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| An essay on the historic aspects as it affects sextant readings in arctic regions | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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A review of web published refraction data | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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I have recently come across a web site dealing with "eh2r," or
extremely high horizontal refraction, in which the author, Wayne Davidson,
claims that his observations of the effect of refraction on the apparent
shape of the sun demonstrate that historians judged Cook's north pole claim
too hastily, and "easily prove Peary's data as unobservable." In fact, as
discussed below, it is Cook's data that are "unobservable," and Davidson's
observations shed little if any light on Peary's claim. BACKGROUND Because light rays bend when passing through the earth's atmosphere, as its density and temperature change, celestial objects appear higher above the horizon than they actually are (except when straight overhead). This effect is relatively minor for objects high up in the sky, but increases as an object gets near the horizon. It is dramatic when the object is very close to or below the horizon, so much so that even when the sun is a degree or more below the horizon, it may be seen above the horizon. The angular measure of the sun's diameter changes depending on the season (since the earth's orbit is not round), but is generally about 32' of arc. Thus the sun's upper edge (limb) is about 32' of arc higher than its lower limb. When the sun is near the horizon, its lower limb (being closer to the horizon) is raised more by refraction than its upper limb, resulting in a squashing, or vertical compression of the sun. Davidson's thesis comes down to this: If a person claims to have seen a nice round sun sitting low in the sky, he is lying. If he claims to have seen a severely squashed sun sitting low in the sky, he may be telling the truth. DAVIDSON'S CLAIMS ABOUT FREDERICK A. COOK I am always skeptical of arguments advanced by people who support Cook's claim of polar attainment in 1908. In my experience, such people have abandoned common sense in favor of knee-jerk anti-establishment inclinations, or they are utterly ignorant of the facts, or both. Davidson's assertion that the vertical compression of the sun observed by Cook is "theoretically possible" is either a joke (that will go unappreciated as such by the non-expert who may be exposed to it) or the most foolish statement1 I have seen in the Cook controversy (and I have seen some foolish ones). Cook published two sun observations in his come-back work My Attainment of the Pole that showed both upper limb and lower limb observations. (Typically upper limb and lower limb observations are averaged to determine the altitude of the center of the sun.) Cook claimed to have used a plate glass artificial horizon, so his observations would have measured the angle between the sun and the reflected image of the sun in the horizontal glass plate. Thus the measured angle in each case would be two times the sun's altitude (angle above the horizon). In the first (1911) edition of his book, he reported the following upper and lower limb double altitude observations—
Averaging the upper and lower limb readings gives the right altitude of the sun's center in each case for the position (latitude) that Cook was trying to support—
However, the spread between the upper limb and lower limb readings is far too small. The difference between the altitude of the upper limb and the altitude of the lower limb should be the sun's diameter, approximately 32'. Cook apparently applied this difference in fabricating double altitude readings (raw data), forgetting that the resulting difference between the altitudes of the respective limbs of the sun (after dividing by 2) would be one-half that amount. To illustrate this, the raw data reported above are converted to observed altitudes (as opposed to double altitudes) by dividing by 2 (ignoring the 2' correction for index error):
The differences shown are the sun's observed diameter, in each case about half what it should have been. Cook's published results brought forth howls of derision, since the only possible explanation for such an error in the sun's observed diameter was that the sights were fakes, worked backwards from the desired result for the sun's center. In later editions of My Attainment of the Pole, Cook attempted to fix this problem. However, he apparently felt constrained from simply throwing out the sights and starting over again. To retain the same result (for latitude), he needed to add about 16 minutes to the upper reading and subtract about 16 minutes from the lower reading. However, this would have changed the degrees reported in some cases. Also, this would require a change to two digits in the minutes column on every reading. Apparently Cook thought it more credible to change only one digit on each reading and not change the degrees column. Thus, he added 10 minutes to each upper limb and subtracted 10 minutes from each lower limb. Of course, the "Cook apologists" will claim that there were four errors in reading, recording or transcribing Cook's observations, each of which just happened to involve a 10' shift in the direction indicated. The next edition of My Attainment of the Pole (1912) curiously showed both the originally reported data and the revised data for one sight, but revised data only for the other sight. Whether this was done out of candor (in one of two cases) or simply a printer's error is not clear. In any event, a 1913 edition of My Attainment of the Pole showed only the revised data, which is repeated below. |
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| ...people who support Cook's claim...have abandoned common sense ...or they are utterly ignorant of the facts... | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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1] The runner-up may be the suggestion by Thomas F. Hall, an early pro-Cook author, that a starving sled dog fed to his teammates could take the place of 60 lb. of pemmican. |
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| Cook (forgot) that the resulting difference between the altitudes of the respective limbs of the sun . . . would be one-half that amount. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Cook's published results brought forth howls of derision, since the only possible explanation for such an error in the sun's observed diameter was that the sights were fakes, worked backwards from the desired result for the sun's center. |
Compared to a solar diameter of 32', these show vertical compression of the sun's observed diameter of 6' 25" and 4' 52", respectively. The correct value for the vertical compression of the sun's diameter at the claimed altitudes, based on U.S. Navy refraction tables, discussed below, is about 15". In other words, the dubious hypothesis that Cook's revised "observations" might possibly be genuine is disproved by the fact that the observations show 20 to 25 times more compression than would be expected.
Davidson, apparently aware only of the revised "observations," defends them as "theoretically possible," noting that Cook "observed the sun with about 5 atmospheres [of pressure] and a surface temperature of -31 to -44 F, a great deal of cold air." Again, this may be a joke, but no "great deal of cold air" could ever result in a natural atmospheric pressure of 5 atmospheres (i.e., five times ordinary atmospheric pressure). Changes of even as little as a few tenths from ordinary atmospheric pressure (1 atmosphere) at sea level are unheard of. A pressure of 5 atmospheres could be found about 120 feet under water, but there Cook would have had other difficulties with his observations. Davidson's results at solar altitudes in the range of 10 to 11 degrees (discussed below) show vertical compression in the range of about 20" to 2', and thus do not come even close to the amount shown in Cook's revised "observations." And, of course, if one goes back to Cook's original "observations," which show three times as much vertical compression as the revised ones, the idea that the results are "theoretically possible" is even more ridiculous. In short, Davidson's explanation of Cook's results on the basis of "a great deal of cold air" is a great deal of hot air, and there is nothing about Cook's reported observations that is even remotely helpful to Cook's claim. |
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| Cook's revised "observations" are disproved by the fact that they show 20 to 25 times more compression than would be expected. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Cook proposed that his Eskimo sundial was "indisputable proof
of being on the North Pole . . . even if all other observations fail."
While this may have had appeal with the uninformed audiences to whom it was directed, shadow observations are, of course, even easier to fake than sextant observations and are far less accurate. |
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| Davidson's explanation of Cook's results on the basis of "a great deal of cold air" is a great deal of hot air... | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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DAVIDSON'S CLAIMS ABOUT PEARY Davidson argues that Peary should have observed a solar diameter considerably less than 32', and that the average of Peary's readings (about 31.8') shows so little vertical compression that the observations must be fakes. However, Davidson overstates the amount of vertical compression that one would reasonably predict for Peary's sights and understates the degree of uncertainty about any such prediction. As Davidson's own data show, the amount of vertical compression that can exist in genuine sights is sufficiently variable that the fact that Peary's sights had (on average) a bit less vertical compression than one would predict is not particularly significant. Davidson's Data Vertical compression of the sun is predicted to be relatively little at the true altitudes observed by Peary (about 6.6 to 6.8 degrees). Davidson's data for sun altitudes between 5 and 12 degrees shows considerably higher vertical compression than would be predicted by theory and shows a great deal of variability. This is illustrated by a tabulation of the data, arranged in order of increasing altitude (Table 1). The first column shows the true altitude of the sun (as reported by Davidson) and the second column shows the vertical diameter of the sun as he measured it, by assuming the horizontal diameter to be 32'. Although the horizontal diameter surely was not precisely 32' for theses photos, the assumption of 32' gives a sufficiently accurate measure of the observed vertical compression of the sun. The difference between 32' and the measured vertical diameter, multiplied by 60, gives the vertical compression in arc seconds, and is shown in the third column. The fourth column shows the predicted vertical compression, in arc seconds, based on refraction data published by the U.S. Naval Observatory in the Nautical Almanac. The data used include the correction for the coldest temperature for which data are provided, (about 0˚ to -20˚ Fahrenheit, depending on atmospheric pressure), further extrapolated for 10˚ colder temperatures. Changes in temperature of plus or minus 10 or 20 degrees would not materially alter these results. The predicted vertical compression is determined for any given solar altitude (center of sun) as the difference between the predicted refraction at a true altitude equal to the solar altitude plus 16' and the predicted refraction at a true altitude equal to the solar altitude minus 16'. (Note that the refraction data published by the U.S. Navy is tabulated by observed altitude, since it is used to correct observations, and needs to be correlated to true altitude -- observed altitude minus refraction -- for present purposes.) Column 5 in Table 1 shows the difference between Davidson's measured values and the theoretical prediction. Two things are striking about column 5. First, Davidson's data show considerably more vertical compression than would result from the predicted refraction, and second, his data show quite large variations. On average, Davidson shows about 47 seconds more vertical compression than the amount predicted in column 4, and this difference occurs pretty much at all solar altitudes, being somewhat greater at the low range of altitudes, and a bit less at the high range. The standard deviation of the difference between Davidson's data and the predicted vertical compression is about 45 seconds of arc. Thus variations of 45 seconds from Davidson's reported results would be expected to be very common, and even variations of as much as 90 seconds of arc would not be extraordinarily unlikely. |
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| Davidson's data show considerably more vertical compression than would result from the predicted refraction, and second, his data show quite large variations. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Of course, Davidson's data could be explained by poor measurement technique2, incorrectly reported solar altitudes, distortion in photo processing or (and I certainly don't believe this to be the case) outright falsification. But assuming Davidson's results to be reliable, one thing they tell us for sure is that the phenomenon of vertical compression is highly variable. The second thing that they tell us (if they are reliable) is that under the range of conditions observed by Davidson, the U.S. Navy tables are not a good predictor of vertical compression. The 1909 North Pole Expedition Sights Both Commander Robert E. Peary and his assistant Professor Ross Marvin took sights from which the measured vertical diameter of the sun can be determined. Marvin's sights were taken at latitudes of 85˚ 48' and 86˚ 38', and are generally accepted as genuine. Peary's sights were taken in the vicinity of his final camp and are disputed by some individuals. If genuine (and I certainly believe they are), they show that he was within about 5 miles of the North Pole. Marvin and Peary used slightly different techniques in taking their sights, and the analysis of the sights is slightly different. Marvin's Latitude Sights. Marvin's latitude sights consist in each case of eight observations of the sun: two upper limb observations followed by four lower limb observations an then two more upper limb observations. The roof glass over the mercury artificial horizon was reversed twice so that it was in one orientation for all upper limb sights and in the opposite orientation for all lower limb sights. The observations were taken in quick succession after Marvin had determined, by continuous observation, that the sun had stopped rising.3 The sun's measured vertical diameter (doubled) is determined by the difference between upper and lower limb observations. In the case of Marvin's sights, the first two upper limbs are averaged and compared to the average of the first two lower limbs, and the average of the third and fourth lower limbs is compared to the average of the third and fourth upper limbs. This results in comparisons based on sights taken as nearly as possible at the same time.
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| 2] Presumably Davidson is aware that even if the sun appeared perfectly round, its image on a photographic plate would be elliptical unless the film was perpendicular to the line from the nodal point of the lens to the center of the sun's image—a condition that generally applies only if the sun is at the center of the photo. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3] They were not "culmination sets" taken over a long period of time to demonstrate the rising and setting of the sun and the time of occurrence of the sun's highest altitude, as Dennis Rawlins has suggested. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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...Vertical compression about 30" average vs. predicted amount on the
order of 55". Davidson reports 96" average. |
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...Vertical compression about 61" average vs. predicted amount on the
order of 51". Davidson reports 111" average. |
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To summarize Marvin's sights, one set of sights shows about 25" less vertical compression of the sun than theory would predict, and the other shows about 10" more. They show from 50" to 66" less vertical compression than the average of Davidson's data for comparable solar altitudes. Marvin's Index Error Sights Prior to each sight, Marvin took a series of measurements to determine the index error of his instrument (which is slightly different with each use, due to the effects of cold, bumping, etc.). Marvin did this each time by taking four observations of the diameter of the sun (by placing the image from the index mirror just above the image from the horizon mirror), and four observations of minus the diameter of the sun (i.e., by reversing which image was on top). The average of the eight readings should be zero, and the difference between the average and zero is the index error. To simplify the computation, Marvin measured all readings in the positive direction on the vernier. That is, for the first set, the readings would be, for example, 0˚ 32'. For the second set of readings, instead of recording, e.g., -31', he would record 359˚ 29'. Then he added 1˚ to all readings, so that the first set was, for example, 1˚ 32' and the second set was, for example, 0˚ 29'. At the end of the computation, he subtracted 1˚ from the result. The vertical dimension of the sun can be determined by the difference between the average two sets of four readings for each observation, as follows:
These show a vertical diameter of the sun for the March 22 sights of 31' 15" (about 51" of vertical compression) and a vertical diameter of the sun for the March 25 sights of 30' 41" (about 84" of vertical compression). Without knowing how far in advance of the noon observation the index error measurements were taken, it is impossible to know the sun's altitude, except to note that it would have been somewhat lower than the noon altitude. However, they suggest that there was in fact more vertical compression for the second set of sights (as the noon observations indicated), indicating variability between March 22 and March 25 in the range of 33". |
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Commander Peary's Sights Peary used a slightly different method to take sights. For each of his complete observations— 1) He took a lower limb sight 2) He took an upper limb sight 3) He reversed the roof glass and took a second lower limb and a second upper limb. Thus each observation contains two measurements of double the sun's vertical diameter, determined by the differences between the first and second, and third and fourth observations, respectively. The data for Peary's three complete sights are tabulated below:
In an e-mail, Davidson states that Peary should have observed solar diameters of about 30'. This is clearly mistaken. Only 3 of Davidson's 12 observations for solar altitudes in the range 6.5 to 7.0 meet (or come close to) this standard. The other 9 show considerably larger solar diameters. Did Davidson fake them? In fact, theory would predict that Peary should have observed a solar diameter of about 31' 22" (35" of vertical compression). Davidson's data show vertical compression from 25" to 137" (average 69") for the 12 observations in the 6.5˚ to 7.0˚ range. Again, Davidson's data show much more vertical compression (in most cases) than would be predicted by theory. Until some reason to cast aside the U.S. Navy data for refraction can be given, I will concentrate on the disparity between Peary's data and the theoretical amount of vertical compression that would be predicted. The sun's actual diameter on April 6-7 was about 31'57", so that the actual amount of vertical compression reflected in each of Peary's measured diameters and the difference from the theoretical amount was:
These data show an average of 27" too little vertical compression, with a standard deviation of 12.4". How significant is this? Bear in mind, that one of Marvin's sets of observations, 8 sights versus Peary's total of 12, showed 25" too little vertical compression. Thus, unless one is ready to claim that Marvin's March 22 sight is a fake, it is hard to claim that this amount of disparity from theory shows that Peary's sights were fake. |
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What sources of error could explain Peary's results? Basically there are three: measurement error, which is random, personal error, which would be nonrandom, and variation in the physical phenomenon of vertical compression itself. Measurement error for a single sextant observation would be predicted to be about 20" (standard deviation). This is the result of the fact that the human eye is normally thought capable of resolving to about 1' of arc. Since Peary's sextant included a telescope of about 3 power, one would expect the resolving power to be improved by 1/3 to about 20".4 A study of Marvin's index error sights shows that groups of sights that should have been identical showed variability (standard deviation) in the range of 13" to 22" for small samples, indicating that 20" is at least a reasonable estimate.5 The sun's diameter determined by Peary in each case was one-half the difference between two double altitudes, so the formula from two sights would be: Diameter = 1/2 A - 1/2 B From well know principles of statistics, the standard deviation for the diameter would be given by the formula: Std. dev. diameter = Sqrt [(1/4 (std. dev. A)2 + 1/4 (std. dev. B)2] Assuming the standard deviation of each measurement (A and B) is 20", this would indicate a standard deviation in the measured diameter of about 14". Interestingly, this is very close to the standard deviation shown by Peary's six measurements of vertical diameter, so that, although Peary's average estimate is high (compared to theory), the amount of random error is about what would be expected. Other than random measurement error, there could also be a "personal error" reflected in Peary's sights. This is a constant error that reflects the observer's tendency to gauge incorrectly the position at which two bright images of the sun are just touching. The bright images tend to leave an apparent dark space, by contrast, between the two images of the sun, and there is a range from the position at which the two disks are clearly overlapping to the position at which they are clearly separated. How much of error could be introduced is hard to say. With the telescope sharply focused, and the right combination of index mirror and horizon mirror shades to darken the images of the sun (and preferably give them different colors), I would expect this error to be small. However, Peary may very well have used only a telescope eye-piece filter, rather than using index mirror and horizon mirror shades. This is generally thought to be more accurate, since the horizon mirror or index mirror shades can introduce errors in the measured angles if their faces are not perfectly parallel (not really a problem in a good instrument). Most sextants include only one eye-piece shade that is dark enough for use with the sun, and this could result in images that are brighter than would ideally be desired. Also, the two images of the sun would be the same color, which I have found to be a disadvantage. |
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4] Of course, Peary could have used the sighting tube instead of a
telescope, in which case the likely error for each sight would be about 1'
of arc. |
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| 5] This should not be confused with the precision of Peary's sextant, which allowed readings to the nearest 10". Precision does not necessarily equate to accuracy. Also, it should be noted that Peary's accuracy might be expected to be somewhat less than Marvin's, since Peary apparently did not attempt to read the vernier to its maximum precision, but instead tended to look to the nearest whole minute. This is illustrated by the disproportionately large number of zero second readings in Peary's sights, compared to the fairly uniform distribution in Marvin's sights. Anyone who has tried to read a 10" vernier, on which the various pairs of lines over a range of possible readings all seem to line up almost equally well, would understand the temptation to simply pick the set of lines corresponding to an even number of minutes. It is worth noting that 10" on the limb of the sextant corresponds to about 2 ten thousandths of an inch. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Although not technically part of "personal error," if the telescope on
Peary's sextant were not perfectly focused, it is possible that slightly blurred edges of
the sun could tend to give it a larger diameter. The telescopes are
generally focused by sliding the eye-piece tube, held by friction, to the
proper distance. It is not hard to believe that getting a perfect focus with
a cold instrument and hands in mittens would be tedious, and Peary might
have settled for less than perfection here. Any error of this nature would
be likely to be constant for the two measurements in any one set of sights
(since it would be unusual to refocus the telescope during a series of
sights). Whether the error would be random between series of sights depends
on whether the telescope had to be collapsed for storage. This may or may
not have been required depending on which of the telescopes Peary used. If
not, Peary might not have refocused the telescope for each set of sights.6
More investigation is needed to quantify the possible effect of poor focus. In addition to measurement errors, random or otherwise, the deviations from theory that appear in Peary's sights could be the result of variations in the physical environment. The theoretical amount of refraction is based on changes in the temperature and density of the earth's atmosphere as a function of altitude, based on typical conditions. Even when corrected for temperature and pressure at sea level, there are variations in the profile of temperature and density versus altitude, particularly in lower altitudes. Davidson's data indicate a great deal of variability. This could be measurement error, but one would expect that with careful procedure measurement errors would produce much less than the variability shown. More likely, these data show that the physical phenomenon itself is highly variable. Marvin's data also seem to show variability in the physical phenomenon of vertical compression between March 22 and March 25. As noted above, he showed from 25" less to 10" more compression than theory would predict. Variations in the physical phenomenon of refraction would be caused by variations in the profile of refraction versus solar altitude. It would seem that near-surface atmospheric conditions (temperature, humidity and pressure) could have cause such variations, but considerably more study is required before any conclusions can be drawn. CONCLUSION The fact that Peary's sights showed some vertical compression, on average, but about 25" less than would be predicted by theory is not particularly significant, given the uncertainty of the physical phenomenon and the difficulties in measuring such small angles with absolute accuracy. Certainly more study of the phenomenon, using the instruments used by Peary, is needed before one can draw any firm conclusions about how likely or unlikely it is that Peary's data were "observable." Certainly at this point, Davidson's conclusion that the data were "not observable" rests on no better foundation than his claim that Cook's data were "theoretically possible." |
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| 6] It is possible that Peary used the sighting tube, rather than a telescope. If so, the sharpness of focus would depend on Peary's eyesight. Also, the random measurement error, discussed previously, would have been much greater due to the lack of magnification. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| "...Peary's sights showed some vertical compression...but ...less than would be predicted by theory... not particularly significant, given the uncertainty of the physical phenomenon and the difficulties in measuring such small angles with absolute accuracy." | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| © 2003 by Douglas R. Davies. All rights reserved. No part of this text may be used without written permission from Douglas R. Davies. Email request | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
