Page:Popular Science Monthly Volume 10.djvu/429

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DISTANCE AND DIMENSIONS OF THE SUN.

sun's distance will be greatly reduced as the result of this work; and yet there are some grounds for anxiety lest the photographic data prove as intractable and inconsistent as those derived from contact observations. Time only can positively settle the question.

One of the best methods of determining the solar parallax is based upon the careful observation of the motions of the moon. It will be recollected that the first suspicion as to the correctness of the then received distance of the sun was raised in 1854 by Hansen's announcement that the moon's parallactic inequality led to a smaller value than that deduced from the transit of Venus—a conclusion corroborated by Leverrier four years later. It seems at first sight strange, but it is true, as Laplace long since pointed out, that the skillful astronomer, by merely watching the movements of our satellite, and without leaving his observatory, can obtain the solution of problems which, attacked by other methods, require tedious and expensive expeditions to remote corners of the earth. Our scope and object do not require us to enter into detail respecting this lunar method of finding the sun's parallax; it must suffice to say that the disturbing action of the sun makes the interval from new moon to full a little longer than that from full to new; and this difference depends upon the ratio between the diameter of the moon's orbit and the distance of the sun in such a manner that, if the inequality is accurately observed, the ratio can be calculated. Since we know the distance of the moon, this will give that of the sun. The results obtained in this way, according to the most recent investigations, fix the solar parallax between 8.83" and 8.92".

There is still another lunar method, mentioned in the synopsis; but its results are much less reliable—subject, that is, to a much larger probable error, though not at all contradictory to those just given.

But the method by which ultimately we shall obtain the most accurate determination of the dimensions of our system is that proposed by Leverrier, making use of the secular perturbations produced by the earth upon her neighboring planets, especially in causing the motions of their nodes and perihelia. These motions are very slow, but continuous; and hence, as time goes on, they will become known with ever-increasing accuracy. If they were known with absolute precision, they would enable us to compute, with absolute precision also, the ratio between the masses of the sun and earth, and from this ratio we can calculate[1] the distance of the sun by either of two or three different methods.

1. One method of proceeding is as follows: Let M be the mass of the sun, and m that of the earth; let R be the distance of the sun from the earth, and r that of the moon; finally, let T be the number of days in a sidereal year, and t the number in a sidereal month. Then, by elementary astronomy,

${\displaystyle \scriptstyle M:m={\tfrac {R^{3}}{T^{2}}}:{\tfrac {r^{3}}{t^{2}}};}$ whence ${\displaystyle \scriptstyle {R^{3}}={r^{3}}\left({\tfrac {T^{2}}{t^{2}}}\right)\left({\tfrac {M}{m}}\right)}$ ;

or, in words, the cube of the sun's distance equals the cube of the moon's distance, multiplied