The stars are then arranged in columns of fours; the images of the planet, owing to its rapid motion, are in echelon. Now each exposure gives very accurately the place of the planet with reference to the group of stars around it, and for merely parallax purposes the ideal would be to have pairs of such photographs taken at the same instant at stations widely separated. By a very simple use of the measures of some ten or twelve stars suitably disposed about the planet, it would be possible to allow almost automatically for the differences of refraction, orientation, scale value, etc., which make the plates not immediately comparable, and to find at once the parallactic displacement. We have been speaking of the displacement as very large, and so it is when compared with the displacements dealt with in previous determinations. But look at it this way: We saw that the earth as seen from Eros subtended an angle of 53 seconds. The scale of the Cambridge plates is such that if we draw a circle having a diameter of a little over 14 mm. we represent the apparent diameter on our plate of the earth as seen from Eros; and within that small circle the whole of the parallactic displacement must necessarily lie. About a millimeter was the average displacement obtained in a favorable combination of observations, and when we consider that we are trying to measure that with a resultant accuracy of 1 in 1,000 it does not seem so very great after all. We have put the problem heretofore in its very simplest form. In actual fact the exposures at different stations were not simultaneous. Early morning observations made at Cambridge might be combinable with evening observations at Lick more or less simultaneous, or with evening observations ten or twelve hours before at (say) Oxford; or they might have to stand alone. Any general method of utilizing all the results must secure the possibility of reducing each plate, so to speak, on its own merits, to allow it to contribute its quota, be it large parallactic displacement or none at all, to the general collection of equations of condition. This requires that we shall know the relative places of all those stars which are to be used as comparison stars for the planet, right along the whole track of the planet. And this derivation of a standard star system is by far the most delicate and difficult part of the whole work. One must start with a foundation of stars observed with the meridian circle, and fill in the fainter stars from the photographs themselves, taking care to provide at the same time the places of all those faint comparison stars which have been used by the visual observers. And all these places of stars must be tied together, so to speak, by the overlapping of the photographs, so that the system may run smoothly throughout its length. Absolute errors of zero there no doubt will be, and must be, but it is required that there shall be no sudden jumps in the errors exceeding one or two hundredths of a second of arc. Now, when one comes to face a problem like this, one must inquire very carefully what is the real accuracy of the photograph. There is no doubt that the ordinary photographic telescope properly worked will repeat itself very well; it will take two plates of the same region which agree with one another excellently. But the question is, How would two plates of the same region taken with different telescopes agree? We know that individual observers have peculiarities of their own which they can repeat almost ad infinitum. Does a photographic telescope do the same, or has it at last conquered that bad habit of idiosyncrasy which has made so much trouble in all refined astronomical work of the older kinds? When we started on the Eros campaign there was practically no information to be obtained upon this point. Almost all the photographic telescopes at work had been engaged upon their own zone of the chart, and almost nothing was known of how the results from different instruments would combine. But in our parallax problem this question is fundamental, and must be answered as soon as possible. I therefore ventured to propose to myself to undertake the reduction of a small section of the photographic results, for a period of nine days in November, 1900, having before me two objects: Firstly, to discover how far it is possible to combine photographs taken at different observatories, how far, in fact, photographic telescopes are giving really accurate results or merely reproducing their own errors; secondly, to obtain a provisional value of the solar parallax, with a probable error if possible as small as that found by Sir David Gill with the heliometer, and to find out provisionally whether Eros was going to confirm that result or to join in the secession from the adopted value. Perhaps I may venture to think that the results of this enterprise have been in some measure successful. I found that as a general rule the results from different telescopes do not combine directly as well as could be hoped and that there are many precautions which must be taken in using them if we are to avoid serious systematic error and a ruination of the parallax determination; but I believe that it is possible to avoid these difficulties and that the photographs properly treated will give a determination of the parallax of far greater accuracy than has hitherto been obtained. I found also that the 300 exposures in that period of nine days, contributed altogether by nine different observatories, gave a value of the solar parallax, 8-797 seconds± 0·0047 second, which is in such nice agreement with Gill's 8.802 seconds ± 0·005 second that one may feel in one's heart (though of course must never express the feeling so prematurely as this) some hope that, in adopting 8.80 seconds as the official value of the solar parallax, the conference of 1896 was not so wrong as some people have been prepared to believe. From heliometer observations of Victoria, Iris, and Sappho. Distance of sun in miles. =8802 seconds± 0·005 second. From 295 photographs of Eros. x=8-797 seconds±0.0047 second. 92,875,000 I can not refrain from calling your attention to a by-product of this work which has for me a singular interest, because it seems to exhibit in a favorable light the accuracy which we may obtain with these photographic methods. After Eros had been under observation for some time it was discovered that its light was varying in a short FIG. 5.-Light curve of Eros. Whole period 5 h. 16 m. period, which was at first thought to be 2h. 38m. Afterwards it was found that the alternate minima of light was unequal, and that the true period should be reckoned as 5h. 16m., two equal maxima and two unequal minima being included within that space of time (fig. 5). The variation appears to be continuous, without sensible pause, which precludes the idea that the planet is double and that the minima are due to eclipses of one body by the other. We must find some other cause. There are two which suggest themselves quite naturally-irregularity of shape and irregularity of surface bright +050. -050. FIG. 6.-Residuals grouped in eighths of the whole period. ness. For our purpose the important point is this-that either of these causes might produce an apparent oscillation in the place of the planet. To discover if this were so, I grouped the residuals in my equations of condition according to their epoch into eight columns, corresponding to successive eighths of the whole period of 5h. 16m., and took the means for each column. If there were a sensible oscillation in a period of 5h. 16m., these would lie on a sine curve. They obviously do not; there is no sensible oscillation in that period (fig. 6). But notice that if we add together the first and fifth, the second and sixth, and so on--that is to say, if we search for the half period of 2h. 38m.—we get quite strong evidence of periodicity (fig. 7). Now, the semiamplitude of the oscillation is only 0.03 second, a quantity so small that one can not but feel doubts as to its reality. At first I was myself inclined to disbelieve; but when a new distribution of the residuals, starting from a different zero of time, gave a similar periodicity, it began to look as if there were something in it. The more I look at it the more I believe that it is a reality, and that the photographs have shown themselves accurate enough to detect an inequality of 0.03 second, corresponding to a shift of 5 miles at a distance of 25,000,000 miles-by far the smallest inequality in the motion of a planet ever brought to light by observation. It is this circumstance that encourages one to believe in the accuracy of the photographs. There are altogether 10,000 separate exposures of the planet which will within the next few years be measured and made available for discussion. If 300 give a P. E. of 0·005 second, what will 10,000 give, added to 6,000 or 7,000 sets of FIG. 7.-Residuals grouped in quarters of the half period. visual observations? It would be going too far to apply the simple rules of probability and say a good deal less than 0.001 second. But I fully believe that if this great array of observations is ever submitted to complete discussion the probable error of the result will not be much above 0.001 second. And supposing that it should support with its greater weight the value 880 seconds which has been assailed, I believe that we should be justified in saying that the solar parallax is 8-80 seconds, and in maintaining the proposition that the determination of the solar parallax is a problem of geometry and celestial surveying, and that upon the sponsors of the indirect methods lies the onus of showing cause for their disagreement. This opens up an interesting prospect. Suppose that in course of time there should come to be a clear and definite agreement among the values found for the constant of the aberration of light, and that its value was (let us say) 2054 seconds, corresponding, as this table shows, to a parallax of 877 seconds, not 880 seconds, on the assumption at least that the velocity of light is exactly determined, as it seems to be, and that the simple theory of aberration is correct. And suppose that by that time we are prepared to say quite definitely that the geometrical value is not 877 seconds but 880 seconds. The most obvious solution of the difficulty would be to conclude that the simple theory of aberration is not true, and to hand over the problem to the mathematical physicists, who might in the result find that a definite geometrical determination of the solar parallax had provided just the criterion which they required to settle certain vexed questions in dynamics. Again, should further investigation confirm the conclusion that 8.76 seconds is the only value of the solar parallax which will reconcile the existing theory of the motion of the planet with the observed value of the constant of gravitation, it may be that the contradiction between the direct and the indirect methods will at last enable the dynamical astronomers to lay a finger upon that flaw which exists somewhere or other in the theory and makes it impossible to say at the present time that all the motions of the solar system can be completely explained. I have ventured to point out that the determination of the solar parallax is a problem of wide interest, since it throws upon so many different people the task of keeping up their particular end against an attack whose accuracy is gradually becoming more and more deadly. The dimensions of the earth, as obtained by geodetic operations, are necessarily beyond the reach of any criticism derived from solar parallax results. Extreme range of these determinations is only 1 in 4,000. 3,962-76 3,963-31 3,963-28 3, 963-37 3,963-29 |