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Thanks to all of you the programming of the function for calculating
refractions in the altitude range between -3deg and 90deg and for elevations
up to 35'000ft has now been completed and the program works. Yes it works,
but I can not really say that I am happy about it without knowing how the
values of the table were derived. But on the other hand I trust the
published data more than the results of the integration program to which I
do not have a description on what it actually does.



The way this refraction function has been programmed can really be described
as the hard way. Two polynomial fits are used for each of the height levels,
one for the altitude range -3deg to 0deg and a second one from 0deg to +5deg
plus a second set for the reverse function. Above +5deg altitude the
formulae from Bennett and Saemundsson are used. To apply table 6 values also
for the lower range of positive altitudes comes from the observation that
the values for 0deg altitude still show a distinct dependency from the
corresponding height level which the formulae from Bennett and Saemundsson
do not account for.



As a navigator you may be able to take into account specific local
conditions when doing measurements. A computer program is however
predictive, atmospheric conditions for the location, day and time in
question are (generally) unknown. I tried to overcome this deficiency by
introducing to the program a global model of average temperatures at sea
level with the diurnal anomalies. All other temperature and air pressure
data are derived from there using the behaviour of the standard atmosphere.



George wrote


> Unfortunately, there is no TRUE value for refraction at low angles of
> altitude, close above the horizon. Pulkova observatory near St Petersburg
> has been running a programme of measurements over many years, that as far
> as I'm aware still continues. Every now and then, a revised publication
> emerges with improved and updated results. Because refraction  at low
> angles varies with the local weather (and not just the air density at the
> observer), quoted values are average results, over a long time. On any day
> the actual refraction can differ, as distortions in the apparent disc of a
> low Sun clearly indicate.. Correcting for local temperature and pressure
> will do something to iron out those variations, but significant
> differences
> will remain.

It is clear that refraction values for altitudes near the horizon can not be
more than indicative, but so are also all data from the standard atmosphere
which represent some statistical average. It is known that the atmospheric
parameters vary much more near the earth's surface than higher up; thus
resulting also in a much lower probability of actually observing those
averaged conditions. I can very well imagine that the Pulkova measurements
aim to improve existing averages as they are used e.g. for the standard
atmosphere.

George wrote

> Bennet has provided a  formula which is an empirical attempt to fit that
> averaged data. At large angles of altitude, it becomes proportion to the
> tan of the zenith angle, as Snell's law requires. Near the horizon, where
> refraction rises sharply, the divergence from Snell's law shows up in
> correction terms which turn out to be remarkably simple. However,  I doubt
> whether those terms have any backing in terms of the physics of the
> refraction process; more likely, they are just empirical attempts to get
> as
> good a fit as possible, compatible with a simple calculation. It was
> devised in the days before everyone had a computer / calculator.

Bennett really found a simple formula to describe the refraction values with
high accuracy over the whole range from 0deg to 90deg. I just was wondering
to what type of measured or calculated data they were fitted to. But I guess
this is described in his paper of which I do unfortunately not have copy.

> So it's no surprise that tabulated refraction values agree well with
> Bennett. His formula was devised to replicate those values. In some
> publications, such as the Nautical Almanac, it appears that Bennett's
> formula itself is used as the basis for the refraction tables (though the
> constants have recently been tinkered-with a bit to improve the fit to
> recent Pulkova data) so it's not surprising that it shows good agreement.

Do you have the latest versions of Bennett's and Saemundsson's formulae
available? I have only those as mentioned by Meeus. As you indicate, they
seem to have been improved in the mean time.

Marcel

   

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