period in which the angle nt + L0 goes through a revolution. In this case the variation will be simply periodic.
The value of any element of the planet’s motion will generally be represented by the sum of an infinite series of such periodic quantities, having different periods. For example
U = a sin (nt + L0) + b sin (mt + L1) + c sin (kt + L2) &c.
In this case the motion of U, while still periodic, is seemingly irregular, being much like that of a pitching ship, which has no one unvarying period.
In the problems of celestial mechanics the angles within the parentheses are represented by sums or differences of multiples of the mean longitudes of the planets as they move round their orbits. If l be the mean longitude of the planet whose motion we are considering, and l′ that of the attracting planet affecting it, the periodic inequalities of the elements as well as of the co-ordinates of the attracted planet, may be represented by an infinite series of terms like the following:—
a sin (l′ − l) + b sin (2l′ − l) + c sin (l′ − 2l) + &c.
Here the coefficients of l and l′ may separately take all integral values, though as a general rule the coefficients a, b, c, &c. diminish rapidly when these coefficients become large, so that only small values have to be considered.
The most interesting kind of periodic inequalities are those known as “terms of long period.” A general idea both of their nature and of their cause will be gained by taking as a special case one celebrated in the history of the subject—the great inequality between Jupiter and Saturn. We begin by showing what the actual fact is in the case of these two planets. Let fig. 3 represent the two orbits, the sun being at C. We know that the period of Jupiter is nearly twelve years, and that of Saturn a little less than thirty years. It will be seen that these numbers are nearly in the ratio of 2 to 5. It follows that the motions of the mean longitudes are nearly in the same proportion reversed. The annual motion of Jupiter is nearly 30°, that of Saturn a little more than 12°. Let us now consider the effect of this relation upon the configurations and relations of the two planets. Let the line CJ represent the common direction of the two planets from the sun when they are in conjunction, and let us follow the motions until they again come into conjunction. This will occur along a line CR1, making an angle of nearly 240° with CJ. At this point Saturn will have moved 240° and Jupiter an entire revolution + 240°, making 600°. These two motions, it will be seen, are in the proportion 5 : 2. The next conjunction will take place along CS1, and the third after the initial one will again take place near the original position JQ, Jupiter having made five revolutions and Saturn two.
The result of these repetitions is that, during a number of revolutions, the special mutual actions of the two planets at these three points of their orbits repeat themselves, while the actions corresponding to the three intermediate arcs are wanting. Thus it happens that if the mutual actions are balanced through a period of a few revolutions only there is a small residuum of forces corresponding to the three regions in question, which repeats itself in the same way, and which, if it continued indefinitely, would entirely change the forms of the two orbits. But the actual mean motions deviate slightly from the ratio 2 : 5, and we have next to show how this deviation results in an ultimate balancing of the forces. The annual mean motions, with the corresponding combinations, are as follows:—
|5n′ − 2n||= 0°.40758|
If we make a more accurate computation of the conjunctions from these data, we shall find that, in the general mean, the consecutive conjunctions take place when each planet has moved through an entire number of revolutions + 242.7°. It follows that the third conjunction instead of occurring exactly along the line CQ1 occurs along CQ2, making an angle of nearly 8° with CQ1. The successive conjunctions following will be along CR2, CS2, CQ3, &c., the law of progression being obvious.
The balancing of the series of forces will not be complete until the respective triplets of conjunctions have filled up the entire space between them. This will occur when the angle whose annual motion is 5n′ − 2n has gone through 360°. From the preceding value of 5n′ − 2n we see that this will require a little more than 883 years. The result of the continued action of the two planets upon each other is that during half of this period the motion of one planet is constantly retarded and of the other constantly accelerated, while during the other half the effects are reversed. There is thus in the case of each planet an oscillation of the mean longitude which increases it and then diminishes it to its original value at the end of the period of 883 years.
The longitudes, latitudes and radii vectores of a planet, being algebraically expressed as the sum of an infinite periodic series of the kind we have been describing, it follows that the problem of finding their co-ordinates at any moment is solved by computing these expressions. This is facilitated by the construction of tables by means of which the co-ordinates can be computed at any time. Such tables are used in the offices of the national Ephemerides to construct ephemerides of the several planets, showing their exact positions in the sky from day to day.
We pass now to the second branch of celestial mechanics viz. that in which the planets are no longer considered as particles, but as rotating bodies of which the dimensions are to be taken into account. Such a body, in free space, not acted on by any force except the attraction of its several parts, will go on rotating for ever in an invariable direction. But, in consequence of the centrifugal force generated by the rotation, it assumes a spheroidal form, the equatorial regions bulging out. Such a form we all know to be that of the earth and of the planets rotating on their axes. Let us study the effect of this deviation from the spherical form upon the attraction exercised by a distant body.
We begin with the special case of the earth as acted upon by the sun and moon. Let fig. 4 represent a section of the earth through its axis AB, ECQ being a diameter of the equator. Let the dotted lines show the direction of the distant attracting body. The point E, being more distant than C, will be attracted with less force, while Q will be attracted with a greater force than will the centre C. Were the force equal on every point of the earth it would have no influence on its rotation, but would simply draw its whole mass toward the attracting body. It is therefore only the difference of the forces on different parts of the earth that affects the rotation.
Let us, therefore, divide the attracting forces at each point into two parts, one the average force, which we may call F, and which for our purpose may be regarded as equal to the force acting at C; the others the residual forces which we must superimpose upon the average force F in order that the combination may be equal to the actual force. It is clear that at Q this residual force as represented by the arrow will be in the same direction as the actual force. But at E, since the actual force is less than F, the residual force must tend to diminish F, and must, therefore, act toward the right, as shown by the arrow. These residual forces tend to make the whole earth turn round the centre C in a clockwise direction. If nothing modified this tendency the result would be to bring the points E and Q into the dotted lines of the attraction. In other words the equator would be drawn into coincidence with the ecliptic. Here, however, the same action comes into play, which keeps a rotating top from falling over. (See Gyroscope and Mechanics.) For the same reason as in the case of the gyroscope the actual motion of the earth’s axis is at right angles to the line joining the earth and the attracting centre, and without going into the details of the mathematical processes involved, we may say that the ultimate mean effect will be to cause the pole P of the earth to move at right angles to the circle joining it to the pole of the ecliptic. Were the position of the latter invariable, the celestial pole would move round it in a circle. Actually the curve in which it moves is nearly a circle; but the distance varies slightly owing to the minute secular variation in the position of the ecliptic, caused by the action of the planets. This motion of the celestial pole results in a corresponding revolution of the equinox around the celestial sphere. The rate of motion is slightly variable from century to century owing to the secular motion of the plane of the ecliptic. Its period, with the present rate of motion, would be about 26,000 years, but the actual period is slightly indeterminate from the cause just mentioned.
The residual force just described is not limited to the case of an ellipsoidal body. It will be seen that the reasoning applies to the case of any one body or system of bodies, the dimensions of which are not regarded as infinitely small compared with the distance of the attracting body. In all such cases the residual forces virtually tend to draw those portions of the body nearest the attracting centre toward the latter, and those opposite the attracting centre away from it. Thus we have a tide-producing force tending to deform the body, the action of which is of the same nature as the force producing precession. It is of interest to note that, very approximately, this deforming force varies inversely as the cube of the distance of the attracting body.
The action of the sun upon the satellites of the several planets and the effects of this action are of the same general nature. For the same reason that the residual forces virtually act in opposite directions upon the nearer and more distant portions of a planet