Star-land: Being Talks With Young People About the Wonders of the Heavens

by Robert S. Ball

THE PARABOLA.

Generally speaking, great comets come to us once and are then never seen again. Such bodies do not move in closed ovals or ellipses, they follow another kind of curve, like that represented in [Fig. 74]. It is one that every boy ought to know. In fact, in one of his earliest accomplishments he learned how to make a parabola. It is true he did not call it by any name so fine as this, but every time a ball is thrown into the air it describes a part of the beautiful curve which geometers know by this word ([Fig. 74]). In fact, you could not throw a ball so that it should describe any other curve except a parabola. No boy could throw a stone in a truly horizontal line. It will always curve down a little, will always, in fact, be a portion of a parabola.

Fig. 74.—The Path of a Projectile is a Parabola.

There are big parabolas and there are small ones. One of the shells which are thrown into a town when bombarded from a distance describes, as it rises and then slopes down again, part of a mighty parabola. So does a tennis ball thrown by the hand or struck by the racket; though here, indeed, I admit that a spin may be given to the ball which will somewhat detract from the simplicity of its movement. In playing baseball, a large part of the skill of the pitcher consists in throwing the ball in such a way that it shall not move in a parabola, but in some twisting curve by which he hopes to baffle his adversary. Setting aside these exceptions, and such another as the case of a body tossed straight up or dropped straight down, we may assert that the path of a projectile is a parabola.

Fig. 75.—The Lighthouse Reflector.

There are some remarkable applications of the same curve for practical purposes. From our lighthouses we want to send beams off to sea, so as to guide ships into port. If we merely employed a lamp without concentrating its rays, we should have a very imperfect lighthouse, for the lamp scatters light about in all directions. Much of it goes straight up into the air, much of it would be directed inland; in fact, there is only an extremely small part of the entire number of rays that will naturally take the useful direction. We therefore require something round the lamp which shall catch the truant rays that are running away to idleness and loss, and shall concentrate them into the direction in which they will be useful to the mariner. An effective way of doing this is to furnish the lamp with a reflector. On its bright surface ([Fig. 75]) all the rays fall which would otherwise have gone astray, and each of them is properly redirected, where the sailors can see it. It is essential that the mirror shall do this work accurately, and this it will only do when it has been truly shaped so as to be a parabola.

You will remember, also, how I described to you the reflector which Herschel made for his great telescope. The shape of the mirror must be most accurately worked, and it, too, must have a parabola for its section; so that you see this curve is one of importance in a variety of ways.

But the grandest of all parabolas are those which the comets pursue. Unlike the ellipse, the parabola is an open curve; it has two branches stretching away and away forever, and always getting further apart. Of course, in the examples of this curve that I have given it is only a small part of the figure that is concerned. When you throw a stone it only describes that part of the parabola that lies between your hand and the spot where the stone hits the ground. It is just a part of the curve in the same way that a crescent may be a bit of a circle. It is to comets that we must look for the most complete illustration of the ample extent of a parabola.

The shape of this grand curve will explain why so many comets only appear to us once. It is quite clear that if you begin to run round a closed racecourse, you may, if you continue your career long enough, pass and repass the starting-post thousands of times. Thus, comets which move in ellipses, and are consequently tracing closed curves, will pass the earth times without number. For this reason we may see them over and over again, as we do Encke’s comet or Halley’s comet. But suppose you were travelling along a road which, no matter how it may turn, never leads again into itself, then it is quite plain that, even if you were to continue your journey forever, you can never twice pass the same house on the roadside. That is exactly the condition in which most of the comets are moving. Their orbits are parabolas which bend round the sun; and, generally speaking, the sun is very close to the turning-point. The earth is also, comparatively speaking, close to the sun; so that while the comet is in that neighborhood we can sometimes see it. We do not see the comet for a long time before it approaches the sun, or for a long time after it has passed the sun. All we know, therefore, of its journey is that the two ends of the parabola stretch on and on forever into space. The comet is first perceived coming in along one of these branches to whirl round the sun; and after doing so, it retreats along the other branch, and gradually sinks into the depths of space.

Why one of these mysterious wanderers should approach in such a hurry, and then why it should fly back again, can be partially explained without the aid of mathematics.

Let us suppose that, at a distance of thousands of millions of miles, there floated a mass of flimsy material resembling that from which comets are made. Notwithstanding its vast distance from the sun, the attraction of that great body will extend thither. It is true the pull of the sun on the comet will be of the feeblest and slightest description, on account of the enormously great distance. Still, the comet will respond in some degree, and will commence gradually to move in the direction in which the sun invites it. Perhaps centuries, or perhaps thousands, or even tens of thousands, of years will elapse before the object has gained the solar system. By that time its speed will be augmented to such a degree, that after a terrific whirl around the sun, it will at once fly off again along the other branch of the parabola. Perhaps you will wonder why it does not tumble straight into the sun. It would do so, no doubt, if it started at first from a position of rest; generally, however, the comet has a motion to begin with which would not be directed exactly to the luminary. This it is which causes the comet to miss actually hitting the sun.

It may also be difficult to understand why the sun does not keep the comet when at last it has arrived. Why should the wandering body be in such a hurry to recede? Surely it might be expected that the attraction of the sun ought to hold it. If something were to check the pace of the comet in its terrific dash round the sun, then, no doubt, the object would simply tumble down into the sun and be lost. The sun has, however, not time to pull in the comet when it comes up with a speed 20,000 times that of an express train. But the sun does succeed in altering the direction of the motion of the comet, and the attraction has shown itself in that way.

I can illustrate what happens in this manner. Here is a heavy weight suspended from the ceiling by a wire; it hangs straight down, of course, and there it is kept by the pull of the earth. Supposing I draw the weight aside and allow it to swing to and fro, then the motion continues like the beat of a pendulum. The weight is always pulled down as near to the earth as possible, but when it gets to the lowest point, it does not stay there, it goes through that point, and rises up at the other side. The reason is that the weight has acquired speed by the time it reaches the lowest point; and that, in virtue of its speed, it passes through the position in which it would naturally rest, and actually ascends the other side in opposition to the earth’s pull, which is dragging it back all the time. This will illustrate how the comet can pass by and even recede from the body which is continually attracting it.

Just a few words of caution must be added. Suppose you had an ellipse so long that the comet would take thousands and thousands of years to complete a circuit, then the part of the ellipse in which the comet moves during the time when we can see it is so like a parabola, that we might possibly be mistaken in the matter. In fact, a geometer will tell us that if one end of an ellipse was to go further and further away, the end that stayed with us would gradually become more and more like this curve. Therefore, some of those comets which seem to move in parabolas may really be moving in extremely elongated ellipses, and thus, after excessively long periods of time, may come back to revisit us.