What would be the effect if the body starting at P moved directly away from 1?
Fig. 20.—Different kinds of orbits.
The student should not fail to observe that the sun's attraction tends to pull the body at P forward along its path, and therefore increases its velocity, and that this influence continues until the planet reaches perihelion, at which point it attains its greatest velocity, and the force of the sun's attraction is wholly expended in changing the direction of its motion. After the planet has passed perihelion the sun begins to pull backward and to retard the motion in just the same measure that before perihelion passage it increased it, so that the two halves of the orbit on opposite sides of a line drawn from the perihelion through the sun are exactly alike. We may here note the explanation of Kepler's second law: when the planet is near the sun it moves faster, and the radius vector changes its direction more rapidly than when the planet is remote from the sun on account of the greater force with which it is attracted, and the exact relation between the rates at which the radius vector turns in different parts of the orbit, as given by the second law, depends upon the changes in this force.
When the velocity is not too great, the sun's backward pull, after a planet has passed perihelion, finally overcomes it and turns the planet toward the sun again, in such a way that it comes back to the point P, moving in the same direction and with the same speed as before—i. e., it has gone around the sun in an orbit like P 6 or P 4, an ellipse, along which it will continue to move ever after. But we must not fail to note that this return into the same orbit is a consequence of the last line in the statement of the law of gravitation (p. 54), and that, if the magnitude of this force were inversely as the cube of the distance or any other proportion than the square, the orbit would be something very different. If the velocity is too great for the sun's attraction to overcome, the orbit will be a hyperbola, like P 2, along which the body will move away never to return, while a velocity just at the limit of what the sun can control gives an orbit like P 3, a parabola, along which the body moves with parabolic velocity, which is ever diminishing as the body gets farther from the sun, but is always just sufficient to keep it from returning. If the earth's velocity could be increased 41 per cent, from 19 up to 27 miles per second, it would have parabolic velocity, and would quit the sun's company.
The summation of the whole matter is that the orbit in which a body moves around the sun, or past the sun, depends upon its velocity and if this velocity and the direction of the motion at any one point in the orbit are known the whole orbit is determined by them, and the position of the planet in its orbit for past as well as future times can be determined through the application of Newton's laws; and the same is true for any other heavenly body—moon, comet, meteor, etc. It is in this way that astronomers are able to predict, years in advance, in what particular part of the sky a given planet will appear at a given time.
It is sometimes a source of wonder that the planets move in ellipses instead of circles, but it is easily seen from [Fig. 20] that the planet, P, could not by any possibility move in a circle, since the direction of its motion at P is not at right angles with the line joining it to the sun as it must be in a circular orbit, and even if it were perpendicular to the radius vector the planet must needs have exactly the right velocity given to it at this point, since either more or less speed would change the circle into an ellipse. In order to produce circular motion there must be a balancing of conditions as nice as is required to make a pin stand upon its point, and the really surprising thing is that the orbits of the planets should be so nearly circular as they are. If the orbit of the earth were drawn accurately to scale, the untrained eye would not detect the slightest deviation from a true circle, and even the orbit of Mercury ([Fig. 17]), which is much more eccentric than that of the earth, might almost pass for a circle.
