As the same atmosphere is doing the refracting in both cases, you can
think of terrestrial refraction as the *part* of the astronomical
refraction caused by the atmosphere between you and some object;
obviously, this is not the whole atmosphere, so terrestrial refraction is
always less than astronomical refraction at the same
*altitude* above the
horizon.
On the other hand, as distant objects are never very far from the horizon,
and refraction generally is largest near the horizon, even
the terrestrial refraction can be quite large. Astronomers usually have
the luxury of observing objects well above the horizon, where even
the refraction due to the whole atmosphere is fairly small.

There's more detail about the relation between terrestrial and astronomical refractions on another page.

While terrestrial refraction is often imperceptible to the naked eye, it's
huge compared to the errors of measurement in surveying, which are a few
seconds of arc, or less. It turns out to
be a more serious problem for
*geodesy*
than the astronomical refraction is
for astronomy — especially because the refraction near the horizon is
extremely variable, while astronomical refraction is well behaved at
altitudes above 10 or 15 degrees.

This good behavior of the astronomical refraction over most of the sky was
proved mathematically by
**Barnaba Oriani**
in 1787, so this rule is sometimes called
“Oriani's Theorem”.
(However, many other people had already noticed that very different atmospheric
models gave almost identical results within about 75° of the zenith.)

In fact, it can be shown
that the refraction near the horizon depends mostly on the local
temperature *gradient*,
which is much more important than the local temperature itself.
For this reason, all the refraction phenomena near the horizon —
mirages, dip, terrestrial refraction, etc., as well as the astronomical
refraction — are very sensitive to the temperature gradient; and
they all vary a great deal more than does the astronomical refraction
well up in the sky.

This sensitivity to temperature gradients, which vary a great deal from
day to day, is the reason for the apparent “capriciousness” of green
flashes (to use the term introduced by
**Willard J. Fisher**.)

Given a model atmosphere, we can
calculate the refraction at any apparent
zenith distance (or altitude, which is the complement of the zenith
distance). As the apparent (refracted) altitude h_{app} is just
the sum of the true
(geometrical) altitude h_{t} and the astronomical refraction R,
the calculated table of R(h_{app}) is easily transformed into a
table of
h_{t} = h_{app} − R. This function
h_{t} ( h_{app} ) is often called the
“transfer
function”
for astronomical refraction.

Once the transfer function is tabulated, it can be used to map the true
shape of the low Sun (a small circle in the geometric sky) to the apparent
shape we actually see, distorted by refraction.
This is not quite as straightforward as it appears, because the
calculation gives true altitude as a function of apparent. So we really
have to do the mapping in reverse, and figure out what part (if any!) of
the Sun appears at a given altitude in the sky. Really, it's necessary to
do the calculation this way, because h_{t} is a single-valued
function of h_{app}; but, in miraging conditions, h_{app}
can be a multiple-valued function of h_{t}. That is, the multiple
images of mirages mean that the *same* part of the Sun appears in two or
more *different* places in the sky.

To do the green-flash simulations, it's necessary to repeat this calculation for several different wavelengths, and then combine the distorted images of different colors in a way that resembles what is actually seen in the sky. The details of how the simulations are made are given on a separate page.

For more technical information about astronomical refraction, see the page on understanding astronomical refraction.

Copyright © 2002 – 2008, 2012 Andrew T. Young

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