real navigators and/or real celestial navigation enthusiasts
just ignore the navigation out in the opean ocean?? Just from the anecdotal
evidence of talking to people, I find that real people try for accuracy as
often as possible.

And you wrote:
"And those are exactly the sites from which Schaefer and Liller have timed
sunsets, at sea, in temperate climes."

I guess not. As has become apparent in later messages, many of these
observations were made from the site of the observatory at Cerro Tololo,
atop a 7000 foot peak in the foothills of the Andes 50 miles from the
Pacific. I will be very blunt here: observations from an inland mountaintop
have ALMOST ZERO RELEVANCE to celestial navigation at sea. Calculate the dip
there. It's around a degree and a half below the true horizon. That means,
in the first place, that the MEAN value of the refraction there will be
significantly greater than the usual 34 minutes at the horizon (this is
partially offset by the fact that the observatory is in thin air due to its
altitude). Much more important is the fact that we are looking down into the
atmosphere; this has a large effect on refraction from high altitude
locations (as Marcel has already described) and leads to some strange and
rather beautiful phenomena. Additionally, this location is a special case
where there will be significant horizontal variations in the atmosphere.
 From Cerro Tololo, you're looking out across a dozen valleys towards the
Pacific, fifty miles away (and the horizon far beyond that). Some areas will
already be in shade at sunset and cooling off as night approaches. Other
areas will still be in direct sun. The air over the sea will have yet
another temperature profile This is a much more complicated and much more
extreme refraction situation than one would ever encounter doing celestial
navigation at sea.


Again referring to the advice in Bowditch, you wrote:
"Not just weak. Absurd."

Ummm, no. That is an exaggeration.

When I noted that the variability in refraction doesn't matter above about
three degrees altitude, you wondered:
"Without disputing that assertion, I would like to see some evidence to
support it. What precision is implied by "exact for all practical
navigation"?"

I'm saying that from around three degrees, even if you look at significant
variations in the low-altitude temperature profile, you will get the same
refraction values to the nearest tenth of a minute of arc starting at about
three degrees altitude (above the TRUE horizon). For example, suppose the
temperature lapse rate (Lr) is -8 deg C per km from 1km to 13km altitude
(constant temp above that, but it doesn't matter at such high altitudes).
Then consider varying the lapse rate from -10deg/km to +10deg/km in the
lowest kilometer of the atmosphere in steps of 5 deg/km. When you do the
integrations, you'll find this table of refraction from 0 to 4 degrees:
alt(deg): refr(min) for each Lr (-10, -5, 0, +5, +10)
0: 33.2, 35.0, 36.9, 38.8, 40.7
1: 24.1, 24.4, 24.8, 25.1, 25.4
2: 18.2, 18.3, 18.4, 18.5, 18.6
3: 14.4, 14.4, 14.4, 14.5, 14.5
4: 11.7, 11.8, 11.8, 11.8, 11.8

I should emphasize that these various lapse rates (the columns in the table)
are not all equally probable. At sea, the lapse rate is typically close to
the lower limit of this table, and sure enough, the values in the Nautical
Almanac are very close to the left-most column. The positive lapse rates
(implying the air warms with altitude) are relatively rare at sea but quite
common on large landmasses, especially when the ground is very cold (e.g. in
the early morning in winter).

Probably, the next thing that one might worry about is more complicated
atmospheric structure. For example, what happens if the temperature falls
for a few hundred meters, then inverts sharply at a higher altitude, and
then falls again (as it always inevitably does, up to the tropopause). It is
not difficult to run integrations of those more complicated situations. The
results are qualitatively similar and differ only in the details. You can
trust the standard tables for altitudes above three degrees (true altitude).

I wrote:
" (incidentally, Greg's "6 minutes" is close to the geometric mean of these
other claims, for whatever that's worth )."

And you replied:
"Not much."

You probably know this, but to avoid misunderstanding... That little
notation "" is an old online thing meaning "grin", equivalent to the
slightly cheesier "smilie" :-). Point being, I was making a little JOKE. 

I wrote previously,
"By the way, there are sometimes temperature inversions even at sea. Calm
weather especially is associated with them. It's worth knowing that the
same conditions which would lead to anomalous low altitude refraction at sea
will also often lead to "sea fog"."

And you dismissed:
"Can Frank reliably deduce, from absence of sea fog, that there's therefore
some limit on the variability of refraction? If not, that comment is of
little use."

Well, George, you seem to have interpreted my phrase 'it's worth knowing' to
mean 'it's a law of nature'. Of course, that would be a most anomalous
interpretation. It is indeed "worth knowing" that sea fog is often
associated with the same conditions that can lead to unusual refraction, but
it's no guarantee. In satellite photos in winter, you can often sea feathery
bands of the stuff in the western Atlantic. They start right on the edge of
the Gulf Stream as cold air passes over that warm water. There's no
guarantee you'll have anomalous refraction there, but it's something to
watch out for. And again, the best clue is the appearance of the
rising/setting Sun itself.

And:
"I have often seen a highly distorted Sun disc, at altitudes of well over 3
degrees, that leads me to question it as a general rule."

I'm going to doubt you on one detail here, George. Three degrees is six Sun
diameters. Now think back... have you REALLY "often" seen a "highly
distorted Sun" at six full Sun diameters away from the horizon?? In my
experience, and yes, I DO look for these phenomena every chance I get, all
the action occurs at much lower angular altitudes.


And you wrote:
"Evidence requested to support that assertion, please."

See the table above.

And:
"I understand (personal communication from Schaefer) that "our
atmosphere always have rapidly and wildly varying thermal structure (i.e.,
temperature inversions come and go on all time scales and are generally
present at some level)." "

Well, that's just plain hand-waving. All time scales?? Micro-seconds to
decades?? And consider the observational evidence that we have ALL SEEN: the
setting or rising Sun. Is it a crazy swirling mass of changing appearances
like a mirage?? No. There are small changes that occur on a time scale of a
minute or so, but the overall appearance is relatively constant (for a fixed
altitude --that is, as the Sun passes through the half-degree altitude line,
it might develop a "pinched" appearance, and that pinch will rise up the
Sun's disc as the Sun sets). And every once in a while, it looks really odd.

And:
"In that, he does not distinguish between over-land and over-sea, though no
doubt such effects are significantly greater over land."

First of all, maybe he doesn't know or care about effects at sea. The
phenomena one will see from mountaintops is much more fun. Second, it sounds
like you're just assuming something here from the fact that he did not
include a bunch of detailed caveats.

I wrote previously:
"Incidentally, I base these conclusions, in part, on a large number of
refraction integrations that I ran back in 2005 (I think) when Marcel got
me interested in the production of refraction tables. It's quite
straight-forward to take an observed temperature profile and generate a
refraction table just for those special conditions."

And you replied:
"Indeed, Schaefer himself published such an integration
procedure (which I haven't seen) in Sky and Telescope, 77, 311 (1989), and
the procedure Frank used may have been that one, or one based on it. It was
to test the unexpectedly large fluctuations that it predicted, that
Schaefer's survey of horizontal refraction was undertaken."

No, no connection with the code I wrote. The code published in S&T back in
1989 is trivial (this was the era of the 80386 and the wonderous 387 "math
coprocessor"!). Here's just a short sample from that column to remind
everyone of the good old days (by then, very few people were using
line-oriented Basic, but the editors of S&T insisted on it, which I told
them was a mistake then and I still think it was a mistake today. Would you
like sauce with your spaghetti?):
"350 REM  REFRACTION SUBR
360 REM
370 N=N0: DH=S/200: IF H<0 THEN DH=-DH
380 H1=0
390 T=T1: GOSUB 1110: T1=T
400 IF (H1-H)/DH>=0 THEN 450
410 H1=H1+DH: GOSUB 1110
420 EX=-1-(T0/(S*LR))
430 N=1+(N-1)*((T/T1)^EX)
440 T1=T: GOTO 400
450 REM  INITIALIZE PARAMETERS
460 RF=0: L=0: LZ=0: R=R0+H
470 L1=LR: Z=Z0
480 REM"

And you wrote:
"Schaefer's survey involved 116 observations at 4 different sites,
worldwide. If Frank PREFERS simulation evidence from integrating observed
temperature profiles" etc. etc.

Ya know, George, atmospheric refraction is a completely solvable physics
problem. There is today still a bit of new science to tease out from the
issue of atmospheric refraction simply because we have far more
computational power today, and that's why there are a few specialists
working on a few interesting problems. We can easily pick whatever profiles
we want, including complicated horizontal variations, if you really want to
get fancy (though that requires a different integration procedure).

And you wrote:
"to such direct measurements, then perhaps he will explain to us the
statistics of the population on which his conclusions were based. Just how
many temperature profiles did he consider? What was their geographical
distribution? How many were taken at the relevant times of sunrise/set? How
many measured air temperatures some tens of miles out from the coast,
relevant to Schaefer's survey and to navigators' needs? And what was the
resulting scatter that Frank calculated?"

Well, George, since you're the one making the assertion that the statement
in Bowditch is "absurd" (I agree that it could use clarification), then
maybe you should be the one digging up data and calculating "scatter". Have
you looked at any temperature profiles? Did you try looking at the balloon
data for Bermuda that I've mentioned several times? And no, before you ask
the obvious, I did not look at JUST Bermuda --there are quite a few oceanic
sites with temperature profile data.

In reply to Peter, you wrote:
"On those occasions when the sky was clear right down to the horizon, with
no sign of distant cloudbanks, the time of the last (or first) glimpse of
the Sun could be logged to the GMT second, from a well-known height of eye,
with a precise GPS position at that moment, with a note of the datum in use.
With the closeness of horizon that can be seen from a small craft, it could
only work reliably in calm conditions without significant swell. How about
that for a project on the next cruise, one that calls for no special
instrumentation? It wouldn't be hard to collect that data together from a
number of navlist members, and boil it down. Perhaps that's been done
already, but if it has I haven't heard of it."

Sure. That sounds like fun. Now where do I sign up for my free ocean voyage?


 -FER


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