# Understanding Astronomical Refraction

## Preamble

The textbooks present atmospheric refraction as a complicated subject, replete with arcane formulae, bizarre series expansions, and approximations that completely hide the underlying physics. In fact, refraction is basically quite simple, though there are a few real complications to be dealt with.

Most of the apparent complications are baggage from the 19th Century, when computing was done by means of logarithm tables, and numerical quadratures were frowned upon. Unfortunately, that century's analytical methods are a poor match to the refraction problem, because they had to approximate the structure of the whole atmosphere by some analytical form. But the real atmosphere is full of fine structure; forcing a model to represent everything from ground to space is not only unrealistic, but makes it impossible to see what parts of the atmosphere produce mirages, green flashes, and other refraction phenomena.

Finally, the goal of representing refraction accurately from the zenith to the horizon is illusory. On the observational side, refraction cannot be measured accurately near the horizon; and from a practical point of view, the near-horizontal refraction is so sensitive to the structure of the lowest atmospheric layers — which are highly variable — that it is pointless to try to represent refraction at low altitudes accurately at all. In fact, thermal structure too small to measure directly can produce quite large variations in near-horizontal refraction, so that tables and formulae based on mean atmospheric models are not only useless, but deceptive. Refraction at the horizon commonly varies by many minutes of arc, and occasionally exceeds nominal values by a degree or more.

Therefore, this page will attempt to lay out the basic physics clearly; show why some simple approximations are very good in the region where astronomical observations (particularly positional measurements) are usually made; and explain why refraction near the horizon is so variable, and so different from its behavior over most of the sky. The approach is necessarily quite different from the standard textbook treatments.

## Outline

This section describes how to proceed. Links go to nearby pages where each topic is discussed in detail. The basic ideas involved are:

By applying these basic ideas to a spherical atmosphere, one can immediately find:

None of this requires anything more than simple trigonometry. The only physics involved is Snel's law of refraction. All else is simple geometry.

Wegener's principle then allows you to see why Cassini's model is poor at the horizon. It also leads directly to Biot's theorem about the magnification at the horizon, which requires only an understanding of the relation between atmospheric temperature gradients and ray bending; in turn, this allows you to find the flattening of the setting Sun. (Note that all this can be done without calculus.)

This leads to a consideration of why refraction behaves so differently near the horizon than it does over most of the sky.

(Now comes the calculus part.)

Differentiating the refractive invariant, plus a little geometry, gives the differential equation of a ray in the atmosphere. This equation is needed to find the refraction in more realistic cases, and to discover why refraction over most of the sky depends only on conditions at the observer, rather than the detailed state of the atmosphere: Oriani's Theorem.

Only after these general properties are understood is it worth worrying about atmospheric models — which is where most of the effort was expended in the 19th Century. The game played in the 19th Century was to find approximate formulae for the structure of the atmosphere that were both (a) reasonably close to the average state of the real atmosphere, and (b) simple enough to allow the refraction integral to be evaluated. Neither of these conflicting requirements was compatible with the complexity of real atmospheric structure.

Furthermore, nearly all of this effort was expended in developing complicated series expansions of the refraction integral — series that, as Ivory pointed out, were only semi-convergent at best; and even that over only a restricted interval of zenith distance. Worse, these series approximations completely obscured the underlying physics. These efforts were directed toward constructing refraction tables that would be valid everywhere.

One remarkable exception was Biot's re-casting of the refraction integral in a form that allowed it to be evaluated directly from observed atmospheric soundings. As an example, he used the data from a manned balloon flight made by Gay-Lussac. The whole approach was so modern that it was ignored by his contemporaries, and re-discovered a century and a half later by Auer and Standish. It is now the recommended method of calculating refraction near the horizon.

However, it is worth emphasizing that Cassini's simple model is more accurate than is needed for all astronomical purposes to at least 74° Z.D., and remains good to better than a second of arc to 81° — provided that the refractivity of air and the height of the homogeneous atmosphere are accurately evaluated for actual conditions.

From a 20th-Century perspective, one should include boundary-layer meteorology in the model, because that is needed to understand the structure in the part of the atmosphere that is important for variable refraction, mirages, and other phenomena near the horizon. So numerical integrations are required only within a few degrees of the horizon, where the variability of the refraction is largely due to the variable lapse rate in the boundary layer.

Finally, we might also take note of Laplace's useful Extinction Theorem, which relates refraction and extinction, and so is useful for understanding the color and brightness of the low Sun.

A more formal introduction to refraction, using the approach outlined above, appears in the April, 2006, issue of The Observatory:

Understanding astronomical refraction
The Observatory 126, 82–115 (2006).

For more technical details of astronomical refraction near the horizon, see my earlier paper

Sunset science.  IV. Low-altitude refraction
Astronomical Journal 127, 3622–3637 (2004).

It's available from the A.J.'s website in both HTML and PDF form. The links given here are from the ADS website.