the eccentricity of the lunar orbit when the mean distance of the moon from the earth is one. 674. In consequence of the action of the sun, the perigee of the lunar orbit has a direct motion in space. Its mean motion in one hundred Julian years, deduced from a comparison of ancient with modern observations, was 4069°.0395 in 1800, with regard to the equinoxes, which by simple proportion gives 32314.4751 for its tropical revolution, and 32324.5807, or a little more than nine years for its sidereal revolution; hence its daily mean motion is 6' 41". These motions change on account of the secular variation in the motion of the perigee. 675. The anomalastic revolution of the moon is her revolution with regard to her apsides, because the moon moves in the same direction with her perigee; after separating from that point, she only comes to it again by the excess of her velocity. That excess is 4771980.69184 in one hundred Julian years; therefore by simple proportion, the moon's anomalastic year is 27.5546. 676. The nodes of the lunar orbit have a retrograde motion, which may be computed from observation, in the same manner with the motion of the perigee. The mean tropical motion of the nodes in 1800 was 1936°.940733, which gives 67884.54019 for their tropical revolution, and 6793a.42118 for their sidereal revolution, or 3' 10".64 in a day; hence the moon's daily motion, with regard to her node, is 13° 13′ 45′′.534. The motion of the perigee and nodes arises from the disturbing action of the sun, and depends on the ratio of his mass to that of the earth; this being very great, is the reason why the greater axis and nodes of the lunar orbit move so much more rapidly than those of any other body in the system. Lunar Inequalities. 677. The moon is troubled in her motion by the sun; by her own action on the earth, which changes the relative positions of the bodies, and thus affects her motions; by the direct action of the planets; by their disturbing action on the earth, and by the form of the terrestrial spheroid. 678. Previous to the analytical investigations, it may perhaps be of use to give some idea of the action of the sun, which is the principal cause of the lunar inequalities. S The moon is attracted by the sun and by the earth at the same time, but her elliptical motion is only troubled by the difference of the actions of the sun on the earth and on herself. Were the sun at an infinite distance, he would act equally and in parallel straight lines, on the earth and moon, and their relative motions would not be troubled by an action common to both; but the distance of the sun although very great, is not infinite. The moon is alternately nearer to the sun and farther from him than the earth; and the straight line Sm, fig. 96, which joins the centres of the sun and moon, makes angles more or less acute with SE, the radius vector of the earth. Thus the sun acts unequally, and in different directions, on the earth and moon; whence inequalities result in the lunar motions, depending on mES, the elongation of the sun and moon, on their distances and the moon's latitude. E ከዚ fig. 96. When the moon is in conjunction at m, fig. 97, she is nearer the sun than the earth is; his action is position at m', the earth is nearer to the sun than the moon is, and therefore the sun attracts the earth more powerfully than he attracts the moon. The difference of these actions tends also to diminish the moon's gravitation to the earth. In quadratures, at Q and q, the action of the sun on the moon resolved in the direction of the radius vector QE, tends to augment the gravitation of the moon to the earth; but this increment of gravitation in quadratures is only half of the diminution of gravitation in syzigies; and thus, from the whole action of the sun on the moon in the course of a synodic revolution, there results a mean force directed according to the radius vector of the moon, which diminishes her gravity to the earth, and may be determined as follows: 679. Let M, fig. 98, be the moon in her nearly circular orbit nMN; E and S the earth and sun in the plane of the ecliptic; nmN the moon's orbit projected on the same. Then Mm is the tangent of the moon's latitude, and Em her curtate distance. Let SE, Em, be represented by r' and r, and the angle AEm by x, m' being the mass of the sun. The attraction of the sun on the moon at M is m' (SM)2 This force may be resolved into three; one in the direction Mm, which troubles the moon in latitude; another in mE, which, being directed towards the centre E, increases the gravity of the moon to the earth, and does not disturb the equable description of areas; and into a third in the direction mS', the excess of which above that by which the sun attracts the earth disturbs the relative position of the moon and earth. The inclination of the lunar orbit is so small that it may omitted at first, and then the force fig. 99, is resolved into m' (Sm) 2' be Let two, one in the direction mE, which only increases the gravity of the moon, and the other in mS', which disturbs her motion. ma represent this last force, and suppose it resolved into mb and mc. The force mb accelerates the moon in the quadrants CA and DB, and retards her in the other two; the force mc lessens the gravity of the moon. 680. The analytical expression of these forces is readily found. For the action of the sun on the moon in the direction Sm, is but on account of the great distance of the sun, hence the action of the sun on the moon in Sm is m' (pl ―r cos x)=' which, resolved in the direction SE, is m' (Sm)=' is the same resolved in mb. But the force in mE which inm'r creases the moon's gravity to the earth, is evidently -; hence the whole force by which the sun increases or diminishes the gravity of the moon to the earth is, force in mE force in mc, or - m'r (1-3 cos. x). In syzigy x = 0°, or 180°, and cos2 + 1; thus the action of 2m'r the sun in conjunction and opposition is In quadratures x = 90°, or 270°; hence cos x = 0, and the sun's action at these force acting on the moon in the direction of the radius vector. 681. In order to have the ratio of this mean force to the gravity of the moon, we must observe that if E and m be the masses of the earth and moon, E+m is the force that retains the moon in her is the force that retains the earth in its orbit. But these which are the radii vectores of the moon and earth divided by the squares of their periodic times, whence and thus it appears that the mean action of the sun diminishes the gravity of the moon to the earth by its 358th part, for 682. In consequence of this diminution of the moon's gravity by its 385th part, she describes her orbit at a greater distance from the earth with a less angular velocity, and in a longer time than if she were urged to the earth by her gravity alone; but as the force is in the direction of the radius vector, the areas are not affected by it; hence, if her radius vector be increased by its 358th part, and her angular velocity diminished by its 179th part, the areas described will be the same as they would have been without that action. The |
