Oriani's theorem deals with a coefficient in the
**Lambert's
series expansion**
of astronomical refraction in powers of **tan Z**,
where Z is the apparent zenith distance at the observer.
Although
**Oriani himself**
found a rather inconvenient proof, a much more
elegant one was discovered by E. T. Whittaker and published in
**Sir Robert Ball**'s
“Treatise on Spherical Astronomy,” pp. 123–124. As this
reference is no longer readily available, the derivation is reproduced here, in
a slightly simpler notation. (I have followed a similar line in my
tutorial paper (2006)
on atmospheric refraction.)

To avoid a lot of subscripts, I'll use upper-case (“capital”) letters for the values of variables at the observer. That makes them parameters of the problem. The lower-case letters will be used for the variables themselves.

In printed works, mathematical symbols are usually set in *italic* type.
That looks a little pale on the computer screen, so I'm using **boldface**
here instead, in the running text. Equations that are set off by
themselves don't need this enhancement, and will be set in the normal
face.

= ∫

where the value of

If we write **R** for the observer's distance from the center of
curvature of the atmosphere, and **(R+h)** for the local radius at
some point along the ray, the
refractive invariant
is

at every point along the ray, where **N** is the value of the
refractive index **n** at the observer. (Recall that I'm using upper-case
symbols for the values of **n** and **z** at the observer, to avoid
using subscripts.)

Then, at an arbitrary point along the ray, **sin z** is just
**I/n(R+h)**; or, in terms of the values at the observer,

Unfortunately, this becomes rather messy when substituted into
**tan z = sin z/cos z**, because we have to
use **sqrt(1−sin ^{2}z)** for

so that

So, set

in the refraction integral, and keep only the first-order terms in

We can then cancel factors of **R** in both numerator and denominator
of the expression for **tan z**.
The refraction integrand, **(tan z) dn/n**, becomes

The argument of the square root can be rearranged as

then the whole refraction integrand becomes

This looks messy. But what we have done is to produce a fairly
simple function of **n** (the left expression in square brackets),
multiplied by a less well-behaved function (the right-hand square bracket)
of a form amenable to binomial expansion. We'll expand that latter
function and integrate termwise to get something reasonable at the end.

The first term is an elementary integral, whose value

is exactly the refraction for the plane-parallel atmosphere. Thus the second integral can be regarded as the first-order correction for atmospheric curvature.

This correction term can be evaluated by setting both **n** and **N**
to unity in its integrand. (The error made is of higher order, as this
term is already of order **s**.) Then the correction term becomes

where

where the upper limit

Furthermore, the factor
**sin Z / cos ^{3}Z = tan Z sec^{2}Z**;
and if we replace

where the coefficients

**Oriani himself**
stated that “This expression depends on no hypothesis about
either the law of heat in the atmosphere or about the density of the
air at various distances from the surface of the Earth.”
**Laplace**
provided a more rigorous and complete proof of Oriani's theorem.

Obviously, these power series in **tan Z** must diverge at the horizon.
However, they in fact diverge *everywhere*! It seems to have first
been understood by
**James Ivory**
that this is only a “semi-convergent” series: the terms get
smaller for a while, and then grow bigger and bigger without limit,
*even near the zenith*.
But the smallest term in the series is small enough that this approach is
useful for numerical purposes out to about 80° zenith distance.

The divergence of the series emphasizes that Oriani's result is restricted to small and moderate zenith distances. Physically, it applies only to the extent that the ray can be regarded as nearly a straight line; that is, the path of the ray through the atmosphere is not appreciably altered by refractive bending. And indeed, the region of validity is where the refraction is no more than a few minutes of arc.

Near the horizon, the refractive ray curvature is comparable to the Earth's curvature. Then, contrary to Oriani's result, the refraction is extremely sensitive to the details of atmospheric structure.

Another problem occurs when the model atmosphere contains water vapor.
The mixing ratio of water vapor is much lower in the upper troposphere and
the stratosphere than near the ground; but, as the refractivity of water
differs from that of dry air, this means that the “constant”
in the Gladstone-Dale formula isn't really constant — and so
**dn** cannot be converted to **c dρ**, which was the trick used
above to make integration by parts possible.

Copyright © 2003 – 2008, 2012 Andrew T. Young

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