# Terrestrial and astronomical refractions

## Introduction

Mirages are generally images of terrestrial objects, so we think of terrestrial refraction  as being responsible for them. On the other hand, the ray bending involved in terrestrial refraction is just a part of the whole astronomical refraction — the part produced by the lowest layers of the atmosphere. So we'd like to see how these phenomena fit together.

In addition, there's a curious difference between the two refractions, with respect to their magnifications. In astronomical refraction, the magnification at the horizon, which explains the flattening of the setting Sun, depends mainly on the lapse rate  below the observer. But in terrestrial refraction, a constant lapse rate is usually considered to produce only a displacement, not a change in the angular size of objects — a result first obtained in 1759 by Lambert.

So the lapse rate affects the apparent size of astronomical objects, but not  the apparent size of terrestrial ones. But if the terrestrial refraction is just part of the astronomical refraction, shouldn't the two behave similarly?

## The prism analogy

People often compare the action of the refracting atmosphere to that of a prism. A prism that produces deviation comparable to the horizontal refraction (about half a degree) is just a thin wedge of glass, so it hardly produces any detectable distortion. That makes the prism behave like the terrestrial refraction, not the astronomical refraction; that's why I don't use this comparison in discussing astronomical refraction.

If, instead of using the prism analogy, you think of atmospheric refraction in terms of bending, you can see that the amount of bending is proportional to the distance the ray travels through the dense lower atmosphere. In particular, if you know that Wegener's principle makes the contribution to astronomical refraction from air above eye level almost constant near the horizon, you can see that the increase in astronomical refraction below the astronomical horizon — and hence, its rate of change with apparent altitude, which is the astronomical magnification — depends entirely on the increasing path length in the air below eye level with increasing depression below the astronomical horizon. (This relation between path length and total bending underlies Laplace's extinction theorem.)

But, when we think of terrestrial refraction, we're usually dealing with an object (like a lighthouse, or a distant mountain) whose extent along the line of sight is much smaller than its height. In effect, we're thinking in terms of a vertical target.

## Vertical objects

Now, the distance from the eye to all parts of a vertical object is nearly the same for all parts of the object. That means that the angular bending of the rays is nearly the same for both the top and bottom of the object. And that  means that the object is displaced vertically by terrestrial refraction, but not distorted. In other words, the prism analogy works for terrestrial refraction, even though it doesn't work for astronomical refraction.

In fact, this argument was used by Lambert to argue that mirages are impossible! He claimed that a constant density gradient in the lower atmosphere could only displace, but not distort, the images of distant terrestrial objects.

Of course, Lambert assumed a fixed density gradient. And this just produces a fixed bending, which produces (nearly) the same refractive displacement for all parts of a terrestrial object, because they're all at essentially the same distance from the observer. If, instead, we assume a density gradient that varies with height, we'll have a vertical displacement that varies with height, and consequently a distorted image. If the size varies, the magnification differs from unity. And if the lapse rate varies enough, we can even get negative magnification — i.e., the inverted image of a mirage.

So one feature that makes terrestrial refraction behave differently from astronomical refraction is just the amount of bending atmosphere in the line of sight. Remember Laplace's theorem: near the horizon, the astronomical refraction is proportional to the air mass in the line of sight. For terrestrial objects, the amount of air we look through is nearly the same for all parts of the object, and so is the refraction.

## Behavior of terrestrial refraction

So, how does the terrestrial refraction depend on atmospheric structure? A constant lapse rate (i.e., a linear temperature profile) produces a constant angular displacement  for all parts of a vertical target. A parabolic temperature profile produces a constant magnification  of that target. (As Biot showed, this can already be enough to produce the inverted image of a mirage.) And a more complicated temperature profile produces more complicated distortions.

A constant lapse rate corresponds to a constant terrestrial refraction coefficient, which is the ratio k  of curvatures of the ray and the Earth. Surveyors use this to correct their observed angles for refraction. And, of course, their correction is proportional to the distance of the object — i.e., to the amount of air in the line of sight.

## Astronomical and terrestrial refractions

But just a minute. If the terrestrial refraction is part  of the astronomical refraction, shouldn't we be able to use the surveyor's refraction coefficient to correct the astronomical refraction near the horizon as well? Indeed, shouldn't such a correction be required ?

This notion has already occurred to several people. For example, Hervé Faye suggested in 1854 that astronomical refraction could be improved near the horizon by making use of the terrestrial refraction coefficient. This suggestion was extremely controversial, being criticized first by Biot and then by several other astronomers, after Faye refused to listen.

In 1976, the same idea was raised by Livieratos, who proposed that it be used the other way around: use astronomical observations of refraction to improve geodesy! Rather than repeat all the arguments, let's look at what's missing here.

The problem is that astronomical refraction below about 15° altitude depends on the lapse rate in the lower troposphere — and more so, the closer to the horizon you look. But terrestrial refraction depends only on the part of the atmosphere near eye level and below: the line of sight to a terrestrial object passes through only  the lowest levels, but the line of sight to an astronomical object passes through the whole  atmosphere. So there's a region in the lower atmosphere that affects astronomical refraction at low altitudes, but has no effect whatsoever on the terrestrial refraction.

If there are no superior mirages, the terrestrial refraction is unaffected by air above both the observer and the objects observed; so unless those objects are tall mountains, we get no information about the lower atmosphere from the terrestrial refraction. Even when we have superior mirages, so that the eye is inside a duct, the terrestrial refraction is not usually affected by air above the top of the duct.

If it were permissible to extrapolate the lapse rate near the ground into the lower troposphere, we could fill in the gap. But the lapse rate near the ground is restricted to the bottom of the boundary layer; the air above that is basically unconnected to the air lower down. As the boundary layer is usually several hundred meters thick, there's a kilometer or two of air above it that strongly affects astronomical refraction near the horizon, but has no influence on terrestrial refraction.

So there's no way to infer the temperature distribution above the boundary layer from terrestrial refraction observations. And that's what's needed to calculate astronomical refraction near the horizon, and hence to connect terrestrial and astronomical refractions.

## An incomplete connection

To be sure, there is a unique relation between horizontal ray curvature and the local density gradient (which, for a given temperature and pressure at eye level, depends only on the local lapse rate), and this unique relation extends to the terrestrial refraction coefficient. And the local lapse rate also tells you the magnification of astronomical objects at the horizon, through Biot's magnification theorem. So the gradient  of the astronomical refraction at the horizon is uniquely determined; and it's connected to the terrestrial refraction, through the local lapse rate.

Unfortunately, this isn't enough to determine the actual amount  of astronomical refraction at the horizon. That also depends on the thickness  of the boundary layer through which this special lapse rate extends. And of course, the rate at which the perturbed refraction blends into the region (above about 15° altitude) where the astronomical refraction becomes insensitive to the lapse rate, and depends only on the temperature and pressure at the observer, also depends on the thickness of this boundary layer, as well as on the temperature distribution for some distance above the boundary layer.

You can see some examples of how the refraction near the horizon depends on the thickness of the boundary layer, and the lapse rate in it, in my 2004 paper on refraction near the horizon. (I adopted the standard-atmosphere lapse rate above the boundary layer.)

## Extracting the temperature profile?

Because the refraction is uniquely determined by the temperature profile, it's tempting to think we could use observations of astronomical refraction to determine the temperature profile. But that turns out to be an ill-posed problem above the astronomical horizon: the temperature distribution is so smeared out in the refraction as a function of zenith distance that errors in the observations are enormously amplified in the retrieved temperature distribution. So it's impractical to try to infer the temperature profile from refraction observations above the horizon.

On the other hand, below  the astronomical horizon, the layer where the ray is horizontal is so heavily weighted in the refraction integral that the problem becomes well-posed — but only  for the part of the profile below eye level. Then a retrieval of the temperature distribution below eye level is possible (see the papers by Bruton and Kattawar in 1997 and 1998 for an example, and the history of the problem, respectively.)

However, it's just  the part of the profile below eye level that can be determined this way. That's enough to determine the terrestrial refraction of objects that are also below eye level, but not objects above it. So this isn't a general solution to the terrestrial-refraction problem.

## A slim possibility

However, as it is at least possible to infer the lapse rate below eye level from the gradient of astronomical refraction at the horizon — e.g., from the flattening of the Sun or Moon — you might imagine that we could use these flattenings to provide the geodesists and surveyors with the refraction coefficient for terrestrial refraction.

But the lapse rate near the ground is changing rapidly near sunrise and sunset, because of the rapid transition between solar heating of the surface during the day and radiative cooling of the surface at night. And there's not much else that's regularly observable at the horizon: the Moon's flattening is difficult to measure accurately because of phase effects, and stars aren't usually observable in the daytime against the bright horizon sky.

About all that leaves is the possibility of estimating the terrestrial refraction at night from stellar observations. But it's hard to see terrestrial features at night. (Livieratos tried using lasers as terrestrial markers at night.)

## Another complication

A further problem in connecting terrestrial and astronomical refraction is the effect of distance. When we view astronomical objects, which are optically at infinity, any ray bending anywhere in the atmosphere produces an equal angular displacement of the object's apparent position. But when the object is at a finite distance, the angular displacement is smaller than the amount of ray bending. The closer the object, the less is the resulting displacement.

This effect is often called parallactic refraction, a term apparently invented in 1958. It's important in reducing observations of artificial satellites and sounding rockets. You'll find just a few references to it in the bibliography. (A brief recent discussion is available at https://doi.org/10.5802/crphys.125 .)

The geodesists usually assume the ray path is a circular arc, which means the bending is uniformly distributed between target and observer. Then only half  of the total ray bending appears as angular displacement of the target. So, while the whole bending is the same as this part of the astronomical refraction, only half of the bending between observer and target is the angular displacement — the terrestrial refraction — of the terrestrial target. It's as though all the bending were concentrated at the point halfway between target and observer.

If the bending occurs even closer to the target, its angular displacement is still less. That's why a ship appears undistorted against a highly distorted sunset in Plate XIX of O'Connell's book: the Sun is distorted by the atmospheric layers in which the ship is embedded, but the ship is not.

This same effect plays an important role in mirages: objects are more distorted, the farther they are beyond the horizon. Those at or within the horizon are not distorted appreciably at all. So, while terrestrial objects beyond the horizon can be distorted, the setting Sun is usually distorted even more.

So we need to know the exact distribution of the bending along the line of sight to compute its effects on terrestrial objects, because the displacement of terrestrial objects depends on their distance from the distorting medium.

## Sunsets and terrestrial refraction

You can regard the dip of the horizon as the sum of the geometric dip and the terrestrial refraction at the apparent horizon. The dip depends on the average lapse rate between eye level and the surface at the apparent horizon.

Of course, the dip affects the time of sunset. But notice that the dip is much less affected by refraction than is the apparent position of the Sun. First, the line of sight to the setting Sun traverses the atmosphere below eye level twice, while the line of sight to the apparent terrestrial horizon only goes through it once. Second, the refractive displacement of the horizon is only about half of the total bending in this part of the path, because of the parallactic effect discussed above. So the refractive displacement of the horizon is only about one fourth of the part of the Sun's refraction produced by the layers below eye level. (Of course, the Sun is also refracted by the air above  eye level, which does not affect the sea horizon at all.)

So, even though refraction raises both the Sun and the apparent horizon above their geometric positions, these effects don't cancel out. But, even though the refractive displacement of the horizon is small compared to the Sun's, one should take these variations into account in comparing observed and computed sunset times.

## Conclusion

As a practical matter, we may as well forget about trying to tie terrestrial and astronomical refraction together quantitatively, although terrestrial mirages at the horizon certainly promise even greater distortions of the sunset.

As the geodesists have worked out rules of thumb for the variation of their refraction coefficient as a function of time of day, and the boundary-layer meteorologists have also some understanding of the diurnal variations in the boundary layer, it's not beyond the realm of possibility that someone might put all these pieces together, and eventually come up with a way to predict the astronomical refraction near the horizon with improved accuracy. But, until then, astronomers are content to stay away from the horizon, and work in the region where Oriani's theorem promises a refraction that depends only on the local temperature and pressure.

Just don't expect average refraction tables to be very close to the actual refraction at a particular time and place, when you're looking within a few degrees of the horizon.