Relatório Especial 14: a posição do Sol e o teste do qui-quadrado mostram que os "desconhecidos" não se comportavam como os casos explicados

Position of the Sun Relative to the Observer The first thing that was done was to calculate the angle of elevation of the sun above the horizon and its bearing from true north as seen by the observer at the time of each sighting. With this information, it could then be determined whether there was a possibility that the reported object could have been illuminated by light from the sun. In addition, it could be determined whether an object could be a mock sun (sun dog) or whether there was a possibility of specular reflection from an aircraft at the position of the object, which would give the appearance of a "flying disc". A program of computation was set up and carried out to obtain the angle of elevation and the bearing of the sun for each sighting. All information needed for this calculation was available on the deck of IBM cards.

This information consisted of: (1) Time and date of observation in Greenwich Civil Time (2) Latitude and longitude of the observer at the time of observation. Figure 39 shows a celestial sphere on which Z represents the observer's zenith, S represents the sun, and N represents the north celestial pole. Using the date and time of the observation, the longitude and declination (S) of the sun were obtained from an ephemeris of the sun and corrected for the equation of time. The difference between the longitudes of the sun and the observer was taken, and called the hour angle (HA on Figure 39). Then, using the declination of the sun (S), the latitude of the observer (lat), and the hour angle (HA), the angle (ZS) between the observer's zenith and the sun can be calculated from the law of cosines of spherical trigonometry. Thus, cos ZS = cos (90 - lat) cos (90 - S) + sin (90 - lat) sin (90 - S) cos (HA). Since the angle ZS is measured from the observer's zenith, the angle of elevation of the sun above the horizon for daytime sightings was found by taking 90 - ZS. When the sun was below the horizon, the angle of depression of the sun below the horizon was found by taking ZS - 90. Having found the angle ZS, the bearing of the sun (angle B) was obtained from the formula: sin (B) / sin (90 - S) = sin (HA) / sin (ZS) All of the above calculations were made with IBM equipment. Sines, cosines, and their inverses were obtained from a deck of 9,000 IBM cards on which seven-place Peter's tables of the sines, cosines, and tangents of angles had been punched for each 0.01 of a degree from 0 to 90 degrees. Upon completion of these calculations, the cards representing OBJECT SIGHTINGS were sorted on the sign of the sine of the bearing angle. This separated the cards into two groups: (1) sightings which occurred between noon and midnight, for which the sine of the bearing angle was positive; and (2) sightings between midnight and noon, for which the sine of the bearing angle was negative. Then each of these groups was sorted into groups for intervals of 10° in angle of elevation of the sun from -90° to +90°. A count was made of the number of cards in each group and from this a histogram was constructed (Figure 40). The UNKNOWN OBJECT SIGHTINGS were then sorted out, counted in the same manner, and a histogram was made (again see Figure 40).

FIGURE 39 DIAGRAM OF A CELESTIAL SPHERE. Labels: Z; N; S; 90-lat; Ha; 90-δ; B; Equator; Earth; Horizon. A-7535

FIGURE 40 FREQUENCY OF OBJECT SIGHTINGS BY ANGLE OF ELEVATION OF THE SUN, INTERVALS OF 10 DEGREES OF ANGLE. Vertical axis: Number of Object Sightings. Horizontal axis: Angle of Elevation of The Sun, -90 to +90 and back to -90. Labels: Sun rise; Sun set; All object sightings; Unknown object sightings. A-7536

The following points should be carefully noted about these histograms: (1) The negligible number of sightings when the sun is within 10° of the zenith and nadir (angle of elevation of the sun = ±90°) of the observer is due to the fact that the southernmost latitude of the U. S. is greater than the declination of the sun at the summer solstice, so that it would be impossible for the sun to reach the zenith or nadir of any observer in the U. S. (where most of the sightings were made). (2) The time of day at which a particular angle of elevation of the sun occurs does not remain fixed but varies from day to day. Consider, for example, the variation in sunrise and sunset times over the course of a year. Thus, there are only two inferences to be made from this histogram: (1) the high peak of sightings soon after sunset, and (2) the lack of increase in the UNKNOWNS relative to the KNOWNS near either sunset or sunrise. This would seem to discount the possibility that atmospheric phenomena such as mock suns were the primary cause of the unknown reports, since such phenomena usually occur when the sun is near the horizon. The Local Sun Time was computed as a step in the calculation of the angle of elevation of the sun. It is related to the hour angle by the equation: Local Sun Time (L. S. T.) = HA/15 + 12.00, where L. S. T. is in hours and HA in degrees. The cards were grouped on the basis of L. S. T. in intervals of one hour, and the number of cards in each interval was counted. Again the UNKNOWNS were sorted out and similarly treated. Histograms were constructed with the results of these tabulations of OBJECT SIGHTINGS (Figure 41). Here, again, there is a peak in the early evening hours. The cards were then broken up into seven groups on the basis of the angle of elevation of the sun, as follows: Group 1 - Daylight sightings for which the sun was more than 10° above the horizon. Group 2 - Sunset sightings for which the sun was between 0° and 10° above the horizon. Group 3 - Sunset sightings for which the sun was between 0° and 10° below the horizon. Group 4 - Evening sightings for which the sun was between 10° and 40° below the horizon.

FIGURE 41 FREQUENCY OF OBJECT SIGHTINGS BY LOCAL SUN TIME, INTERVALS OF ONE HOUR. Vertical axis: Number of Object Sightings. Horizontal axis: Local Sun Time, 0 to 2400. Labels: Midnight; Noon; Midnight; All object sightings; Unknown object sightings. A-7537

Group 5 - Night sightings for which the sun was more than 10° below the horizon and which were not included in Group 4. Group 6 - Sunrise sightings for which the sun was between 0° and 10° below the horizon. Group 7 - Sunrise sightings for which the sun was between 0° and 10° above the horizon. These group numbers were punched on the cards and incorporated into the coding system. The number of OBJECT SIGHTINGS in each group for each identification was then tabulated and is given in Table I. TABLE I OBJECT SIGHTINGS Angle of Elevation Group Identification 1 2 3 4 5 6 7 Balloon 156 17 28 83 40 0 2 Astronomical 52 6 43 236 118 9 6 Aircraft 187 23 49 144 60 5 2 Light phenomena 8 2 4 25 7 0 0 Insufficient information 72 12 26 76 28 2 0 UNKNOWN 134 14 25 150 86 6 7 Other 64 8 12 50 36 3 7 Total 673 82 187 764 375 25 24 According to this table, a large majority of the KNOWN OBJECT SIGHTINGS in Group 1 (343 out of 467) were either aircraft or balloons. In Groups 4 and 5 combined, a large majority (681 out of 899) were either balloons, aircraft, or astronomical. Accordingly, a re-evaluation of the UNKNOWNS in these three groups was planned with the objective of determining which of the UNKNOWNS in Group 1 might possibly be aircraft or balloons and which of the UNKNOWNS in Groups 4 and 5 might possibly be balloons, aircraft, or astronomical objects. More will be said of this project later. Statistical Chi Square Test In the meantime, mirror graphs had been constructed from the frequency tabulations which seemed to show that, when the KNOWNS (total less UNKNOWNS) and the UNKNOWNS were grouped according to one of six characteristics, the percentage of KNOWNS and the percentage of

UNKNOWNS in each characteristic group showed the same general trend. In other words, on the basis of these graphs, it looked as though there was a good possibility that the UNKNOWNS were no different from the KNOWNS, at least in the aggregate. It was decided to investigate this by the use of a statistical procedure called the "Chi Square Test". The Chi Square Test is a statistical test of the likelihood that two distributions come from the same population, that is, it gives the probability that there is no difference in the make-up of the two distributions being measured. The method is outlined as follows: (1) Adjust the distributions by multiplying the KNOWNS in each characteristic group by the ratio of the total number of UNKNOWNS to the total number of KNOWNS. (The Chi Square Test is applicable only to distributions which have the same total number of elements.) (2) Take the difference between the number of UNKNOWNS and the adjusted number of KNOWNS in each characteristic group. (3) Square the remainder from Step 2. (4) Divide the result of Step 3 by the corresponding number of adjusted KNOWNS. This is the chi square for the particular group. Summing the individual chi squares over the groups of a characteristic gives the chi square for that characteristic. This number is then compared with a table of the distribution of chi square which can be found in many texts on elementary statistics. It will be noted that chi square is tabulated in terms of degrees of freedom which in this case is one less than the number of groups of sightings for each characteristic. The tabulations of KNOWNS and UNKNOWNS against the six characteristics and the Chi Square Test as it was applied are shown in Tables II through VII. In each case, the number of degrees of freedom is given, as is the value of chi squares corresponding to probabilities of 5 per cent and 1 per cent that two distributions with this number of degrees of freedom come from the same population. Since the greater the value of chi square the smaller the probability of homogeneity of two distributions, a calculated value of chi square greater than either the 5 per cent or 1 per cent values will indicate a probability less than 5 per cent or 1 per cent, respectively, that the two distributions are homogeneous. The term homogeneity is used here to indicate that two distributions could have come from the same population.

TABLE VI CHI SQUARE TEST OF KNOWNS VERSUS UNKNOWNS ON THE BASIS OF SPEED. Columns: Speed; Number of KNOWNS; Adjusted Number of KNOWNS (K); Number of UNKNOWNS (n); x2, (K-n)2/K. Stationary 249, 61, 53, 1.05. Less than 100 mph 154, 38, 26, 3.79. 100 to 400 mph 181, 45, 58, 3.76. Over 400 mph 403, 99, 145, 21.37. Meteor-like 83, 20, 16, 0.80. Not stated 695, 171, 136, 7.16. Total 1765, 434, 434, 37.93. Degrees of freedom 5. 5%: 11.1. 1%: 15.1.

Totals of Tables II to XIII (chi square; degrees of freedom; 5% and 1% limits). KNOWNS versus UNKNOWNS (1765 KNOWNS adjusted to 434): II Color 26.15 (8; 15.5; 20.1). III Number 40.73 (4; 9.5; 13.3). IV Shape 29.05 (6; 12.6; 16.8). V Duration of observation 49.49 (7; 14.1; 18.5). VI Speed 37.93 (5; 11.1; 15.1). VII Light brightness 5.85 (5; 11.1; 15.1). REVISED KNOWNS (astronomical removed, 1286 adjusted to 434) versus UNKNOWNS: VIII Color 13.40 (7; 14.1; 18.5). IX Number 23.73 (4; 9.5; 13.3). X Shape 11.46 (5; 11.1; 15.1). XI Duration 14.34 (7; 14.1; 18.5). XII Speed 43.71 (4; 9.5; 13.3). XIII Light brightness 10.68 (4; 9.5; 13.3).

In five of the six cases, the probability is less than 1 per cent that the distributions are the same. In the sixth case, Light Brightness, the classifications are too nebulous to be of real value. However, these tests do not necessarily mean that the UNKNOWNS are primarily "flying saucers" and not aircraft, balloons, or other known objects or natural phenomena. The UNKNOWNS might still be unidentified KNOWNS if either of the following cases occurred: (1) The characteristics which were observed for the UNKNOWNS were different from those observed for the KNOWNS because of the psychological make-up of the observer or because of atmospheric distortion. This assumes the distribution of objects in KNOWNS and UNKNOWNS is the same. (2) The UNKNOWNS may be known objects in different proportions than the group identified as KNOWNS. (That is, a greater percentage of the UNKNOWNS could be aircraft than the percentage of aircraft in the identified KNOWNS.) The second case is the more probable one. In this connection, it is interesting to note the factors which contributed to a large chi square result in the tests made above: (1) Color The major contribution to chi square in color is from the color green. There is a large excess of green sightings among the KNOWNS over the UNKNOWNS. Of the 130 known objects in this classification, 98 are astronomical, and are due mostly to the green fireballs reported from the Southwest U. S. (2) Number The large chi square is due to a greater proportion of UNKNOWNS in the multiple object classification. Apparently these are harder to identify. (3) Shape In this case, there is a higher percentage of UNKNOWNS in the rocket-aircraft-shape classification. These might be familiar objects for which unusual maneuvers were reported. There is a higher percentage of KNOWNS in the flame and in the meteor- or comet-shape category, which in both cases appears to result mainly from excesses of astronomical sightings.

(4) Duration of observation Here there is an excess of KNOWNS in the less-than-5-second group. Again, the majority of KNOWNS in this group are astronomical. The greater proportion of UNKNOWNS in the 31- to 60-second and 61-second to 5-minute groups cannot be explained. (5) Speed The major contribution to chi square for this characteristic is due to a large excess of UNKNOWNS in the over 400-mph class. It can be assumed that some of the excessive speeds are inaccuracies in estimates by observers. However, some radar sightings, which are practically impossible to identify, show objects with speeds of 1,000 to 2,000 mph and over, and these reports account for a number of these UNKNOWNS. (6) Light brightness Since this chi square was not significant, it is not necessary to discuss it here. An examination of these discrepancies thus brings up a very interesting point. In every case for which there is a significant excess of KNOWNS over UNKNOWNS, the excess can be attributed to an excess of identifiable astronomical phenomena. This would seem to lead to the conclusion that astronomical phenomena are easy to identify and there are very few left in the UNKNOWNS. Accordingly, the astronomical object sightings were deleted from the KNOWN object sightings and the Chi Square Test was again applied. The results are shown in Tables VIII through XIII, where in this case the KNOWNS do not contain astronomical sightings. It will be noted that some groups were combined when the adjusted number of KNOWNS was ten or less, except for the case for which the number of objects per sighting was the characteristic studied. These were borderline cases, and no good combination of groups existed. It is apparent that the deletion of astronomical sightings gives a better fit, although the decision is not clear cut, since for two cases (light brightness and speed), the chi square increased. However, it can again be pointed out that the reporting of these two characteristics is highly subjective and is open to question. The estimation of speed is especially open to question because of the impossibility of accurately determining it visually.

Another interesting aspect of these new tests is that there are only two large discrepancies in all of the groups. These are for the 11 or more groups in the classification by number of objects per sighting and for the over-400-mph and meteor-like group for the classification by speed. The first was relatively unchanged by deletion of the astronomical sightings principally because of the concentration of sightings in the single-object category. The second was slightly increased by the removal of the astronomical sightings from the meteor-like classification. However, the main discrepancy, that of the excess of UNKNOWNS in the over-400-mph class, was little changed. The results of these tests are inconclusive since they neither confirm nor deny that the UNKNOWNS are primarily unidentified KNOWNS, although they do indicate that relatively few of the UNKNOWNS are actually astronomical phenomena. It was decided that this process would not be carried to its logical conclusion (that is, the determination of a linear combination of KNOWNS that would give a negligible chi square when compared with the UNKNOWNS), since it was felt that the inaccuracies in the reports would give a distorted and meaningless result.