Page images
PDF
EPUB

CHAP. XI. ANALYSIS OF DISTURBING FORCES.

343

P in the direction P M, and on S in the direction S M. And the disturbing part of M's attraction, being the difference only of these forces, will have no fixed direction, but will act on P very differently, according to the configurations of P and M. It will therefore be necessary, in analyzing its effect, to resolve it, according to mechanical principles, into forces acting according to some certain directions; viz., along the radius vector S P, and perpendicular to it. The simplest way to do this, is to resolve the attractions of M on both S and P in these directions, and take, in both cases, their difference, which is the disturbing part of M's effect. In this estimation, it will be found then that two distinct disturbing powers originate; one, which we shall call the tangential force, acting in the direction P Q, perpendicular to S P, and therefore in that of a tangent to the orbit of P, supposed nearly a circle-the other, which may be called the radial disturbing force, whose direction is always either to or from S.

(541.) It is the former alone (art. 419.) which disturbs the equable description of areas of P about S, and is therefore the chief cause of its angular deviations from the elliptic place. For the equable description of areas depends on no particular law of central force, but only requires that the acting force, whatever it be, should be directed to the center; whatever force does not conform to this condition, must disturb the areas.

(542.) On the other hand, the radial portion of the disturbing force, though, being always directed to or from the center, it does not affect the equable description of areas, yet, as it does not conform in its law of variation to that simple law of gravity by which the elliptic figure of the orbits is produced and maintained, has a tendency to disturb this form; and, causing the disturbed body P, now to approach the center nearer, now to recede farther from it, than the laws of elliptic motion would warrant, and to have its points of nearest approach and farthest recess otherwise situated than they would be in the undisturbed orbit, tends to

derange the magnitude, excentricity, and position of the axis of P's ellipse.

(543.) If we consider the variation of the tangential force in the different relative positions of M and P, we shall find that, generally speaking, it vanishes when P is at A or C, see fig. to art. 540. i. e. in conjunction with M, and also at two points, B and D, where M is equidistant from S and P (or very nearly in the quadratures of P with M); and that, between A and B, or D, it tends to urge P towards A, while, in the rest of the orbit, its tendency is to urge it towards C. Consequently, the general effect will be, that in P's progress through a complete synodical revolution round its orbit from A, it will first be accelerated from A up to B-thence retarded till it arrives at C-thence again accelerated up to D, and again retarded till its re-arrival at the conjunction A.

(544.) If P's orbit were an exact circle, as well as M's, it is evident that the retardation which takes place during the description of the arc A B would be exactly compensated by the acceleration in the arc DA, these arcs being just equal, and similarly disposed with respect to the disturbing forces; and similarly, that the acceleration through the arc B C would be exactly compensated by the retardation along C D. Consequently, on the average of each revolution of P, a compensation would take place; the period would remain unaltered, and all the errors in longitude would destroy each other.

(545.) This exact compensation, however, depends evidently on the exact symmetry of disposal of the parts of the orbits on either side of the line CS M. If that symmetry be broken, it will no longer take place, and inequalities in P's motion will be produced, which extend beyond the limit of a single revolution, and must await their compensation, if it ever take place at all, in a reversal of the relations of configuration which produced them. Suppose, for example, that, the orbit of P being circular, that of M were elliptic, and

CHAP. XI. EFFECTS OF THE TANGENTIAL FORCE. 345

that, at the moment when P set out from A, M were at its greatest distance from P; suppose, also, that M were so distant as to make only a small part of its whole revolution during a revolution of P. Then it is clear that, during the whole revolution of P, M's disturbing force would be on the increase by the approach of M, and that, in consequence, the disturbance arising in each succeeding quadrant of its motion, would over-compensate that produced in the foregoing; so that, when P had come round again to its conjunction with M, there would be found on the whole to have taken place an over-compensation in favour of an acceleration in the orbitual motion. This kind of action would go on so long as M continued to approach S; but when, in the progress of its elliptic motion, it began again to recede, the reverse effect would take place, and a retardation of P's orbitual motion would happen; and so on alternately, until at length, in the average of a great many revolutions of M, in which the place of P in its ellipse at the moment of conjunction should have been situated in every variety of distance, and of approach and recess, a compensation of a higher and remoter order, among all those successive over and under-compensations, would have taken place, and a mean or average angular motion would emerge, the same as if no disturbance had taken place.

(546.) The case is only a little more complicated, but the reasoning very nearly similar, when the orbit of the disturbed body is supposed elliptic. In an elliptic orbit, the angular velocity is not uniform. The disturbed body then remains in some parts of its revolution longer, in others for a shorter time, under the influence of the accelerating and retarding tangential forces, than is necessary for an exact compensation; independent, then, of any approach or recess of M, there would, on this account alone, take place an over or under compensation, and a surviving, unextinguished perturbation at the end of a synodic period; and, if the conjunctions always took place on the same point of P's ellipse, this

cause would constantly act one way, and an inequality would arise, having no compensation, and which would at length, and permanently, change the mean angular motion of P. But this can never be the case in

the planetary system. The mean motions (i. e. the mean angular velocities) of the planets in their orbits, are incommensurable to one another. There are no two planets, for instance, which perform their orbits in times exactly double, or triple, the one of the other, or of which the one performs exactly two revolutions while the other performs exactly three, or five, and so on. If there were, the case in point would arise. Suppose, for example, that the mean motions of the disturbed and disturbing planet were exactly in the proportion of two to five; then would a cycle, consisting of five of the shorter periods, or two of the longer, bring them back exactly to the same configuration. It would cause their conjunction, for instance, to happen once in every such cycle, in the same precise points of their orbits, while in the intermediate periods of the cycle the other configurations kept shifting round. Thus, then, would arise the very case we have been contemplating, and a permanent derangement would happen.

(547.) Now, although it is true that the mean motions of no two planets are exactly commensurate, yet cases are not wanting in which there exists an approach to this adjustment. And, in particular, in the case of Jupiter and Saturn,—that cycle we have taken for our example in the above reasoning, viz. a cycle composed of five periods of Jupiter and two of Saturn,-although it does not exactly bring about the same configuration, does so pretty nearly. Five periods of Jupiter are 21663 days, and two periods of Saturn 21518 days. The difference is only 145 days, in which Jupiter describes, on an average, 12o, and Saturn about 5°, so that after the lapse of the former interval they will only be 5° from a conjunction in the same parts of their orbits as before. If we calculate the time which will exactly bring about, on the average, three conjunctions

CHAP. XI. THEORY OF JUPITER AND SATURN.

347

of the two planets, we shall find it to be 21760 days, their synodical period being 7253-4 days. In this interval Saturn will have described 8° 6' in excess of two sidereal revolutions, and Jupiter the same angle in excess of five. Every third conjunction, then, will take place 8° 6' in advance of the preceding, which is near enough to establish, not, it is true, an identity with, but still a great approach to the case in question. The excess of action, for several such triple conjunctions (7 or 8) in succession, will lie the same way, and at each of them the motion of P will be similarly influenced, so as to accumulate the effect upon its longitude; thus giving rise to an irregularity of considerable magnitude and very long period, which is well known to astronomers by the name of the great inequality of Jupiter and Saturn.

(548.) The arc 8° 6' is contained 444 times in the whole circumference of 360°; and accordingly, if we trace round this particular conjunction, we shall find it will return to the same point of the orbit in so many times 21760 days, or in 2648 years. But the conjunction we are now considering, is only one out of three. The other two will happen at points of the orbit about 123° and 246° distant, and these points also will advance by the same arc of 8° 6' in 21760 days. Consequently, the period of 2648 years will bring them all round, and in that interval each of them will pass through that point of the two orbits from which we commenced: hence a conjunction (one or other of the three) will happen at that point once in one third of this period, or in 883 years; and this is, therefore, the cycle in which the " great inequality" would undergo its full compensation, did the elements of the orbits continue all that time invariable. Their variation, however, is considerable in so long an interval; and, owing to this cause, the period itself is prolonged to about 918 years.

(549.) We have selected this inequality as a proper instance of the action of the tangential disturbing force,

« PreviousContinue »