A Text-Book of Astronomy

by George C. Comstock

We have seen that the moon moves around the sun in a path differing but little from the smooth curve shown in [Fig. 53], with arrows indicating the direction of motion, and it would follow absolutely such a smooth path were it not for the attraction of the earth, and in less degree of some of the other planets, which swing it about first to one side then to the other. But action and reaction are equal; the moon pulls as strongly upon the earth as does the earth upon the moon, and if earth and moon were of equal mass, the deviation of the earth from the smooth curve in the figure would be just as large as that of the moon. It is shown in the figure that the moon does displace the earth from this curve, and we have only to measure the amount of this displacement of the earth and compare it with the displacement suffered by the moon to find how much the mass of the one exceeds that of the other. It may be seen from the figure that at first quarter, about July 7th, the earth is thrust ahead in the direction of its orbital motion, while at the third quarter, July 22d, it is pulled back by the action of the moon, and at all times it is more or less displaced by this action, so that, in order to be strictly correct, we must amend our former statement about the moon moving around the earth and make it read, Both earth and moon revolve around a point on line between their centers. This point is called their center of gravity, and the earth and the moon both move in ellipses having this center of gravity at their common focus. Compare this with Kepler's First Law. These ellipses are similarly shaped, but of very different size, corresponding to Newton's third law of motion ([Chapter IV]), so that the action of the earth in causing the small moon to move around a large orbit is just equal to the reaction of the moon in causing the larger earth to move in the smaller orbit. This is equivalent to saying that the dimensions of the two orbits are inversely proportional to the masses of the earth and the moon.

By observing throughout the month the direction from the earth to the sun or to a near planet, such as Mars or Venus, astronomers have determined that the diameter of the ellipse in which the earth moves is about 5,850 miles, so that the distance of the earth from the center of gravity is 2,925 miles, and the distance of the moon from it is 240,000 - 2,925 = 237,075. We may now write in the form of a proportion—

Mass of earth : Mass of moon :: 237,075 : 2,925,

and find from it that the mass of the earth is 81 times as great as the mass of the moon—i. e., leaving kind and quality out of account, there is enough material in the earth to make 81 moons. We may note in this connection that the diameter of the earth, 7,926 miles, is greater than the diameter of the monthly orbit in which the moon causes it to move, and therefore the center of gravity of earth and moon always lies inside the body of the earth, about 1,000 miles below the surface.

95. Density of the moon.—It is believed that in a general way the moon is made of much the same kind of material which goes to make up the earth—metals, minerals, rocks, etc.—and a part of the evidence upon which this belief is based lies in the density of the moon. By density of a substance we mean the amount of it which is contained in a given volume—i. e., the weight of a bushel or a cubic centimeter of the stuff. The density of chalk is twice as great as the density of water, because a cubic centimeter of chalk weighs twice as much as an equal volume of water, and similarly in other cases the density is found by dividing the mass or weight of the body by the mass or weight of an equal volume of water.

We know the mass of the earth ([§ 45]), and knowing the mass of a cubic foot of water, it is easy, although a trifle tedious, to compute what would be the mass of a volume of water equal in size to the earth. The quotient obtained by dividing one of these masses by the other (mass of earth ÷ mass of water) is the average density of the material composing the earth, and we find numerically that this is 5.6—i. e., it would take 5.6 water earths to attract as strongly as does the real one. From direct experiment we know that the average density of the principal rocks which make up the crust of the earth is only about half of this, showing that the deep-lying central parts of the earth are denser than the surface parts, as we should expect them to be, because they have to bear the weight of all that lies above them and are compressed by it.

Turning now to the moon, we find in the same way as for the earth that its average density is 3.4 as great as that of water.

96. Force of gravity upon the moon.—This number, 3.4, compared with the 5.6 which we found for the earth, shows that on the whole the moon is made of lighter stuff than is the body of the earth, and this again is much what we should expect to find, for weight, the force which tends to compress the substance of the moon, is less there than here. The weight of a cubic yard of rock at the surface of either earth or moon is the force with which the earth or moon attracts it, and this by the law of gravitation is for the earth—

W = k · (m m')/(3963)²;

and for the moon—