In discussing the uncertainty affecting the atmospheric refraction at low altitudes Bowditch, Ed. 1917, Art. 298, sets forth that under certain conditions of the atmosphere a very extraordinary deflection occurs in rays of light which reach the observer’s eye from low altitudes the amount of which is not covered by the ordinary corrections for temperature and pressure. The error thus created is described under Dip (Art. 301) ; on account of it altitudes less than 10° should be avoided.” There follows under Art. 301 the statement that “ reliable observations have frequently placed this error above 10' and values as high as 32' have been recorded.”
If the Dip is of uncertain amount to anything like the extent here suggested its uncertainty must affect in equal amount all altitudes measured from the sea horizon, whether they be more or less than 10°, and the precept to avoid low altitudes seems to be a non sequitur from the cause alleged. The uncertainty in the amount of the Dip correction arises from abnormal refraction suffered by the rays of light that come from the horizon to the observer and it appears to be inferred that rays coming from the sun or a star and passing near the horizon will be in like manner abnormally refracted by an additional amount comparable with that above set forth. In consequence an observation of such a body would be uncertain on account of abnormal refraction of its own light in addition to abnormal refraction of the horizon and one of these sources of error could be escaped by avoiding low altitudes.
To the writer, as an astronomer, these alleged abnormal refractions appear exaggerated, both in amount and in frequency of occurrence. At least they seem to call for evidence in their support, and as such evidence is not found in books readily accessible I have sought to obtain it from observations, e. g., risings and settings of celestial bodies observed at the sea horizon under conditions that will permit a determination of the amount of refraction actually present and its comparison with the corresponding amounts furnished by the refraction theory and tables. I have been unable to find suitable published material of this kind, although it doubtless exists, and a considerable amount of inquiry for unpublished material has brought only one return, viz., an excellent series of 27 observations of the time of sunset made at Key West between January 10 and April 16, 1911, by Captain E. E. Hayden, U. S. N. These observations were kindly placed at my disposal by the Hydrographic Office.
The physical conditions under which these observations were made are distinctly favorable to the occurrence of abnormal refraction, since, owing to the sun’s change in declination, the observations were made over fairly deep water (the Gulf Stream) during January, while later the horizon behind which the sun set lay over extensive coral reefs covered by shallow water. While the refraction tables take no account of such circumstances, theory indicates that the actual refractions should be larger in the early part and smaller in the latter part of this series of observations than the tabular values, even when corrected for temperature. The observed facts are in agreement with theory.
I have used each observed time of sunset in the Key West series to compute the actual amount of refraction suffered by the sun’s disappearing edge and I have compared this result with the corresponding value computed directly from Radau’s theory of the refraction (Annales, Observatoire de Paris, Tome 19) since this appears to me the best theory available at this time. It is certainly much better than Bessel’s theory, which is employed in Bowditch, although under ordinary conditions the numerical results of the two theories are not very different. I have not used for this purpose the numerical tables given by Radau, but have computed values directly from the differential equation for the refraction, taking into account the division of the atmosphere into troposphere and stratosphere and the observed temperature gradients established by the meteorologists subsequent to Radau’s work. The results of the comparison between fact and theory may be arranged as follows:
- In the Key West observations the average deviation of the observed from the computed refraction at the horizon, taken without regard to sign, is 2'.2.
- These deviations are distinctly seasonal in character. When the sun sets over deep water the computed and observed refractions are in substantial agreement. When sunset is over the shallow waters of the coral reef the observed refractions are smaller than the computed ones.
- The constant coefficients that enter into the refraction theory are based upon observations of stars made at night, and Radau indicates, loc. cit., that the refraction suffered by the setting sun is probably less, under normal conditions, than that of the stars. Following this suggestion I have determined from the Key West observations themselves a mean value of the coefficient of refraction and substituting this in place of the conventional value employed above I find for the probable error of a computed horizontal refraction (supposing the observed refraction to be absolutely free from error) p. e.= ± 1'.0.
- In any given case the actual refraction is a physical product of the air and the manner in which its density varies along the path of the ray of light in question. To examine into this distribution of density and its change from day to day, I have computed the refractions not only for the normal atmospheric conditions for sub-tropical waters, indicated by modern meteorological research, but also for two abnormal conditions lying on opposite sides of the normal state, but well within the range of observed variation from that state. One of these abnormal conditions approximates to that which results in a thunder storm, i. e., the bottom stratum of the air (within two kilometers of the earth’s surface) is strongly heated so that the vertical fall of temperature is very rapid, but not sufficiently rapid to result in dynamical instability of the atmosphere. The second abnormal condition is defined by a vertical temperature gradient comparable with that which obtains over a snow-clad soil at night, i. e., there is a slight rise of temperature with increasing height up to a few hundred feet (inverted gradient) but this rise is kept well within the limits furnished by observation. At an assumed temperature and barometric pressure corresponding to the mean of the Key West observations, i. e., 25° C. and 763 mm., the theoretical refraction at the horizon, h=0°, corresponding to these three states of the atmosphere is
| h=0° | h=4° |
First abnormal state | 27’ 23” | 11’ 2.6” |
Normal state | 31’ 10” | 11’ 2.7” |
Second abnormal state | 36’ 29” | 11’ 3.2” |
Bowditch values | 34’ 28” | 11’ 6.0’ |
It appears from these numbers that the refraction at the horizon is so largely influenced by local and transitory conditions in the lower atmosphere that it may differ by ten or even twenty per cent from the mean value furnished by the tables. Since these conditions can rarely be taken into account by the observer the figures may be looked upon as furnishing a measure of the risk he runs in using an observed altitude of o°. Nevertheless, the Key West observations show that such large deviations from the tabular amount of the refraction are the exception rather than the rule, quite like in this respect to the abnormal variations in the dip of the horizon noted in Bowditch.
With one exception all of the Key West observed refractions fall well within the limits fixed by the atmospheric conditions assumed above, and their entire range of values is sufficiently accounted for by conditions falling inside these limits. But in one case, January 12, the observed horizontal refraction is only 23'.4, which corresponds to an atmospheric gradient that should promptly result in a tornado. As no such result was noted this abnormal refraction must be attributed to some other cause, possibly to an error of twenty seconds in the observed time of sunset due to the sun’s disappearance behind a distant cloud bank instead of behind the horizon.
- The unpredictable variations in the amount of the horizontal refraction, which are found above, may well lead to caution in the use of observations made at the horizon, although at a pinch they will furnish a line of position (Sumner line) whose probable error will be approximately 2'. A more important result of this inquiry is that even large errors in the horizontal refraction, such as are found under IV, disappear very rapidly with even a slight increase in the altitude of the body observed. The last column of the table in IV shows for an altitude of 40 the computed refraction corresponding to the several states of the atmosphere that produce at the horizon the results there shown. These variations in the refraction at 40 altitude are wholly insignificant for any purpose of navigation, but it should be borne in mind that these variations arise wholly from anomalous conditions in the lowest two kilometers of the earth’s atmosphere. Anomalous variations of density are found in the atmosphere up to a height of some 15 km., but their effect upon the horizontal refraction is relatively small, e. g., an extreme range of 100 seconds at the horizon and 1" or possibly 2" at an altitude of 40. It appears, therefore, that at so low an altitude as 40 the computed refractions are sufficiently reliable for all purposes of navigation. The Bow- ditch limit of io°, below which observations should not be made, is a needless restriction upon the navigator. Particularly in the winter season when the sun cannot be observed upon the prime vertical, it may be observed, and should be observed, much lower and therefore much nearer to the prime vertical than is done under current practice.
While experience must determine the lower limit of altitude to which observations may be extended, theory would indicate that the refraction furnished by the best tables for an altitude of 20 is reliable to within a fraction of a minute.
Additional data with which to compare and control these conclusions would be very gladly received by the author. Such data should comprise the observed time of rising or setting, at the sea horizon, of any bright celestial body, supplemented with the elevation of the observer’s eye above the sea, the temperature and barometric pressure of the atmosphere, and if possible the relative humidity or vapor tension. The latitude and longitude of the place of observation should be given to within a fraction of a minute. The observed times are best expressed in Greenwich mean time.