OPTICAL
REFRACTION
CURVATURE
IN THE
TROPO[SPHERE]
Optical Refraction Curvature in the Tropo[sphere]: The Optical Coefficient of Refraction (\(k\))
Nota Bene
As stated in the first paragraph of our web page titled, Refraction Overview:
The purpose of the REFRACTION section is to (a) present to the reader, the underlying rationale for (spheroidal earth) geodesists using (alleged) tropo[spheric] refraction to explain (in many cases) the visibilty of subjects of observation that should otherwise be below the horizon (i.e., below and normal to the observer’s horizon line of sight), and (b) give the benefit of the doubt to the fallacious spheroidal earth paradigm by allowing readers to assume the impact of maximum tropo[spheric] refraction on their field observation calculations (see Introduction to the (Allegedly Spheroidal) Earth Calculators).
Such calculations are of paramount importance in REVEALING THE PLANAR LARGE-SCALE STRUCTURE OF THE EARTH’S SURFACE.
Carl Friedrich Gauss (1826)
Since at least the early nineteenth century, adherents to the currently accepted model of the large-scale structure of the earth’s surface have been characterizing the curvature of tropo[spheric] refraction in terms of a fraction of the (alleged) curvature of the earth. As stated by F. K. Brunner in relation to the German mathematician, Carl Friedrich Gauss:
In 1826 C. F. Gauss derived a value of 0.13 for the refraction coefficient (i.e. the ratio of the curvature of the earth to that of the light path) of the vertical angle measurements in the geodetic network of Hannover. He assigned an uncertainty of ± 25% to this value.1 […]
Hence, the refraction coefficient, \(k\) (according to Gauss and in terms of uncertainty) is in the range,\[\{k\,|\,[k_{-}=(0.75\times0.13)]\]\[\textrm{ < } (k_{mean}=0.13)\]\[\textrm{ < } [k_{+}=(1.25\times0.13)]\},\]
which gives,
\[\{k\,|\,(k_{-}=0.0975)\]\[ \textrm{ < } (k_{mean}=0.13)\]\[ \textrm{ < } (k_{+}=0.1625)\},\]
or approximately,
\[\{k\,|\,(k_{-}=0.10)\]\[\textrm{ < } (k_{mean}=0.13)\] \[\textrm{ < }( k_{+}=0.16)\}.\]
Encyclopaedia Britannica (1842)
A further nineteenth century exposition of the said characterization of curvature appeared in the Seventh Edition (1842) of The Encyclopaedia Britannica in an article on levelling.2 The article first defines levelling as follows:
LEVELLING may be defined the art which enables us to find a line or surface exactly level, as also to find how much higher or lower any given point on the surface of the earth is than another.3
Beside the left margin caption stating,
Difference between the apparent and true level4
the article states:
Let BD be a small portion of the earth’s circumference, whose centre of curvature is A, and consequently all the points on this arch will be on a level. But a tangent BC meeting the vertical AD in C, will be the apparent level at the point B; and therefore DC is the difference between the apparent and true level at the point B. The distance CD, therefore, must be deducted from the observed height, to have the true difference of level, or the differences between the distances of two points from the surface of the earth, or from the centre of curvature A.5 […]
Figure 1. Difference between the apparent and true level (from The Encyclopaedia Britannica, 1842).
After providing formulae and tables in respect of CD, the article continues:
The preceding formulae and tables suppose the visual ray CB to be a straight line; whereas, on account of the unequal densities of the air at different distances from the earth, the rays of light are incurvated by refraction. The effect of this is to lessen the difference between the true and apparent levels, but in such an extremely variable and uncertain manner, that if any constant or fixed allowance is made for it in formulae or tables, it will often lead to a greater error than what it was intended to obviate. For, though the refraction may at a mean compensate for about a seventh of the curvature of the earth, it sometimes exceeds a fifth, and at other times does not amount to a fifteenth.6 [...]
Hence, the refraction coefficient \(k\) (according to the 1842 edition of The Encyclopaedia Britannica) is in the range,
\[\{k\,|\,(k_{min}=1/15)\]\[\textrm{ < } (k_{mean}=1/7)\] \[\textrm{ < }( k_{max}=1/5)\},\]
or,
\[\{k\,|\,(k_{min}=0.067)\]\[\textrm{ < } (k_{mean}=0.142)\] \[\textrm{ < }( k_{max}=0.20)\},\]
or approximately,
\[\{k\,|\,(k_{min}=\boldsymbol{0.07})\]\[\textrm{ < } (k_{mean}=\boldsymbol{0.14})\] \[\textrm{ < }( k_{max}=\boldsymbol{0.20})\}.\]
D. C. Williams and H. Kahmen (1984)
The above statement from The Encyclopaedia Britannica (1842) remains significant today. In more recent times, Williams and Kahmen7 have determined the influence of refraction at low altitudes to be from about one-fifth (presumably a maximum) of the (alleged) curvature of the earth (in consideration of the atmospheric pressure gradient exclusively) down to about one-seventh of the (alleged) curvature of the earth (if an adiabatic temperature lapse rate8 is assumed), the latter corresponding to the standard value of \(0.14\) traditionally used by surveyors. With respect to refraction magnitudes, Williams and Kahmen specifically conclude:
[…] At low altitudes, the contribution of the pressure gradient term to [the path curvature of the light ray] \(σ\) is \(32\,\textrm{μrad/km}\) downwards, about one fifth of the curvature of the earth (\(1\,\textrm{μrad}=0.21''=0.65^{\textrm{cc}}\)). Further, if the temperature lapse rate \(∂T/∂z\) assumes the adiabatic value \(Γ\), then it is shown in meteorological texts that [their (Eq. (13)]
\[Γ=\dfrac{∂T}{∂z}=\dfrac{g}{C_p}\]\[ =0.010° \textrm {C}\,\textrm{m}^{-1}\tag{13}\]\(C_p\) being the specific heat of air at constant pressure. This reduces \(σ\) to about \(22\, \textrm{μrad/km}\), one seventh of the curvature of the earth, which correspond to the standard coefficient of refraction value of \(0.14\) normally used by surveyors.9 […]
— FINIS —
The historical value for the mean radar (or radio frequency) coefficient of refraction \(k\) is discussed and summarized in the Radar (or RF) Refraction Curvature in the Trops[sphere] subsection and a brief overview of the Effective Earth Radius Model (EERM) developed in respect of tropo[spheric] radar (or RF) refraction is provided.
F. K. Brunner, “A. Overview of Geodetic Refraction Studies,” in Geodetic Refraction: Effects of Electromagnetic Wave Propagation Through the Atmosphere, edited by F. K. Brunner (Berlin: Springer-Verlag, 1984), pp. 1–6.↩️
The Encyclopaedia Britannica or Dictionary of Arts, Sciences, and General Literature, Seventh Edition (Edinburgh: Adam and Charles Black, 1842), Volume XIII, LEVELLING, pp. 259–263 (see https://books.google.ca/books?id=qvhMAQAAMAAJ&pg=PA261&dq=levelling+encyclopaedia+britannica&hl=en&sa=X&ved=0ahUKEwj58q7Hhc_UAhWENj4KHWJDBc4Q6AEILTAB#v=onepage&q=levelling%20encyclopaedia%20britannica&f=false).↩️
Ibid., p. 259.↩️
Ibid., p. 260.↩️
Ibid.↩️
Ibid. The reader should note that the geometry reproduced from p. 260 of the reference as Figure 1 above as well as the tables on p. 260 in respect of that geometry represent approximations of the alleged curvature of the earth based on the Pythagorean theorem and are sufficiently accurate for the distances involved in surveying. The alleged curvature of the earth (for any distance) is represented by the geometry depicted in Figure 1 on our web page titled, (Allegedly Spheroidal) Earth Curvature Geometry and the subsequent web pages listed in the GEOMETRY section. Calculators based on that geometry are provided in the CALCULATORS section.↩️
D. C. Williams and H. Kahmen, “B. Two Wavelength Angular Refraction Measurement,” in Geodetic Refraction: Effects of Electromagnetic Wave Propagation Through the Atmosphere, edited by F. K. Brunner (Berlin: Springer-Verlag, 1984), pp. 7–31.↩️
Joseph M. Moran, Weather Studies: Introduction to Atmospheric Science, Fourth Edition (Boston: American Meteorological Society, 2009), p. 141. Under ADIABATIC PROCESS, the author states “… During an adiabatic process, no heat is exchanged between an air parcel and its surroundings. […] The temperature of an ascending or descending unsaturated air parcel changes in response to expansion or compression only. […] Adiabatic cooling of ascending unsaturated air amounts to 9.8 Celsius degrees per 1000 m (5.5 Fahrenheit degrees per 1000 ft of ascent. This is the dry adiabatic lapse rate […] Latent heat that is released to the environment during condensation or deposition partially counters expansional cooling. Consequently, an ascending saturated (cloudy) air parcel cools more slowly than an ascending unsaturated (clear) air parcel. Rising saturated air parcels cool at the moist adiabatic lapse rate.”↩️
Williams and Kahmen, op. cit., pp. 11–12. It should be noted that while the term \(g\) used by Williams and Kahmen in their Eq. (13) refers to the acceleration due to gravity, and while the relation of that otherwise measurable entity to air density is not disputed, this website does dispute the metaphysical precept (and yes, precept, not concept) of gravity whereas that precept is sophistically integral to the current (heliocentric, spheroidal earth) model of the large-scale structure of the earth’s surface. But notwithstanding that the preceding statement begs the question of the conceptualization of that entity in terms of the proposed (geocentric, planar earth) model, elucidation of the phenomenon of terrestrial acceleration is necessarily secondary to the definitive and ontological ascertainment of the large-scale structure of the earth’s surface within the framework of the scientific method.↩️
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