Extraterrestrial Mirages

When we turn from mirages of terrestrial objects to those of extraterrestrial bodies, most notably the sun and moon, we can no longer pretend that Earth is flat. But we can pretend that its atmosphere is uniform and bounded with a constant refractive index equal to the surface value n0. The integrated refractive index of a vertical ray from the surface to infinity is the same in an atmosphere with an exponentially decreasing molecular number density as in a hypothetical atmosphere with a uniform density equal to the surface value up to the scale height H.

A ray refracted along a horizon path by this hypothetical atmosphere and originating from outside it (Fig. 8.20) must have been incident on it from an angle S below the horizon. From Snel's law we have sin = sin($t + S) = n0 sin i&t = sin cos S + cos i&t sin S,

from which it follows that sin 5 = tan $t (no — cos 5), (8.52)

where

The radius R of Earth is much greater than the scale height of its atmosphere. Given that we expect 5 C 1 because n « 1, we can approximate sin 5 by 5 and cos 5 by 1 in Eq. (8.51) to obtain

A slightly more accurate (about 10%) but more cumbersome expression can be obtained from Eq. (8.51) by truncating the series expansion for the cosine after the second term rather than the first.

According to Eq. (8.54) when the sun or moon is seen to be on the horizon it is actually more than halfway below it, 5 being about 0.34°, whereas the angular width of the sun or moon is about 0.5°.

Extraterrestrial bodies seen near the horizon also are vertically compressed. The simplest way to estimate the amount of compression is from the rate of change of angle of refraction with angle of incidence for a uniform slab, which from Eq. (8.51) is d$t cos $i ¡1 — sin $i

where the angle of incidence is taken to be that for a curved but uniform atmosphere such that the refracted ray is horizontal: R

if we neglect terms of order (H/R)2 and approximate n0 + 1 as 2 and n0 as 1 when it is a multiplicative factor. According to Eq. (8.57) the sun or moon near the horizon is distorted into an ellipse with aspect ratio about 0.87. We are unlikely to notice this distortion, however, because we expect the sun and moon to be circular, and hence we see them that way. But if we compare two photographs of a low sun or moon taken at the same moment, one rotated by 90° relative to the other, the elliptical shape may become obvious.

Our conclusions about the downward displacement and distortion of the sun were based on a refractive-index profile determined mostly by the pressure gradient. That is, the average

Figure 8.21: Atmospheric refraction transformed this low sun into nearly a triangle. The serrations are a consequence of horizontal variations in the atmospheric refractive index.

refractive index gradient for a uniform slab of thickness H is (1 — n0)/H, which is the same as Eq. (8.39) with n = n0 if the temperature gradient term is negligible. But as we noted, near the surface the temperature gradient is the prime determinant of the refractive-index gradient, as a consequence of which the horizon sun can take on shapes more striking than a mere ellipse. For example, Fig. 8.21 shows a nearly triangular sun with serrated edges. Assigning a cause to these serrations provides a lesson in the perils of jumping to conclusions. Obviously, the serrations are the result of sharp changes in the temperature gradient - or so one might think. Setting aside how such changes could be produced and maintained in a real atmosphere, a theorem by Alistair Fraser gives pause for thought: "In a horizontally (spherically) homogeneous atmosphere it is impossible for more than one image of an extraterrestrial object (sun) to be seen above the astronomical horizon [horizontal direction determined by a bubble level]." These serrations on the sun are multiple images. But if the refractive index varies only vertically (i.e., along a radius), no matter how sharply, multiple images are not possible. Thus the serrations must owe their existence to horizontal variations of the refractive index, which Fraser attributes to gravity waves propagating along a temperature inversion.

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