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# Perturbations

## moon, earth, disturbing, orbit, force, body, apse, sun, suns and forces

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PERTURBATIONS, in physical astronomy, are the disturbances produced in the sim ple elliptic motion of one heavily body about another, by the action of a third body, or by the non-sphericity of the principal body. Thus, for instance, were there no bodies in space except the earth and moon, the moon would describe accurately an ellipse about the earth's center as focus, and its radius-vector would pass over equal areas in equal times; but only if both bodies be homogeneous and truly spherical, or have their constit uent matter otherwise so arranged that they may attract each other as if each were col lected at sonic definite point of its mass. The oblateness of the earth's figure, therefore, produces pertubations in what would otherwige be the fixed elliptic orbit of the moon. Again, when we consider the sun's action, it is obvious that in no position of the moon can the sun act equally upon both earth and moon; for at new moon, the'moon is nearer to the sun than the earth is, and is therefore more attracted (in proportion to its mass) than the earth—that is, the difference of the sun's actions on the earth and moon is equiva lent to a force tending to draw the moon away from the earth. At full moon, on the other hand, the earth (in proportion to its mass) is more attracted than the moon is by the•sun; and the perturbing influence of the sun is again of the nature of a force tending to separate the earth and moon. About the quarters, on the-other hand, the sun's attrac tion (mass for mass) is nearly the same in amount on the earth and moon, but the direc tion of its action is not the same on the tweebodies, and it is easy to see that in this case the pertubing force tends to bring the earth and moon nearer to each other. For any given position of the moon, with reference to the earth and sun, the difference of the accelerating effects of the sun on the earth and moon is a disturbing force; and it is to this that the pertubations of the moon's orbit, which are the most important, and among the most considerable, in the solar system, are due. {By the word difference, just em ployed, we are of course to understand, not the arithmetical difference, but the resultant of the sun's direct acceleration of the moon, combined with that on the earth reversed in direction and magnitude; as it is only with the relative motions of the earth and moon that we are concerned.] This disturbing force may be resolved into three components; for iustance, we may have one in the line joining the earth and moon, another parallel to the pane of the ecliptic, and perpendicular to the moon's radius-vector, and a third perpendicular to the plane of the ecliptic. The first component, as we have already seen, tends to separate the earth and moon at new and full, and to bring them closer at the quarters; but during a whole revolution of the moon, the latter tendency is more than neutralized by the former; that is. in consequence of the sun's disturbing force, the moon is virtually lass attracted by the earth than it would have been had the sun been absent.

The second component mainly tends to accelerate the moon's motion in some parts of its orbit, and to retard it at others. The third component tends, on the whole, to draw the moon towards the plane of the ecliptic. We cannot, of course, enter here into complete sketch of the analysis of such a question as this; but we may give one or two very simple considerations which will, at all events, indicate the nature of the grand problem of perturbations.

The method, originally suggested by Newton, which is found on the whole to be the most satisfactory in these investigations, is what is called the Variation of Parameters, and admits of very simple explanation. The path which a disturbed body pursues is, of course, no longer an ellipse, nor is it in general either a plane curve or re-entrant. But it may be to be an ellipse which is undergoing slow modificationg in form, position, and dimensions, by the agency of the disturbing forces. In fact, it is obvious that any small arc of the actual orbit is a portion of the elliptic orbit which the body would pursue for ever afterwards, if the disturbing forces were suddenly to cease as it moved in that arc. The parameters, then, are the elements of the orbit; that is. its major axis, eccentricity, longlitude of apse, longitude of node, inclination to the ecliptic, and epoch; the latter quantity indicating the time at which the body passed through a particular point, as the apse, of its orbit. If these be given. the is completely known. with the body's n position in it at any given instant. If there be no disturbing forces, all these quantities are constant.; and therefore. when the disturbing forces are taken into account, they change very slowly, as the disturbing forces are in most cases very small. To give an instance of the nature of their changes, let us roughly consider one or two simple cases. First, to find the nature of some of the effects of a disturbing force acting in the radius-vector, and tending to draw the dis turbed, from the central, body. Let S be the focus, P the nearer apse, of the midis. turbed elliptic orbit. When the moving body passes the point M, the tendency of the disturbing force is to make it describe the dotted curve iu the figure—i. e., the new direction of motion will make with the line MS an angle more nearly equal to a right angle than before; and therefore the apse Q in the disturbed orbit will be sooner arrived at than P would have been in the undis turbed orbit—that is, the apse nyrtfit.1, or revolves in the contrary sense to that of 31's motion. Similarly, the effect of 31, is also to make the apse regrede to Q,. At 31, and 31,, on the other hand, the tendency is to make the apse progrede. Also, as the velocity is scarcely altered by such a force, the major axis remains unaltered. Thus at 31 the eccentricity is diminished. and at 31, increased, since the apsidal distance is increased at 31, and diminished at M,.

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