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Bill wrote-

>Paul Hirose had posted the formula:
>
>R = 1 / tan (H + 7.31 / (H + 4.4))
>
>R is refraction in minutes
>H is observed height in degrees (but was used as Hc by Paul)

From George-
This is exactly the same formula as was given in the 2001 Nautical Almanac
on page 280. It was there stated to be appropriate for 10 degrees C and for
1010 mbar. It's also the same as given in Meeus' Astronomical Algorithms,
quoting from George Bennett's work.

>I noticed the 2005 Nautical Almanac provides a refraction formula similar to
>the one Paul posted in its sight reduction/direct computation section (page
>280):
>
>Ro = 0.0167 / tan (H + 7.32 / (H + 4.32))
>
>I believe I understand the 0.0167 vs. 1.  The formula Paul posted gives
>refraction in minutes, while the almanac formula gives refraction in degrees
>(1/60 = 0.0167).

Correct.

>I do not understand 7.31 vs. 7.32 or 4.4 vs. 4.32.  Has the formula changed
>slightly over the years, or are there different camps on which (if any) is
>correct?  My only thoughts are that the almanac formula goes on to correct
>Ro to R with a temperature/pressure-refraction correction, while no mention
>of temp/pressure correction is noted in the formula Paul posted.

No, that argument about temperature and pressure doesn't hold water;
assuming that the expression given for refraction on page 280 of the 2005
almanac quotes a temp of 10deg C and 1010 mbar, just as the 2001 almanac
did. I don't have a 2005 almanac so I can't check that.

It's interesting that those constants in the formula have been changed
slightly. I don't know, any more than Bill does, why it has happened. The
constants are empirically adjusted so that refraction, calculated from the
formula, corresponds as well as possible to the mean value of observed
refraction.

The specialists in this matter are at Pulkova observatory, near St
Petersburg,, and every now and again they reassess a long series of
observations and publish the mean values, which change from time to time as
observational techniques improve and more data is accumulated. It seems
likely that the changed values may correspond to a reassessment of
low-level refractions at Pulkova: but I don't know that.

Over most of the angular range, a simple expression for refraction in
minutes of 1 / tan H is reasonably accurate, and the two constants (say,
7.31 and 4.4) have little effect. They only come into play at small
altitudes, and give a mean refraction for zero altitude (truly horizontal
light, at right-angles to the zenith) of 34.5 minutes.

The values of 7.32 and 4.32, on the other hand, predict mean refraction for
horizontal light to be 33.8 minutes. This is a significant difference, but
not, I suggest, a very important one, because of the great VARIABILITY of
low-level refraction, depending on local weather. Whatever the formula
predicts as the MEAN refraction at some angle, there's no guarantee that on
a particular day that's what the ACTUAL refraction will be.

The constants may have been adjusted to fit new observed mean values of
zero-angle refraction: alternatively, there may have been some tinkering
with them to get a better fit to observations over a range of small angles
greater than zero, at the expense of the closeness of fit to zero-degree
refraction.

But I doubt if Bill will find any significant difference in the results of
the two slightly-different expressions, for any observation he might make
if it is at an adequate and sensible angle above the horizontal.

George.

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