another remarkable coincidence in the solar system—the second concord—the direction of the movements of the great planets—the movement of ceres—yet another planet—discovery of eros—the nearest neighbour of the earth—throwing heads and tails—a calculation of the chances—the numerical strength of the argument—an illustration of the probability of the origin of the solar system from the nebula—the explanation of the second concord offered by the nebular theory—the relation of energy and moment of momentum—different systems illustrated—that all the movements should be in the same direction is a consequence of evolution from the prim?val nebula.
we have seen in the last chapter that there is a very remarkable concordance in the positions of the planes of the orbits of the planets, and we have shown that this concordance finds a natural historical explanation in the nebular origin of our system. we have now to consider another striking concord in the movements of the planets in their several orbits, and this also furnishes us with important evidence as to the truth of the nebular theory. the argument on which we are now to enter is one which specially appealed to laplace, and was put forward by him as the main foundation of the nebular theory.
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fig. 48.—illustration of the second concord.
in the adjoining fig. 48 we have a diagram of a portion of the solar system. we shall regard the movements as somewhat simplified. the sun is supposed to be at the centre, turning round once every twenty-five days, on an axis which is supposed to be perpendicular to the plane of the paper. we may also for our present purpose assume that the orbits of the earth and the other planets lie in this same plane.
in the first place we observe that the earth 310might have gone round its track in either direction so far as the welfare of mankind is concerned. the succession of day and night, and the due changes of the seasons, could have been equally well secured whichever be the direction in which the earth revolves. we do, however, most certainly find that the direction in which the earth revolves round the sun is the same as the direction in which the sun rotates on its axis. this is the first coincidence.
we may now consider other planets. look, for instance, at the orbit of jupiter. it seems obvious that jupiter might have been made to revolve round the sun either one way or the other; indeed, it will be remembered that though kepler’s laws indicate so particularly the shape of the track in which the planet revolves, and prescribe so beautifully the way in which the planet must moderate or accelerate its velocity at the different parts of its track, yet they are quite silent as to the direction in which the planets shall revolve in that track. if we could imagine a planet to be stopped to have its velocity reversed, and then to be started in a precisely opposite direction, it would still continue to describe precisely the same path; it would still obey kepler’s laws with unfailing accuracy, so far as our present argument is concerned, and the velocity which it would have at each point of the track would be quite the same whether the planet were going one way or whether it was going the other. it is therefore equally possible for jupiter to pursue his actual track by going round the sun in the same direction as the earth, or by going in the opposite direction. but we actually find that jupiter does take the same direction as the earth, and this, as we 311have already seen, is the direction in which the sun rotates. here we have the second coincidence.
we now take another planet; for example, mars. again we affirm that mars could have moved in either direction, but, as a matter of fact, it pursues the same direction as jupiter and the earth. in the orbital movement of saturn we have the fourth coincidence of the same kind, and we have a fifth in the case of mercury, and a sixth in venus, a seventh in uranus, and an eighth in neptune. the seven great planets and the earth all revolve around the sun, not only in orbits which are very nearly in the same plane, but they also revolve in the same direction.
the coincidences we have pointed out with regard to the movements of the great planets of our system may be also observed with regard to the numerous bodies of asteroids. on the first night of the century just closed, the 1st of january, 1801, the first of the asteroids, now known as ceres, was discovered. this was a small planet, not a thousandth part of the bulk of one of the older planets, and visible, of course, only in the telescope. like the older planets, it was found to obey kepler’s laws; but this we might have foreseen, because kepler’s laws depend upon the attraction of gravitation, and must apply to any planet, whatever its size. when, therefore, the new planet was found, and its track was known, it was of much interest to see whether the planet in moving round that track observed the same direction in which all the older planets had agreed to travel, or whether it moved in the opposite direction. in the orbit of ceres we have a repetition of the coincidence which has been noticed in each of the other planets. the new planets, like all 312the rest, move round the sun in the same direction as the sun rotates on its axis. the discovery of this first asteroid was quickly followed by other similar discoveries; each of the new planets described, of course, an ellipse, and the directions which these planets followed in their movements round the sun were in absolute harmony with those of the older planets.
but, besides the great planets and the asteroids properly so called, there is yet another planet, eros. its testimony is of special value, inasmuch as it seems to stand apart from all other bodies in the solar system, and with, of course, the exception of the moon, it is the earth’s nearest neighbour. but whatever may be the exceptional features of eros, however it may differ from the great planets and the asteroids already known, yet eros makes no exception to the law which we have found to be obeyed by all the other planets. it also revolves round the sun in the same direction as all the planets revolve, in the same direction as the rotation of the sun (fig. 49).
we may pause at this moment to make a calculation as to the improbability that the sun, the earth, the seven great planets, and ceres, numbering altogether ten, should move round in the same direction if their movements had been left to chance. this will show what we can reasonably infer from this concord in their movements. the theory of probabilities will again enlighten a difficult subject.
there are only two possible directions for the motion of a planet in its orbit. it must move like the hands of a watch, or it must move in the opposite direction. the planet must move one way or the other, just as a penny must always fall head or tail.
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fig. 49.—orbits of earth, eros and mars.
we may illustrate this remarkable coincidence in the following manner: suppose we take ten coins in the hand, and toss them all up together and let them fall on the table; in the vast majority of cases in which the experiment may be tried, there would be some heads and some tails; they would not all be heads. but it is, of course, not impossible that the coins should all turn up heads. we should, however, deem it a very remarkable circumstance if it happened: yet it would certainly not be more remarkable than that the ten celestial movements should all take place in the same direction, unless, indeed, it should turn out that there is some sound physical cause which imposes on the planets of the solar system an obligation, restricting 314their movements round the sun to the same direction as that in which the sun itself rotates.
it will be useful to study the matter numerically; and the rules of probabilities will enable us to do so, as we may see by the following illustration: we deem the captain of a cricket team fortunate when he wins the toss for innings. we should deem him lucky indeed if he won it three times in successive matches. if he won it five times running, his luck would be phenomenal; while, if it was stated that he won it ten times consecutively, we should consider the statement well-nigh incredible. for it is easy to calculate that the chances against such an occurrence are one thousand and twenty-four to one. in like manner we may say, that for nine planets and the sun all to go round in the same direction would be indeed surprising if the arrangement of the planets had been determined by chance; there are more than a thousand chances to one against such an occurrence.
but ceres was only the earliest of many other similar discoveries. and as each asteroid was successively brought to light, it became most interesting to test whether it followed the rest of the planets in that wonderful unanimity in the direction of their movements of revolution, or whether it made a new departure by going in the opposite direction. no such exception has ever yet been observed. let us take, then, ten more planets, in addition to those we have already considered, so that we have now nineteen planets all revolving in the same direction as the sun rotates. it is easy to compute the improbability that these twenty movements should all be in the same direction, if, indeed, it were by chance that their directions had been determined. 315it is the same problem as the following: what is the chance that twenty coins, taken together in the hand and tossed into the air at once, shall all alight with their heads uppermost? we have seen that the chances against this occurrence, if there were ten coins, is about a thousand to one. it can easily be shown that if there were twenty coins the chances against the occurrence would be a million to one. we thus see that, even with no more than nineteen planets and the sun, there is a million to one against a unanimity in the directions of the movements, if the determination of the motions was made by chance. we may, however, express the result in a different manner, which is more to the purpose of our argument. there are a million chances to one in favour of the supposition that the disposition of the movements of the planets has not been the result of chance; or we may say that there are a million chances to one in favour of the supposition that some physical agent has caused the unanimity.
we can add almost any desired amount of numerical strength to the argument. the discoveries of minor planets went on with ever-increasing success through the whole of the last century. when ten more had been found, and when each one was shown to obey the same invisible guide as to the direction in which it should pursue its elliptic orbit, the chances in favour of some physical cause for the unanimity became multiplied by yet another thousand. the probability then stood at a thousand millions to one. as the years rolled by, asteroids were found in ever-increasing abundance. sometimes a single astronomer discovered two, and sometimes even more than two, on a single night. in the course of a lifetime a diligent astronomer 316has placed fifty discoveries of asteroids, or even more than fifty, on his record. by combined efforts the tale of the asteroids has now approached five hundred, and out of that huge number of independent planetary bodies there is not one single dissentient in the direction of its motion. without any exception whatever, they all perform their revolutions in the same direction as the sun rotates at the centre. when this great host is considered, the numerical strength of the argument would require about 150 figures for expression. each new asteroid simply doubled the strength of the argument as it stood before.
professor j. j. thomson recently discovered that there are corpuscles of matter very much smaller than atoms. let us think of one of these corpuscles, of which many millions would be required to make the smallest grain of sand which would just be visible under a microscope. think, on the other hand, of a sphere extending through space to so vast a distance that every star in the milky way will be contained within its compass. then the number of those corpuscles which would be required to fill that sphere is still far too small to represent the hugeness of the improbability that all the five hundred planetary bodies should revolve in the same direction, if chance, and chance alone, had guided the direction which each planet was to pursue in moving round its orbit.
the mere statement of these facts is sufficient to show that some physical agent must have caused this marvellous concord in the movements of the solar system. how the argument would have stood if there had been even a single dissentient it is not necessary to consider, for there is no dissentient no reasonable 317person will deny that these facts impose an obligation to search for the physical explanation of this feature in the planetary movements.
as in the last chapter, where we were dealing with the positions of the planes of the orbits, there can here be no hesitation as to the true cause of this most striking characteristic of the planetary movements. the nebular theory is at once ready with an explanation, as has been already indicated in chapter xi. the prim?val nebula, endowed in the beginning with a certain amount of moment of momentum, has been gradually contracting. it has been gradually expending its energy, as we have already had occasion to explain; but the moment of momentum has remained undiminished. and from this it can be shown that the dynamical principles guiding the evolution of the nebula must ultimately refuse permission for any planet to revolve in opposition to the general movement. this point is a very interesting one, and as it is of very great importance in connection with our system, i must give it some further illustration and explanation.
the two figures that are shown in fig. 50 represent two imaginary systems. we have a sun in each, and we have two planets in each. the sun is marked with the letter s, and the two planets are designated by a and b. for simplicity i have represented the orbits as circles, and for the same reason i have left out the rest of the planets; we shall also suppose the orbits of the two planets that are involved to lie exactly in the same plane. in the two systems that i have here supposed, the two suns are to be of the same weight, the planet a in one system is of equal mass to the planet a in the other; and the planets b in the two systems are also equal. 318it is also assumed that the orbit of a in one diagram shall be the same as the orbit of a in the other, and that the orbit of b in one shall be precisely the same as the orbit of b in the other. the sun rotates in precisely the same manner in both, and takes the same time for each rotation. a, in one system, goes round in the same time that a does in the other; and b, in one system, goes round in the same time that b does in the other. there is, therefore, a perfect resemblance between the two systems i have here supposed in every point but one. i have indicated, as usual, the movements of the bodies by arrows, and, while in one of the systems the sun and a and b all go round in the same direction, in the other system the sun and a go round, no doubt, in the same direction, but the direction of b is opposite. we are not, in this illustration, considering the rotations of the planets on their axes. that will be dealt with in the next chapter.
fig. 50.—i. a natural system on the left.
ii. an unnatural system on the right.
there can be no doubt that either of these two 319systems would be possible for thousands of revolutions. there is nothing whatever to prevent a and b from being started in the same direction round the sun as in the first figure, or with a in one direction and b in the opposite direction, as in the second figure. it is equally conceivable that, while a and b revolve in the same direction, both should be opposite to that of the sun. but one system is permanent, and the other is not.
for, as a matter of fact, we do not find in nature such an arrangement as that in the second figure, or as that in which both the planets revolve in opposite directions to the sun’s rotation; what we do find is, that the planets go round in the same direction as the sun. and the explanation is undoubtedly connected with the important principle already illustrated, namely, that natural systems are in a condition in which the total quantity of energy undergoes continuous reduction in comparison with the moment of momentum.
in the arrangements made in the two figures, it will be recollected that the masses of the three bodies were respectively the same, and also their distances apart, and their velocities. as the energy depends only on the masses, the distances, and the velocities, the energies of the two systems must be identical. but the moment of momentum of the two systems is very different, for while in the one case the sum of the moments of momentum of the sun’s rotation and that of the planet a, which is going in the same direction, are to be increased by the moment of momentum of b, the same is not the case in the other system. the moment of momentum of the sun and of a conspire, no doubt, and must be added together; but as b is revolving in the opposite direction, the moment of momentum of this 320planet has to be subtracted before we obtain the nett moment of momentum of the system. hence, we perceive a remarkable difference between the two systems; for, though in each the total energy is the same, yet in the latter case the moment of momentum is smaller than in the former.
it has been pointed out that the effect of the mutual actions of the different bodies of a system is to lessen, in course of time, the total quantity of energy that they receive in the beginning, while it is not in the power of the mutual actions of the particles of the system to affect the sum total of the moment of momentum. hence we see that, so long as the system is isolated from external interference, the tendency must ever be towards the reduction of the quantity of energy to as low a point as may be compatible with the preservation of the necessary amount of moment of momentum. the first of the two systems given in fig. 50 is much more in conformity with this principle than the second. the moment of momentum in the former case must be nearly as large as could be obtained by any other disposition of the matter forming it, with the same amount of energy. but in the second diagram the moment of momentum is much less, though the energy is the same. it follows that the energy of this system might be largely reduced, for if accompanied by a suitable rearrangement of the planets the reduced amount of moment of momentum might be easily provided for. we thus see that this system is not one to which the evolution of a material arrangement would ultimately tend. it is, therefore, not to be expected in nature, and we do not find it. of course, the same would be equally true 321if, instead of having merely two planets, as i have here supposed for the sake of illustration, the planets were much more numerous. the operation of the causes we have been considering will show that, in the evolution of such a system, there will be a tendency for the planets to revolve in the same direction.
it is easy to see how, in the contraction of the original nebula, there must have been a strong influence to check and efface any movements antagonistic to the general direction of the rotation of the nebula. if particles revolve in a direction opposite to the current pursued by the majority of particles, there would be collisions and frictions, and these collisions and frictions will, of course, find expression in the production of equivalent quantities of heat. that heat will, in due course, be radiated away at the expense of the energy of the system, and consequently, so long as any contrary movements exist, there will be an exceptional loss of energy from this cause. thus the energy would incessantly tend to decline. as the shrinking of the body proceeded while the moment of momentum would have to be sustained, this would incessantly tend more and more to require from all the particles a movement in the same direction.
the second concord of the planetary system, which is implied in the fact that all the planets go round in the same direction, need not therefore surprise us. it is a consequence, an inevitable consequence, of the evolution of that system from the great prim?val nebula. we have seen that it would be excessively improbable that even nine or ten planets should revolve round the sun in the same direction, if the directions of their movements had been merely decided by chance. we 322have seen that the movements of the hosts of planets, which actually form our system, would be inconceivable, unless there were some reason for those movements. the chances against such an arrangement having arisen without some predisposing cause is so vast that, even if the chances were infinite, the case would be hardly strengthened. but once we grant that the system originated from the contraction of the prim?val nebula, dynamics offers ready aid, and the difficulty vanishes. not only do we see most excellent reasons why all the planets should revolve in the same direction; we are also provided with illustrations of similar evolutions in progress in other parts of the universe; we learn that the evolving nebula, however erratic may have been its primitive motion, whatever cross currents may have agitated it in the early phases of a possibly violent origin, will ultimately attain a rotation uniform in direction. as the evolution proceeds, the various parts of the nebula draw together to form the planets of the future system, and the planets retain the movement possessed by their component particles. thus we see that the nebular theory not only extricates us from the difficulty of trying to explain something which seemed almost infinitely improbable, but it also shows why no other disposition of the motions than that which we actually find could be expected. the nebular theory explains to us why there is no exception to that fundamental law in the solar system which declares that the orbits of the planets shall all be followed in the same direction.
this wonderful agreement in the movements of the planets, which we have called the second concord, thus affords us striking evidence of the general truth of 323the nebular theory. but there is yet a third concord in the solar system which, like the other two, lends wonderful corroboration to the sublime doctrine of kant and laplace. this we shall consider in the next chapter.
