Popular Science Monthly/Volume 44/November 1893/Laplace's Plan for Perpetual Moonlight

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LAPLACE'S PLAN FOR PERPETUAL MOONLIGHT.
By DANIEL KIRKWOOD.

ONE of the questions considered by Laplace in the early part of the century, and which he thought of sufficient interest to have a place in his System of the World, has dropped almost wholly out of view. I refer to the relation of the moon to the earth—what it is and what it might have been. The subject is not even referred to in any recent text-book on astronomy. The conclusion of Laplace, however, was not hastily reached, and it remained in his hands, without modification, for a number of years. The great name of the author probably prevented astronomers of the day from undertaking any criticism of his conclusions, and especially from the expression of any opinion on a mathematical question different from that of the greatest astronomer of the century. If Laplace himself ever saw his mistake he never mentioned it, as in the case of a mathematical error pointed out by Dr. Bowditch, the translator of the Mécanique Céleste. Dr. Bowditch's letter informing him of the error was never acknowledged. The mistake, however, was rectified in a new edition. But I proceed with the subject.

If moonlight, it has been said, be always pleasant and desirable—if it contribute to the convenience and enjoyment of life, and if its perpetuity be not inconsistent with the laws by which the world is governed—why has its use been so largely denied us? Why has Nature, or the Author of Nature, left us so great a portion of our time in almost total darkness? Such questions have doubtless occurred to thoughtful minds in all ages. The subject is one of interest and curiosity. Let us briefly consider some of the possible relations of a satellite to its primary, including a special case proposed by Laplace.

Sir Isaac Newton, who preceded Laplace by about a century, had found evidence, as he claimed, that the material universe is the work of an all-wise designer. The author of the Mécanique Céleste, the greatest mathematical astronomer of his age, seldom discussed questions of a moral nature; but, not accepting Newton's views on the doctrine of final causes, or the doctrine of design in the material world, he took occasion to point out a so-called failure of Nature in adapting means to ends. If the moon was designed to give light by night, the purpose, he said, had largely failed, and he (Laplace) could suggest a better plan. But the entire passage is quoted as follows:

"Some partisans of final causes have imagined that the moon was given to the earth to afford it light during the night. But in this case Nature would not have attained the end proposed, since we are often deprived at the same time of the light of both sun and moon. To have accomplished this end, it would have been sufficient to have placed the moon at first in opposition to the sun and in the plane of the ecliptic, at a distance from the earth equal to the one hundredth part of the distance of the earth from the sun, and to have impressed on the earth and moon parallel velocities proportional to their distances from the sun. In this case, the moon, being constantly in opposition to the sun, would have described round it an ellipse similar to that of the earth. These two bodies would then constantly succeed each other, and as at this distance the moon could not be eclipsed, its light would always replace that of the sun."[1]

The plan here proposed was one of startling boldness; but without assuming to defend the doctrine of final causes, it must be said in fairness that to afford light by night had never been claimed as the only design for which the moon was given. Other purposes no less important may be readily imagined. Moreover, the moon's light at the distance named by Laplace would have been little more than one twentieth part of that afforded by the full moon at its actual distance, or less than that of our new moon two days after the change. Such moonlight, though perpetual, would have had little comparative value. Again, the tidal effect upon the earth would have been scarcely perceptible. But without further insisting on these points, however important, let us compare the proposed arrangement with that of Nature. Would it have involved nothing inconsistent with the system's stability? or would its adoption have resulted in depriving our world of the moonlight enjoyed in the existing system?

The annexed figure[2] illustrates Laplace's proposed arrangement. The distance at which he would have placed the moon from the earth is about 1,000,000 miles, or a little more than four times the actual distance. An eclipse of the moon is caused by its falling into the earth's shadow. This can extend into space only about 800,000 miles, and, as this is less than the distance of Laplace's proposed moon, the latter, as he remarks, could never be eclipsed.

Let us suppose the distance of the moon from the earth to be increased, what changes would be effected in the observed phenomena? At 478,000 miles, twice the present distance, the length of the lunar month would be seventy-seven days; the quantity of moonlight would be one fourth of what we now enjoy; and the height of tides in the open seas would be but a few inches. At 717,000 miles, three times the present distance, the length of the month would be one hundred and forty-two days, and the apparent size of the moon would be reduced to one ninth of its present value. With increasing distance the phenomena would still further change, till at the orbit named by Laplace the month would be equal to the year, and the moon's enlightened hemisphere would be turned constantly to the earth. But the great astronomer's dream of perpetual moonlight—how long would it be realized?

Another question of vital importance is here involved in the theory under consideration—the variation of the earth's attraction on the moon supposed to be removed to a greater distance. This variation is more rapid than that of the sun's attractive force on the same body, as the distance between the sun and moon Is four hundred times that between the moon and the earth. At what point, then, would our satellite escape from the earth's controlling influence and commence to revolve as an independent planet about the sun? This question, strangely enough, seems never to have received Laplace's consideration; at least his statement was continued without change in a later edition of his Systeme du Monde. This problem touching the moon's limit of stability was not solved until sixteen years after Laplace's death.[3]

The relative distances as well as the direction and force of the impulses necessary to produce the required motions in the scheme of Laplace were given by himself in the paragraph quoted. The state of things at double the moon's distance has also been estimated. At four times the distance, or somewhat more, we find Laplace's position of perpetual moonlight; but just here we find the region where the earth loses its control over the moon's motion. The moon escapes from the earth's influence, and henceforth owns allegiance only to the sun. She becomes a primary planet, with a year somewhat greater than ours and a day of doubtful length. As regards the earth, lunar tides can no longer exist. Moonlight and the moon would forsake us together; and the new condition of things, could it be realized, would be worse than the first.

From the case here considered we may learn (1) that dogmatism in regard to the divine plan in the structure and constitution of the universe is not always wise. Final causes may engage the attention of thoughtful minds, but who shall set limits to their extent or application? "Touching the Almighty," said Elihu, "we can not find him out." (3) The wisdom manifested in the adaptations of material things around us transcends that of man's highest efforts. Attempts to disparage the skill of Nature's handiwork must end in failure and disappointment.

The failure of the theory proposed in the case of the earth and moon is no less striking when applied to Mars, Jupiter, or any other planet. In every instance the position of the satellite assumed to afford permanent moonlight would be one of instability. This striking fact renders the oversight of Laplace the more remarkable. It may be stated, however, that by the arrangement of several moons about the same planet almost, if not entirely, perpetual moonlight might be possible. The system of Jupiter and his moons furnishes a clear illustration.

In conclusion, we have seen, then, that where one of the greatest mathematicians of all time suggested a change—a so-called improvement in the system of the world—the modification would have left us without tides, or, worse still, the earth in the system proposed would have lost control of her satellite, and we would not only have been deprived of moonlight, but also of the moon itself.

 

  1. Système du Monde, Hart's translation, vol. ii, p. 79.
  2. Figure omitted.
  3. The solution was first given by M. Liouville in 1842.