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Figure 3.3. Snell's law: Refraction changes with medium. The relations between the angles of incidence, i, and refraction, r, and the indices of refraction, n, in the two media are n1sin i = n2sin r for the incoming ray, and n2sin i' = n1sin r' for the outgoing ray. Note that if the incoming and outgoing surfaces are not parallel, neither will be the entering and exiting rays. Drawing by E.F. Milone.

Figure 3.3. Snell's law: Refraction changes with medium. The relations between the angles of incidence, i, and refraction, r, and the indices of refraction, n, in the two media are n1sin i = n2sin r for the incoming ray, and n2sin i' = n1sin r' for the outgoing ray. Note that if the incoming and outgoing surfaces are not parallel, neither will be the entering and exiting rays. Drawing by E.F. Milone.

would if viewed from an airless world (see Figure 3.2). It is a consequence of the behavior of light in optical media: Light travels slower through a medium other than vacuum. The ratio of the speed in vacuum to that in the medium is called the index of refraction, n. Snell's law summarizes the bending of light as it passes from a medium of index n1 into a medium of index n2. If a! is the angle with respect to the surface normal of the light incident at the boundary between the two media, and a2 is the angle with respect to the same normal inside medium 2, the relation between the two angles is n1sina1 = n2sin a 2. (3.15)

The relationship is illustrated in Figure 3.3.

If the atmosphere were a plane-parallel layer, and the optical density of the atmosphere were uniform, this expression could be used directly to obtain the refraction. The index of refraction of airless space, n1 = 1, and the index of refraction of air is slightly greater than 1.00, the exact amount depending on temperature and air pressure. Thus, a2 < ai, and the ray bends toward the normal as it enters the atmosphere. The difference between the two angles, for values of a1j2 s 45° (see Tricker 1970, pp. 11-13), is

By comparing the refraction at many successive levels of the atmosphere, one may arrive at a slightly more general result. For relatively small zenith distances, or large altitudes, an approximate correction to the zenith distance for refraction is given by Woolard and Clemence (1966, p. 84):

Expressed as a change in altitude, Ah = -Az. For STP (standard temperature and pressure conditions, T = 0°C and sea-level pressure of 1000mbars), it amounts to about 1 arc-minute at 45° altitude. For substantially higher zenith distances, or lower altitudes, the Pulkovo Observatory tables, first published in 1870, have often been used, but the results are strongly dependent on ambient weather conditions, particularly water vapor content. Garfinkel (1944) published a theoretical treatment that is valid to z > 90° (i.e., below the horizon) but was not easily applied; this work was extended, and a table of extinction at large zenith distances was provided in Garfinkel (1967). Schaefer (1993a, p. 314) cites for the refraction down to the horizon two formulae given by Saemundsson (1986) that contain the true and refracted altitudes, respectively, but contain no explicit terms for temperature or pressure. Thom (1971/1978) measured current values in the British isles and concluded that the refraction varies widely enough to provide a slight but measurable uncertainty in the hour angle and azimuth of rise of objects on the horizon. In §6.2.12, we describe Thom's use of an assumed value of the refraction to determine the date when certain assumed alignments were in use. Thus, refraction affects our interpretation of archeological alignments.

The refraction nominally amounts to ~34' or 0°57 at the horizon, but sometimes can greatly exceed this value. This means that at first gleam, the moment when the upper limb of a rising Sun or Moon is first seen, it is still a full diameter below the horizon; even when it appears fully risen with the lower limb on the horizon, the unrefracted Sun is still below the horizon. The effect of refraction causes the Sun to appear slightly oblate at the horizon. Refraction is slightly greater for the lower limb (~35arcmin) than for the upper (by ~29arcmin); hence, the vertical dimension of the Sun will be smaller than its width (~20%). It does not, however, cause the Sun to appear larger at the horizon than at the zenith. Theoretically, the angular width of the disk diameter at sunrise is slightly smaller due to a slightly greater topocen-tric distance at the horizon compared with the meridian, the diameter being largest for a zenith Sun. In practice, refraction is important in naked eye astronomy only for the Moon, for which the geocentric parallax amounts to ~1°. Measurements consistently confirm this expectation. Refraction phenomena of rising or setting objects can be complicated, however, by the presence of temperature inversions in the atmosphere. Because this commonly occurs along coastlines, particularly in desert areas, unusual shapes and colors sometimes result. See Greenler (1980, Plates 7-10, 7-11, and pp. 160-162) for a series of illustrations of the appearance of the setting Sun under differing refractive conditions and for the refractive effects of differences between air and ground/water temperatures leading to mirages. Greenler (1980, pp. 158-159) notes that inferior mirages, which occur when the air temperture is lower than the ground, are best known as desert mirages but are seen as well in much colder climates over bodies of water. Anomalously high atmospheric refraction has been reported for observations in the high arctic, again attributed to an inversion layer. This optical ducting phenomenon is called the Novaya Zemlya effect, after observations of the midnight Sun from the south coast of Novaya Zemlya (f ~ 76°5) were reported by an expedition led by Willem Barentsz in 1597. The Sun should have been below the horizon at the expedition's location on the dates reported, and this subsequently aroused controversy. Subsequent observations of the effect were reported by Shackleton (1920), Liljequist (1964), and Lehn and German (1981), among others. On May 16,1979, at Tuktoyaktuk in the Canadian arctic, Lehn and German (1981) found a distorted image of the Sun above the horizon when it should have been 94 arc-minutes below the horizon! They were able to model the observations by means of a temperature inversion layer, which creates a kind of light pipe over the altitude range of rapid temperature rise and the horizon. Images of the Sun that show the effect and distortions in solar appearance are reproduced in Figure 3.4.

The common perception that the Moon appears to be larger at the horizon has been attributed (Pernter and Exner 1922) to the human impression that the horizon seems further away from the observer than is the zenith, so that objects of the same angular size will appear larger. Kaufman and Rock (1962a,b) discuss in detail a series of experiments on the Moon illusion and show that the effect disappears when the terrain is blocked from view. The 2nd century a.d. (Han period) Chinese astronomer Chang Heng attributed the phenomenon to an optical atmospheric effect. Rees (1986) summarizes the explanations for the illusion, but there is no consensus for which, if any, is the cause. The current view is that several causes may contribute to the illusion (Schaefer 1993a, p. 340). In so far as physical explanations apply, it is curious that we have seen no reference to a "big Sun" effect; perhaps this in itself lends weight to the psychological explanation for the "big Moon" effect.

The index of refraction is wavelength dependent, which means that it changes with light of different wavelength. Most media bend blue light more strongly than red. This dispersive effect gives rise to a wealth of phenomena, including prism-produced spectra. "Wedge refraction," as the dispersive effect of the Earth's atmosphere is known, causes low-altitude star images to take on the appearance of tiny spectra, with the blue end of the spectrum shifted higher, toward the zenith. This phenomenon can be seen through even small telescopes. Woolard and Clemence state that the red and green images are separated by 2"2 at z = 75°. The dispersive effects of parcels of air on sunrise or sunset images may give rise to a "green flash" (§5.1.2), in which the blue-green components left in the Sun's light are momentarily visible to the observer. Dispersive refraction effects also play a role in the colors of rainbows, which involve refraction in water droplets, and of sundogs, which involve

Figure 3.4. Refracted images of the setting Sun as seen at Tuktoyaktuk, in Canada's Northwest Territories, on May 16, 1979, that demonstrate the Novaya Zemlya Effect. Shown are the Sun's appearances at (a) 1:34 a.m., MDT, h = -35 (b) 1:41:30, h = -46.5 (c) 1:49, h = -57'; (d) 2:06, h = -75'; and (e) 2:44, h = -94 . The altitude, h, refers to the center of the Sun's disk. Photos courtesy of Professor W. Lehn.

Figure 3.4. Refracted images of the setting Sun as seen at Tuktoyaktuk, in Canada's Northwest Territories, on May 16, 1979, that demonstrate the Novaya Zemlya Effect. Shown are the Sun's appearances at (a) 1:34 a.m., MDT, h = -35 (b) 1:41:30, h = -46.5 (c) 1:49, h = -57'; (d) 2:06, h = -75'; and (e) 2:44, h = -94 . The altitude, h, refers to the center of the Sun's disk. Photos courtesy of Professor W. Lehn.

Figure 3.5. The effect on the altitude of the horizon of the combination of dip and refraction. Drawing by E.F. Milone.

Figure 3.5. The effect on the altitude of the horizon of the combination of dip and refraction. Drawing by E.F. Milone.

ice crystal refraction. More thorough discussions of these effects can be found in Minnaert (1954), Meinel and Meinel (1983), and Greenler (1980).

In addition to corrections for refraction, observations need to be corrected for dip and for parallax. The higher we are, the more we can see "over" a level horizon. Observations of objects on the horizon made from a mountain top or from a high shipboard mast must be corrected for the depressed horizon, but dip is already appreciable for an observer standing at the shoreline. Figure 3.5 shows the effect on the altitude of the horizon of the combination of dip and refraction, but without the curvature of the Earth.

A commonly used expression for the dip correction to the observed altitude is tion; for € < ~250ft, k ~ 1. If the observation is being made for navigational purposes, the correction, which is negative, is made to the observed altitude, which is too large. If the observer is depressed below the true, level horizon, the correction is positive. The dip correction is a compensation for a non-level observational horizon and it is strongly affected by, and so must be corrected for, refraction. A classic reference to this treatment is Helmert (1884).

According to Young (1998, private communication), a more correct expression for the dip is

where € is the height of the eye above sea level, examining a sea-level horizon. The constant k, which depends on refraction and indirectly on €, because the degree of refraction depends on €, would be 1.07 in the absence of refrac-

where AT = T (at the observer) - T (at the horizon). Note that in this expression, the temperature correction may be negative, and although Eqn (3.19) holds only for Ah ^ 0, in fact, elevations of several arcminutes may occur. The observed azimuth and altitude must be corrected for dip and refraction, before being used to compute the declination of topocentric geocentric ^ "

Figure 3.6. Topocentric versus geocentric coordinate systems, and the geocentric parallax of the Moon. Drawing by E.F. Milone.

the object [see (2.1)]. There is one more correction to apply, however, for observations of the nearest of the heavenly bodies, the Moon.

The effect of parallax is a familiar one in everyday life: As we blink first one eye then the other, close objects appear to shift against more distant objects. If one moves sideways, the angular shift is larger. It is also larger for the nearer objects. The amount of parallax observed therefore depends both on the distance of the object and on the size of the observer's baseline. In the astronomical context, observations are normally reduced to the center of the Earth because the catalogued positions of objects are given in geocentric coordinates, whereas the observer measures topocentric coordinates in the local (topocentric) coordinate system (see Figure 3.6).

If an observed altitude has been corrected for dip and refraction, the parallax correction to be applied is

where [email protected] is the radius of the earth at the observer and d is the geocentric distance at the moment of observation. The effect of the parallax is to lower the observed altitude compared with that seen by a hypothetical geocentric observer. At the place on the Earth directly under the Moon (the sublunar point), the zenith distance of the Moon is zero. At all other locations from which the Moon is visible, the zenith distance must be greater; therefore, the Moon's altitude is smaller at all of those locations. Therefore, the correction to the observed parallax is positive in all cases. The geocentric parallax of the Moon is larger than that of any other natural object: The average equatorial horizontal parallax (i.e., the parallax at rising or setting for an observer on the equator, at average lunar distance) is about 57 arc-minutes. The solar horizontal equatorial parallax is ~8.8 arc-seconds. For some configurations of the inner planets, Mars, and several asteroids, the parallax is intermediate. Equation (3.20) can be used to calculate the parallax for those cases. Stellar parallaxes are measured with respect to the semimajor axis of Earth's orbit and are less than 1 arc-second.10

The final correction we mention here is that of the semi-diameter. Tabulated positions of the Sun and moon are given (in modern catalogues anyway) for the centers of the disks. If we are interested in the upper limb (edge), the corrected

10 That of a Centauri, the closest star system, is only 0.76arc-sec.

angular radius (or semidiameter as it is called in some catalogues) must be subtracted from the observed altitude to reduce the observation to disk center.

To apply the corrections we have discussed correctly, remember that we have described corrections to observational data. If one is correcting values of the altitude, say, calculated from catalogue positions with the purpose of reproducing observational data, the corrections indicated must all be applied in the opposite sense. Finally, we note that although the corrections noted here suffice for most purposes, the occasional need for greater precision may require more rigorous treatment. Finer corrections for the effects we discuss here as well as other effects can be found in Woolard and Clemence (1966), for example, among other works on positional and practical astronomy.

An empirical study of refraction at the horizon through timings of risings and settings of the Sun, Moon, and Venus and unpublished sextant observations was reported by Schaefer and Liller (1990), who showed that significant variations from the expected shift due to local temperature and pressure conditions can be seen. They correct for the semi-diameter and use a dip correction: D = arccos[1/(1 + A/6378)], where A is the altitude of the site above sea level, in kilometers. They compute the measured refraction: R0 = Zt - 90° - D. On the basis of 144 measurements from seven sites, Hawaii; Viña del Mar, La Serena, and Cerro Tololo (Chile); Nag's Head and Kitty Hawk (North Carolina); and O'Neill, Nebraska, they report (their Fig. 2, and pp. 802-803) an average R0 value of 0°551 over a total range of 0°234 to 1°678, an overall uncertainty figure of ±0°16, and that 97% of their measurements fall within a range of 0°64, and only 4 measurements are outside of the range 0°. 23 to 0°. 87. Although the larger-than-expected variations are attributable to a variety of observing circumstances (timing errors, effects of clouds, etc.), they conclude that differences in the atmospheric temperature profile are responsible. They use these results to argue against the validity of the precision measurements of lunar declination claimed by Thom (cf. §6), even though their results are not based on any measurements made in the British Isles or, of course, necessarily appropriate to atmospheric conditions present in the Megalithic.

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