Part 2: The Bradley Type Experiments
James Bradley was born in 1693, and educated at Balliol College, Oxford.1 His astronomical instructor was one of the finest of that period in England, his uncle, the Rev. James Pound. In 1717, Edmond Halley ushered him into the scientific world and by 1721 Bradley had been appointed to the Savilian chair of astronomy at Oxford. He also lectured in experimental philosophy from 1729 until 1760. Upon the death of Halley in 1742, Bradley succeeded to his position as Astronomer Royal. At that time, the Copernican theory of the earth and planets orbiting the sun was still the subject of some debate. If the idea was correct, the new telescopes held the promise of being able to pick up the apparent displacement of the nearer stars relative to the more distant stars of the stellar background as the earth moved in its orbit. The same effect is seen when a nearby tree changes its position against the background of buildings as you walk near it in the park. The effect is called parallax. Picard, about 1671, had noticed annual variations in the position of the pole star but hesitated to attribute them to parallax or refraction.2 Hooke in 1674 and Flamsteed from 1689 to 1697 decided that parallax was the cause. However, Cassini the Younger and Manfredi, around 1699, carefully noted that the variation was opposite that required for parallax. In December of 1725, Samuel Molyneux decided to settle the question of parallax and started observing at Kew with an excellent instrument built by the famous mechanic George Graham. He chose the star Gamma [γ] Draconis since it passed overhead and no correction was needed for refraction. He looked at the star on December 3, 5, 11, and 12 and was joined by Bradley on the 17th. Bradley noted the star was further south on this occasion than Mosyneux observed it. Suspecting an instrumental error, he observed again on the 20th. It was even further south than on the 17th. They noted that it was in the opposite sense to what parallax predicted, as well as being at the time of the year when parallax changes would be expected to be minimal.3 After continued observation and equipment testing, it was concluded that the effect was not instrumental. The star stopped its southerly movement about the beginning of March 1726 when it was found to be about twenty seconds of arc (written 20”) further south than in December. By definition, 60 arc seconds total 1 arc minute (written 1’), while 60 arc minutes equal 1 degree (1°). A full circle comprises 360 degrees. The star’s declination then started to increase, moving northward, and in June returned to the December value. This motion continued until it again became stationary in September, differing by 39: from its March value. By December 1726, it had returned to its original position. An annual motion of γ Draconis had been established. Its cause was not parallax. (Indeed, because of the immense distances involved, the annual parallax effect is about 100 times smaller for even the nearer stars.) Bradley wondered if nutation might be responsible, that is the oscillation of the axis of the earth under the influence of the Moon. Today we know this to have an 18 year period. Such an observation would have been of importance in establishing Newton’s Law of Gravitation which had been propounded in 1687. But if what Bradley and Molyneux had observed was nutation, all stars would be affected almost equally. Bradley then ordered another instrument from Graham which could examine stars other than on the zenith. In August 1727, he started observing eight stars and found that they described ellipses on the celestial sphere whose major axes were about 40” long and lay parallel to the plane of the earth’s orbit (the ecliptic). However, geometrically speaking, the minor axes of these ellipses were proportional to the sine of the angle between the star’s direction and the ecliptic. Therefore, nutation (or wobble of the earth on its axis) had to be ruled out as the cause of what was being seen as the apparent ellipses of the stars. As it eventuated, about 20 years later, in 1748, Bradley confirmed nutation by observation, and hence, Newton’s Laws.4 It was against this background, and with some perplexity of mind as to the cause of the observed effect, that Bradley was invited to go sailing on the Thames, so the apocryphal story goes. He became intrigued with the behavior of the flags on the sail boats. Why, when the wind was blowing in a constant direction, did the flags swing to a different one when the yachts changed course? The answer came that the cause was the net result of the yacht’s motion plus the wind. The earth also had an annual motion, Bradley reasoned, and the only other factor, equivalent to the wind, was the light from the stars coming in a constant direction towards the earth. As the earth swung around in its orbit, so did the apparent position of the stars, just like the flags on the yachts. It is called aberration, and, in this case, the aberration of light. A more everyday example is that when we walk through vertically falling rain with an umbrella, the faster we walk, the more we have to tilt the umbrella as the rain appears to come at us from an increasing angle. Knowing how fast we are walking and measuring the rain’s angle then tells us how fast the rain is falling. If we walk at a constant rate, then this simple trigonometric relationship tells us that the rain’s speed multiplied buy the angle is always a constant value, no matter what the rain does. If the rain falls faster, the angle gets less. If the rain falls more slowly, the angle gets greater. We may write it this way: (angle) x (rain speed) = constant µ ( walking speed) where the symbol µ means ‘proportional to’. Since the motion of the earth in its orbit is essentially constant, Bradley knew that this same relationship could be applied to the figures tumbling through his excited mind, provided he used the appropriate units. The aberration angle, K,he had measure from the stars. If the speed of light was c, equivalent to the rain speed, then as written above Kc = constant. On the accepted figures today the result is Kc = 6,144,402 where K is in arc-seconds and c in kilometers per second.5 Bradley concluded “that Light moves, or is propagated as far as from the Sun to the Earth in 8 minutes, 12 seconds.”6 Bradley had confirmed not only the Copernican model for the solar system, but also the hotly debated idea of a finite value for c. His discovery was announced on the first of January, 1729. Bradley’s observations of aberration go in three periods. First, from 1726 to 1727 there was his study of γ Draconis, then seven other stars at Kew. Second, from 1727 to 1747, he studied 23 stars at Wanstead in Essex where his uncle had been Rector. (The Rev. James Pound had died during 1724, before his nephew made the all-important discovery for which he had been training him.) Finally, as Astronomer Royal, Bradley made another series of observations on γ Draconis from 1750 to 1754 from Greenwich. Bradley quotes his result for γ Draconis in period one as 20”.2 for the major axis when the 39” was transferred to the ecliptic pole. When measured in radians, the pure mathematical unit used in some calculations, 20.2 arc seconds becomes 1/10210 radians. From the simple trigonometric relationship used in the example with the flags on the boats, 10210 was also equal to the speed of light divided by the earth’s orbit speed. Thus, if V is the orbit speed, Bradley could then write c/V = 10210 But orbit speed is equal to distance traveled divided by time taken. The time taken is one year in seconds, or 365.25 x 24 x 60 x 60. The distance traveled is the circumference of a circle, or 2πr, where r is the radius, or the distance of the earth from the sun. Therefore, he could substitute for V in the first equation and come up with Tc/(2πr) = 10210 However, he did not know the distance from the earth to the sun with any degree of accuracy, so r was in question. Therefore, the simple way out of the problem was to recognize that if the time taken for light to travel across the orbit radius r – or that distance – was t, then this equaled distance r divided by the speed of light, c, or r/c (the inverse of what he already had in the second equation). Thus, he could announce that light took t = r/c = T/(10210 x 2π) = 8 minutes 12 seconds to get from sun to earth. Bradley, comparing the two extreme values of Roemer and Cassini for this ‘light equation,’ commented, “The Velocity of Light therefore deduced from the foregoing Hypothesis, is as it were a Mean betwixt what had at different times been determined from the Eclipses of Jupiter’s Satellites.”7 Bradley noted that the declination of the star was increasing by an arc second in three days, which indicates he could easily measure to 0”.3 as he noted its daily motion. On p. 655, he indicated that the above calculation for t could give a result to about 5 seconds, indicating an error of perhaps 0”.2. Other re-workings in Table 3 below reduce the error to below ± 0”.1. If we take the probable error as being one-tenth of an arc second in 20”.2, then from Kc = 6144402, a value for c of 304,000 ± 1500 Km/s results from Bradley’s initial work. Though the work on γ Draconis in this first period resulted in K = 20”.2, Bradley’s treatment of the other seven stars also examined, then revised, his mean aberration value to 20.25 arc seconds. His results for all eight stars are examined in Table 2.
TABLE 2
MEAN VALUE: 20”.2 LIMIT MEAN: 20”.25 * in the Flamsteed Catalogue. Now in Auriga The mean of these results is again 20”.2, just as for γ Draconis alone. However, Bradley took the extreme limits given by a τ Persei estimate and the α Aurigae (Capella) value that contained other errors, and taking the mean of these gave the final value as 20”.25 for the first period at Kew. Re-processing Bradley’s Results Busch, Auwers and Newcomb re-processed the first γ Draconis observations. Busch finally obtained, after some significant analysis, determination of mean errors and weighting procedures, a value of 20”.2495. Auwers criticized Busch’s treatment, made corrections, took into account collimation and screw errors and gave his final result at 20”.3851. Newcomb applied a further correction and obtained 20”.53 for the aberration constant. A similar procedure gave Busch a value of 20.2050 arc seconds for the second period results, while Auwers’ criticism and re-processing gave a result of 20”.460. The observations of the third period were treated by Bessel and Peters. Both rejected the observations of 20, 21, and 23 February, 1754, as they “disagree with the rest and give large remainders.”9 They respectively obtained K = 20”.475 and 20.522 arc seconds. The final average value, omitting both of Busch’s disputed re-workings, was 20”.437 for a mean date of 1740. A c value of 300,650 Km/s results. Table 3 lists the values.
Table 3
MEAN VALUE: 20”.437 FOR 1740 From Bradley’s final observations in 1754 until the work of Struve at the Pulkova Observatory in 1840, there appears to be only one extant set of observations, namely that of Lindenau. He processed observations of Polaris between 1750 and 1816 to obtain a value for the aberration constant of 20”.45 ± 0.011 for a mean date of 1783.11 This gives a value for c of 300,480 Km/s in 1783. Between 1840 and 1842, Struve studied seven stars from Pulkova with Repsold’s transit instrument in the prime vertical position.12 His value for K, issued in 1845, though with a mean date of 1841, was 20”.445 ± 0.011. Folke re-processed Struve’s observations with five stars as Struve had remarked that “the observations of the two stars in Cassiopeia are less exact due to their great brilliancy which precluded accurate setting on them.”13 This gave a value of K = 20”.458 ± 0.008. In 1853, Struve re-processed his observations allowing for temperature and vibration effects and a zenith correction, issuing a final value of K = 20”.463 ± 0.017.14 This results in a c value of 300,270 ±250 Km/s against the mean date of 1841. Though the value Struve issued in 1845 was the generally quoted definitive value,15 this 1853 correction is to be preferred and will be taken as the best result against the 1841 data. Further work by Struve between 1842 and 1844, again using Repsold’s transit instrument in the prime vertical, gave a further determination of the aberration constant. From this set of observations, Struve set K = 20”.480.16 This results in a value for c of 300, 020 in 1843. After a series of observations using the Pulkova vertical circle, a meridian circle and the transit instrument in prime vertical, and comparing them with those of Struve, Nyren, in 1884 announced that the value of K must be increased to 20”.492 ± 0.006.17 This was set against the date of 1883. Newcomb announced this as the definitive value in 1886, superceding Struve’s value.18 The actual value from the weights applied to his observations was 20.4915 arc seconds. This results in a value for the speed of light in 1883 of 299,850 ± 90 Km/s. In addition to the results outlined above, there are some 58 further values obtained by this method. Table 4 supplies the complete listing.19 The constant value Kc = 6144402 has been adopted. However, many observations, and particularly those from Pulkova and Kazan give high values for K and suggest that their best value is K = 20”.511.20 If this result, also quoted by Whittaker, were used, then c would be above today’s value in nearly all cases in Table 4.14 Newcomb suggested the probably cause of the high value for K from Pulkova even in the days of Nyren.21 As many of the Pulkova observations need to be made in twilight, any star will appear fainter in transit across the eat vertical than when crossing the west vertical an hour or so later if they are evening observations. For morning observations the reverse situation holds. The observer would tend to note the passage of the fainter image systematically too late. Speaking of the effect this has on K, Newcomb commented, “we can not but have at least a suspicion that…values may be slightly too large from this cause.” It appears that all Pulkova observations were affected by this problem, and so as a class should illustrate any general trend shifted into a lower range of c values. The 63 aberration determinations from 1740 – 1930 listed in Table 4 were made with basically the same type of equipment, with essentially the same error margins and substantially the same observational methods. The results from Pulkova Observatory are illustrated in Figure I. A least squares linear fit to all data gives a decay of 5.04 Km/s per year with a confidence of 96.1% that c has not been constant at 299,792.458 Km/s for the period covered by these Bradley-type determinations. These results suggest that the possibility of a decay in c should be examined further.
Table 4 – Bradley Method Values (Aberration)
References1 G. Sarton, “Discovery of the Aberration of Light”, Isis, Vol, 1931, 16, p.233 2 J.B.J. DeLambre, Histoire de L’Astronmie Modern, Vol. II, Paris, 1821, p. 616 3 Bradley’s letter to Halley in Philosophical Transactions, No. 408, vol 35, pp 639-40. One mss copy is dated Jan. 1, 1729, though it was in the Phil Trans. For December 1728. 4 J. Audouze and G. Israel, editors, Cambridge Atlas of Astronomy, Cambridge University Press, Dec. 1985, p. 413 5 The I.A.U. value in 1984 of K = 20”.496 ± 0.001. Now redefined as 20”.49552. With c defined as 299792.458 Km/s, this gives Kc = 6144402. This value is adopted here to be conservative. 6 Bradley’s letter to Halley in Philosophical Transactions, No. 408, vol 35, p.653 7 ibid 9 K.A. Kulikov, Fundamental Constants of Astronomy, (translated from the Russian for NASA by the Israel Program for Scientific Translations), original version Moscow, 1955, pp 81-83 10 ibid 11 Kulikov, op. cit., Table III, No. 15. Errors obtained from a comparison of weights 12 S. Newcomb, Nature, May 13, 1886, p. 30. Also Kulikov, op.cit., p. 83, and Table III, No. 35 13 Kulikov, op.cit., p. 84 and Table III, no. 36 14 E.T. Whittaker, History of Theories of Aether and Electricity, Vo. 1, Dublin 1910, p. 95. also Kulikov, op.cit., p. 85 15 S. Newcomb, op.cit 16 Kulikov, op.cit., Table III, No. 5 17 Kulikov, op.cit., p. 85, and Table III, no. 41 18 S. Newcomb, op.cit. 19 Taken from Kulikov, op.cit., Table III, p. 191 – 195. Dated values only taken from Table and repeats from Table III Parts 1 and 2 that occur in Part 3 are omitted. Also four values from Table 10 not listed in Table III have been included. 20 See Kulikov, op.cit., pp 88, 92, 93. These values of K are systematically high when compared with the USA results. Nevertheless, as a class, they still exhibit the feature of c decay, though shifting the c values into a lower range. 21 S. Newcomb, “The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy”, Supplement to the American Ephemeris and NauticalAlmanac for 1897, Washington, 1895, p. 136. Aberration values quoted here appear in Kulikov’s more comprehensive Table.
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