Astronomy in a nutshell

by Garrett Putman Serviss

Knowing the manner in which the earth attracts, we have the means of determining its entire mass, or, as it is sometimes called, the weight of the earth. The principle on which this is done is easily understood, Suppose, for instance, that a small ball of lead, of known weight, is brought near a large ball, and delicately suspended in such a way that, by microscopic observation, the movement imparted by the attraction of the large ball can be measured. The force required to produce this movement can be compared with the force of the earth's attraction which produces the weight of the ball, and thus the ratio of the mass of the earth to that of the ball is determined. The total mass of the earth has been found to be equivalent to a “weight” of about 6,500,000,000,000,000,000,000 tons. The mean density of the earth compared with that of water is found to be about 5½, that is to say, the earth weighs 5½ times as much as a globe of water of equal size.

Newton did not stop with showing the manner of the earth's attraction upon bodies on or near its surface; he proved that the earth attracted the moon also, and thus retained it in its orbit. To understand this we must notice another fact concerning the manner in which gravitation acts. Its force varies with distance. Experiment followed by mathematical demonstration, has proved that the variation of the attraction is inversely proportional to the square of the distance. This simply means that if the distance between the two bodies concerned is doubled, the force of attraction will be diminished four times, 4 being the square of 2; and that if the distance is halved, the force will be increased fourfold. Increase the distance three times, and the force diminishes nine times; diminish the distance three times, and the force increases nine times, because 9 is the square of 3, and, as we have said, the force varies inversely, or contrarily, to the change of distance. Knowing this, Newton computed what the force of the earth's attraction must be on the moon, and he found that it was just sufficient to keep the latter moving round and round the earth. But why does not the moon fall directly to the earth? Because the moon had originally another motion across the direction of the earth's attraction. How it got this motion is a question into which we cannot here enter, but, if it were not attracted by the earth (or by the sun), the moon would travel in a straight line through space, like a stone escaping from a sling. The force of the attraction is just sufficient to make the moon move in an orbital path about the earth.

Fig. 5. How the Earth Controls the Moon.
Let C be the centre of the earth and M that of the moon. Suppose the moon to be moving in a straight line at such a velocity that it will, if not interfered with, go to A in one day. Now suppose the attraction of the earth to act upon it. That attraction will draw it to M′. Again suppose that at M′ the moon were suddenly released from the earth's attraction; it would then shoot straight ahead to B in the course of the next day. But, in fact, the earth's attraction acts continually, and in the second day the moon is drawn to M″. In other words the moon is all the time falling away from the straight line that it would pursue but for the earth's attraction, and yet it does not get nearer the earth but simply travels in an endless curve round it.

The same principle was extended by Newton to explain the motion of the earth around the sun. The force of the sun's attraction, calculated in the same way, can be shown to be just sufficient to retain the earth in its orbit and prevent it from travelling away into space. And so with all the other planets which revolve round the sun. And this applies throughout the universe. There are certain so-called double, or binary, stars, which are so close together that their attraction upon one another causes them to revolve in orbits about their common centre. In truth, all the stars attract the earth and the sun, but the force of this attraction is so slight on account of their immense distance that we cannot observe its effects. The reader who wishes to pursue this subject of gravitational attraction should consult more extensive works, such as Prof. Young's General Astronomy, or Sir George Airy's Gravitation.

Photograph of a Group of Sun-spots
Similar groups are frequently seen during periods of sun-spot maximum.

3. The Tides. The tides in the ocean are a direct result of the attraction of gravitation. They also involve in an interesting way the principle that a spherical body, like the earth, attracts and is attracted as if its entire mass were concentrated at its centre. The cause of the tides is the difference in the attraction of the sun and moon upon the body of the earth as a rigid sphere, and upon the water of the oceans, as a fluid envelope whose particles, while not free to escape from the earth, are free to move, or slide, among one another in obedience to varying forces. The difference of the force of attraction arises from the difference of distance. Since the moon, because of her relative nearness, is the chief agent in producing tides we shall, at first, consider her tidal influence alone. The diameter of the earth is, in round numbers, 8000 miles; therefore, its radius is 4000 miles. From this it follows that the centre of the earth is 4000 miles farther from the moon than that side of the earth which is toward her at any time, and 4000 miles nearer than the side which is away from her. Consequently, her attraction must be stronger upon the water of the ocean lying just under her than upon the centre of the earth, and it must also be stronger upon the centre of the earth than upon the water of the ocean lying upon the side which is farthest from her. The result of these differences in the force of the moon's attraction is that the water directly under her tends away from the centre of the earth, while, on the other hand, the earth, considered as a solid sphere, tends away from the water on the side opposite to that where the moon is, and these combined tendencies cause the water to rise, with regard to its general level, in two protuberances, situated on opposite sides of the earth. These we call tides.

Fig. 6. The Tidal Force of the Moon.
The solid earth is represented surrounded by a shell of water. The water on the side toward the moon is more attracted than the centre of the earth, C; the water on the opposite side is less attracted. The lines of force from the moon to the parts of the water lying toward A and B are inclined to the direct line between the centres of the earth and the moon, and the forces acting along these lines tend to draw the water in the directions shown by the arrow points. These are resultants of the horizontal and vertical components of the moon's attraction at the corresponding points on the earth, and the force acting along them tends to increase the weight of the water wherever the lines are inclined more toward the centre of the earth than toward the moon. On the side opposite the moon the same effects are produced in reverse, because on that side the general tendency is to draw the earth away from the water. Consequently if the earth did not rotate, and if it were surrounded with a complete shell of water, the latter would be drawn into an ellipsoidal shape, with the highest points under and opposite to the moon, and the lowest at the extremities of the diameter lying at right angles to the direction of
the moon.