Outlines of a mechanical theory of storms

by Thomas Bassnett

In these investigations it is necessary to bear in mind that the whole resisting power of the ether, in disturbing the planetary movements, is but small, in comparison with gravitation. We will, however, show that, in the case of the planets, there is a compensation continually made by this resistance, which leaves but a very small outstanding balance as a disturbing power. If we suppose all the planets to move in the central plane of the vortex in circular orbits, and the force of the radial stream, (or that portion which is not in accordance with the law of gravitation,) to be inversely as the square roots of the distances from the sun, it is evident, from what has been advanced, that an equilibrium could still obtain, by variations in the densities, distances and diameter of the planets. Supposing, again, that the planets still move in the same plane, but in elliptical orbits, and that they are in equilibrium at their mean distances, under the influence or action of the tangential current, the radial stream, and the density of the ether; we see that the force of the radial stream is too great at the perihelion, and too small at the aphelion. At the perihelion the planet is urged from the sun and at the aphelion towards the sun. The density and consequent momentum is also relatively too great at the perihelion, which also urges the planet from the sun, and at the aphelion, relatively too small, which urges the planet towards sun; and the law is the same in both cases, being null at the mean distance of the planet, at a maximum at the apsides; it is, consequently, as the cosine of the planet’s eccentric anomaly at other distances, and is positive or negative, according as the planet’s distance is above or below the mean.

At the planet’s mean distance, the circular velocity of the vortex is equal to the circular velocity of the planet, and, at different distances, is inversely in the sub-duplicate ratio of those distances. But the circular velocity of a planet in the same orbit, is in the simple ratio of the distances inversely. At the perihelion, the planet therefore moves faster than the ether of the vortex, and at the aphelion, slower; and the difference is as the square roots of the distances; but the force of resistance is as the square of the velocity, and is therefore in the simple ratio of the distances, as we have already found for the effect of the radial stream, and centrifugal momentum of the internal ether. At the perihelion this excess of tangential velocity creates a resistance, which urges the planet towards the sun, and at the aphelion, the deficiency of tangential velocity urges the planet from the sun,—the maximum effect being at the apsides of the orbit, and null at the mean distances. In other positions it is, therefore, as the cosines of the eccentric anomaly, as in the former case; but in this last case it is an addititious force at the perihelion, and an ablatitious force at the aphelion, whereas the first disturbing force was an ablatitious force at the perihelion, and an addititious force at the aphelion; therefore, as we must suppose the planet to be in equilibrium at its mean distance, it is in equilibrium at all distances. Hence, a planet moving in the central plane of the vortex, experiences no disturbance from the resistance of the ether.

As the eccentricities of the planetary orbits are continually changing under the influence of the law of gravitation, we must inquire whether, under these circumstances, such a change would not produce a permanent derangement by a change in the mean force of the radial stream, so as to increase or diminish the mean distance of the planet from the sun. The law of force deduced from the theory for the radial stream is as the 2.5 power of the distances inversely. But, by dividing this ratio, we may make the investigation easier; for it is equivalent to two forces, one being as the squares of the distances, and another as the square roots of the distances. For the former force, we find that in orbits having the same major axis the mean effect will be as the minor axis of the ellipse inversely, so that two planets moving in different orbits, but at the same mean distance, experience a less or greater amount of centripulsive force from this radial stream, according as their orbits are of less or greater eccentricity, and this in the ratio of the minor axis. On the other hand, under the influence of a force acting centripulsively in the inverse ratio of the square roots of the distances, we find the mean effect to be as the minor axis of the ellipse directly, so that two planets in orbits of different eccentricity, but having the same major axis, experience a different amount from the action of this radial stream, the least eccentric orbit being that which receives the greatest mean effect. By combining these two results, we get a ratio of equality; and, consequently, the action of the radial stream will be the same for the same orbit, whatever change may take place in the eccentricity, and the mean distance of the planet will be unchanged. A little consideration will also show that the effect of the centrifugal momentum due to the density of ether will also be the same by change of eccentricity; for the positive will always balance the negative effect at the greatest and least distances of the planet. The same remark applies to the effect of the tangential current, so that no change can be produced in the major axes of the planetary orbits by change of eccentricity, as an effect of the resistance of the ether.

We will now suppose a planet’s orbit to be inclined to the central plane of the vortex, and in this case, also, we find, that the action of the radial stream tends to increase the inclination in one quadrant as much as it diminishes it in the next quadrant, so that no change of inclination will result. But, if the inclination of the orbit be changed by planetary perturbations, the mean effect of the radial stream will also be changed, and this will tell on the major axis of the orbit, enlarging the orbit when the inclination diminishes, and contracting it when it increases. The change of inclination, however, must be referred to the central plane of the vortex. Notwithstanding the perfection of modern analysis, it is confessed that the recession of the moon’s nodes does yet differ from the theory by its 350th part, and a similar discrepancy is found for the advance of the perigee.[40] This theory is yet far too imperfect to say that the action of the ethereal medium will account for these discrepancies; but it certainly wears a promising aspect, worthy the notice of astronomers. There are other minute discordancies between theory and observation in many astronomical phenomena, which theory is competent to remove. Some of these we shall notice presently; and, it may be remarked, that it is in those minute quantities which, in astronomy, are usually attributed to errors of observation, that this theory will eventually find the surest evidence of its truth.

KEPLER’S THIRD LAW ONLY APPROXIMATELY TRUE.

But it may be asked: If there be a modifying force in astronomy derived from another source than that of gravitation, why is it that the elements of the various members of the system derived solely from gravitation should be so perfect? To this it may be answered, that although astronomers have endeavored to derive every movement in the heavens from that great principle, they have but partially succeeded. Let us not surrender our right of examining Nature to the authority of a great name, nor call any man master, either in moral or physical science. It is well known that Kepler’s law of the planetary distances and periods, is a direct consequence of the Newtonian Law of gravitation, and that the squares of the periodic times ought to be proportional to the cubes of the mean distances. These times are given accurately by the planets themselves, by the interval elapsing between two consecutive passages of the node, and as in the case of the ancient planets we have observations for more than two thousand years past, these times are known to the fraction of the second. The determination of the distances however, depends on the astronomer, and a tyro in the science might suppose that these distances were actually measured; and so they are roughly; but the astronomer does not depend on his instruments, he trusts to analogy, and the mathematical perfection of a law, which in the abstract is true; but which he does not know is rigidly exact when applied to physical phenomena. From the immense distance of the planets and the smallness of the earth, man is unable to command a base line sufficiently long, to make the horizontal parallax a sensible angle for the more distant planets; and there are difficulties of no small magnitude to contend with, with those that are the nearest. In the occasional transit of Venus across the sun, however, he is presented with a means of measuring on an enlarged scale, from which the distance of the sun is determined; and by analogy the distance of all the planets. Even the parallax of the sun itself is only correct, by supposing that the square of the periodic time of Venus is in the same proportion to the square of the periodic time of the earth as the cube of her distance is to the cube of the earth’s distance. Our next nearest planet is Mars, and observations on this planet at its opposition to the sun, invariably give a larger parallax for the sun—Venus giving 8.5776″ while Mars gives about 10″. It is true that the first is obtained under more favorable circumstances; but this does not prove the last to be incorrect. It is well known that the British Nautical Almanac contains a list of stars lying in the path of the planet Mars about opposition, (for the very purpose of obtaining a correct parallax,) that minute differences of declination may be detected by simultaneous observations in places having great differences of latitude. Yet strange to say, the result is discredited when not conformable to the parallax given by Venus. If then, we cannot trust the parallax of Mars, à fortiori, how can we trust the parallax of Jupiter, and say that his mean distance exactly corresponds to his periodic time? Let us suppose, for instance, that the radius vector of Jupiter fell short of that indicated by analogy by 10,000 miles, we say that it would be extremely difficult, nay, utterly impossible, to detect it by instrumental means. Let not astronomers, therefore, be too sure that there is not a modifying cause, independent of gravitation, which they will yet have to recognize. The moon’s distance is about one-fourth of a million of miles, and Neptune’s 2854 millions, or in the ratio of 10,000 to 1; yet even the moon’s parallax is not trusted in determining her mass, how then shall we determine the parallax of Neptune? It is therefore possible that the effective action of the sun is in some small degree different, on the different planets, whether due to the action of the ether, to the similarity or dissimilarity of material elements, to the temperature of the different bodies, or to all combined, is a question yet to be considered.

As another evidence of the necessity of modifying the strict wording of the Newtonian law, it is found that the disturbing action of Jupiter on different bodies, gives different values for the mass of Jupiter. The mass deduced from Jupiter’s action on his satellites, is different from that derived from the perturbations of Saturn, and this last does not correspond with that given by Juno: Vesta also gives a different mass from the comet of Encke, and both vary from the preceding values.[41]

In the analytical investigation of planetary disturbances, the disturbing force is usually divided into a radial and tangential force; the first changing the law of gravitation, to which law the elliptic form of the orbit is due. The radial disturbing force, therefore, being directed to or from the centre, can have no influence over the first law of Kepler, which teaches that the radius vector of each planet having the sun as the centre, describes equal areas in equal times. If the radial disturbing force be exterior to the disturbed body, it will diminish the central force, and cause a progressive motion in the aphelion point of the orbit. In the case of the moon this motion is very rapid, the apogee making an entire revolution in 3232 days. Does this, however, correspond with the law of gravitation? Sir Isaac Newton, in calculating the effect of the sun’s disturbing force on the motion of the moon’s apogee, candidly concludes thus: “Idoque apsis summa singulis revolutionibus progrediendo conficit 1° 31′ 28″. Apsis lunæ est duplo velocior circiter.” As there was a necessity for reconciling this stubborn fact with the theory, his followers have made up the deficiency by resorting to the tangential force, or, as Clairant proposed, by continuing the approximations to terms of a higher order, or to the square of the disturbing force.

Now, in a circular orbit, this tangential force will alternately increase and diminish the velocity of the disturbed body, without producing any permanent derangement, the same result would obtain in an elliptical orbit, if the position of the major axis were stationary. In the case of the moon, the apogee is caused to advance by the disturbing power of the radial force, and, consequently, an exact compensation is not effected: there remains a small excess of velocity which geometers have considered equivalent to a doubling of the radial force, and have thus obviated the difficulty. To those not imbued with the profound penetration of the modern analyst, there must ever appear a little inconsistency in this result. The major axis of a planet’s orbit depends solely on the velocity of the planet at a given distance from the sun, and the tangential portion of the disturbance due to the sun, and impressed upon the moon, must necessarily increase and diminish alternately the velocity of the moon, and interfere with the equable description of the areas. If, then, there be left outstanding a small excess of velocity over and above the elliptical velocity of the moon, at the end of each synodical revolution, in consequence of the motion impressed on the moon’s apogee by the radial force, the legitimate effect would be a small enlargement of the lunar orbit every revolution in a rapidly-increasing ratio, until the moon would at last be taken entirely away. In the great inequality of Jupiter and Saturn, this tangential force is not compensated at each revolution, in consequence of continual changes in the configuration of the two planets at their heliocentric conjunctions, with respect to the perihelion of their orbits, and the near commensurability of their periods; and the effect of the tangential force is, in this case, legitimately impressed on the major axes of the orbits. But why (we may ask) should not this also be expended on the motion of the aphelion as well as in the case of the moon? Astronomy can make no distinctions between the orbit of a planet and the orbit of a satellite. And, we might also ask, why the tangential resistance to the comet of Encke should not also produce a retrograde motion in the apsides of the orbit, instead of diminishing its period? To the honor of Newton, be it remembered, that he never resorted to an explanation of this phenomenon, which would vitiate that fundamental proposition of his theory, in which the major axis of the orbit is shown to depend on the velocity at any given distance from the focus.

Some cause, however, exists to double the motion of the apogee, and that there is an outstanding excess of orbital velocity due to the tangential force, is also true. This excess may tell in the way proposed, provided some other arrangement exists to prevent a permanent dilation of the lunar orbit; and this provision may be found in the increasing density of the ether, which prevents the moon overstepping the bounds prescribed by her own density, and the force of the radial stream of the terral vortex. In the case of Jupiter and Saturn, their mutual action is much less interfered with by change of density in the ether in the enlarged or contracted orbit, and, consequently, the effect is natural. Thus, we have in the law of density of the ethereal medium a better safeguard to the stability of the dynamical balance of the system, than in the profound and beautiful Theorems of La Grange. It will, of course, occur to every one, that we are not to look for the same law in every vortex, and it will, therefore, appear as if the satellites of Jupiter, whose theory is so well known, should render apparent any deviation between their periodic times and the periodic times of the contiguous parts of the vortex, which would obtain, if the density of the ether in the Jovian vortex were not as the square roots of the distances directly. But, we have shown how there can be a balance preserved, if the tangential resistance of the vortex shall be equal and contrary at the different distances at which the satellites are placed; that is, if these two forces shall follow the same law. These are matters, however, for future investigation.