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May I add to Antoine's words my own pleasure at seeing a contribution to 
this list from George Bennett, long a familiar name to this list from his 
simple and close approximation to refraction, to be found in the Almanac 
and in Meeus.

I have a comment on his attachment about that approximation, and its 
development for use backwards. It concerns nothing more than two little 
symbols.

In the equation for A, why is the number 7.31 expressed as 7.31' ; and in 
the equation for r', why is 9.48 expressed as 9.48' ?

When I intended to try out that equation for r', seeing that the numerator 
of the second term was expressed in arcminutes, and it would have to be 
added to a quantity H in degrees, my first thought was to divide that 9.48 
by 60, to put the two terms on equal terms for adding. That was wrong, I 
soon realised. That is a numerical expression, in which the input valueas 
are in degrees, and the result happens to come out in minutes, if we simply 
treat those constants as numbers. So I suggest that it's misleading to 
attach that arc-minutes label to the constant 9.48.

In the same way, it's misleading to label the numerator of the expression 
for A as 7.31 arc-minutes. A is to be added to h, in degrees. Meeus, in his 
expression 16.4, simply treats 7.31 as a number, which I suggest is the 
right thing to do.

Bennett's simple expressions exploit the accidental fact that the 
refraction for 45 degrees is close to 1 arc-minute. Accidental, because 
there's no fundamental reason why degrees should be divided sexagesimally. 
Bennett's simple expression is about the only thing I can say in favour of 
sexagesimals: I hate them.

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

There are two minor snags that might possibly, but rarely, arise when 
implementing Bennett's formula for refraction, given as
R = cot ( h + 7.31 / (h + 4.4�))

One, mentioned in Meeus, is that R is not quite zero at 90�, which is 
somewhat unphysical; symmetry says that it must be.

Another is that some computers / calculators will not produce a cotangent 
directly, but have to get there via calculating the tangent. And for an 
angle within the working range, at about 89.92�, its tan becomes infinite, 
at which point some calculators will refuse to proceed further.

Both of these possible snags are circumvented by tinkering slightly with 
the Bennett refraction formula, at the expense of some loss of simplicity.

Instead of
R = tan (90� - h - 7.31/(h+ 4.4�)), which is identical to the original 
formula, write
R = tan (90� - .99914h - 7.31/(h+ 4.4�))

This sets the refraction at 90� much more closely to xero, and moves the 
blowup angle out of any possible working range.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

   

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