# Astronomical Refraction

## Terminology

Astronomical refraction” is the angular displacement of astronomical objects from their true or geometrical position, because of the bending of rays in the Earth's atmosphere. It is contrasted with “terrestrial refraction,” which is the corresponding angular displacement of objects on the Earth and in its atmosphere, such as ships, mountains, clouds, etc.

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.

## Amount of refraction

#### Astronomical refraction

To give some rough numbers: the astronomical refraction is about a minute of arc in the part of the sky midway between zenith and astronomical horizon, but is typically over 30 minutes of arc (half a degree) at the horizon. (This is the basis for the common claim that “when you see the Sun just touching the horizon, it has already set.”) However, in the Novaya Zemlya effect, commonly observed at high latitudes, but occasionally seen even as close to the equator as San Diego, the horizontal refraction can exceed two degrees.

#### Terrestrial refraction

Of course, the terrestrial refraction is smaller. But even so, it rarely is less than 1/15 of the angular distance of an object from the observer, as seen from the center of the Earth. Bearing in mind the rule that “a minute [of arc on the Earth's surface] is a [nautical] mile,” you can see that terrestrial refraction is usually several seconds of arc for relatively nearby objects, and can easily amount to several minutes of arc for distant mountains.

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.)

## Variability near the horizon

This variability near the horizon is very surprising to the average astronomer, who has been educated to believe that refraction (in the part of the sky where astronomical measurements are usually made) can be calculated accurately from the local atmospheric temperature and pressure. This is true in the part of the sky where Oriani's Theorem holds, but it breaks down very rapidly on approaching the horizon.

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.)

## Calculation of the refraction

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 happ is just the sum of the true (geometrical) altitude ht and the astronomical refraction R, the calculated table of R(happ) is easily transformed into a table of ht = happ − R. This function ht ( happ ) 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 ht is a single-valued function of happ; but, in miraging conditions, happ can be a multiple-valued function of ht. 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.

## More details

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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