The Story of the Heavens

by Robert S. Ball

We have now wrought the facts of observation into a form which enables us to examine into the cause of a movement so systematic. Why is it that each star should seem to describe a small circular path? Why should that path be parallel to the ecliptic? Why should it be completed exactly in a twelvemonth? We are at once referred to the motion of the earth around the sun. That movement takes place in the ecliptic. It is completed in a year. The coincidences are so obvious that we feel almost necessarily compelled to connect in some way this apparent movement of the stars with the annual movement of the earth around the sun. If there were no such connection, it would be in the highest degree improbable that the planes of the circles should be all parallel to the ecliptic, or that the time of revolution of each star in its circle should equal that of the revolution of the earth around the sun. As both these conditions are fulfilled, the probability of the connection rises to a value almost infinite.

The important question has then arisen as to why the movement of the earth around the sun should be associated in so remarkable a manner with this universal star movement. There is here one obvious point to be noticed and to be dismissed. We have in a previous chapter discussed the important question of the annual parallax of stars, and we have shown how, in virtue of annual parallax, each star describes an ellipse. It can further be demonstrated that these ellipses are really circles parallel to the ecliptic; so that we might hastily assume that annual parallax was the cause of the phenomenon discovered by Bradley. A single circumstance will, however, dispose of this suggestion. The circle described by a star in virtue of annual parallax has a magnitude dependent on the distance of the star, so that the circles described by various stars are of various dimensions, corresponding to the varied distances of different stars. The phenomena of aberration, however, distinctly assert that the circular path of each star is of the same size, quite independently of what its distance may be, and hence annual parallax will not afford an adequate explanation. It should also be noticed that the movements of a star produced by annual parallax are much smaller than those due to aberration. There is not any known star whose circular path due to annual parallax has a diameter one-twentieth part of that of the circle due to aberration; indeed, in the great majority of cases the parallax of the star is an absolutely insensible quantity.

There is, however, a still graver and quite insuperable distinction between the parallactic path and the aberrational path. Let us, for simplicity, think of a star situated near the pole of the ecliptic, and thus appearing to revolve annually in a circle, whether we regard either the phenomenon of parallax or of aberration. As the earth revolves, so does the star appear to revolve; and thus to each place of the earth in its orbit corresponds a certain place of the star in its circle. If the movement arise from annual parallax, it is easy to see where the place of the star will be for any position of the earth. It is, however, found that in the movement discovered by Bradley the star never has the position which parallax assigns to it, but is, in fact, a quarter of the circumference of its little circle distant therefrom.

A simple rule will find the position of the star due to aberration. Draw from the centre of the ellipse a radius parallel to the direction in which the earth is moving at the moment in question, then the extremity of this radius gives the point on its ellipse where the star is to be found. Tested at all seasons, and with all stars, this law is found to be always verified, and by its means we are conducted to the true explanation of the phenomenon.

We can enunciate the effects of aberration in a somewhat different manner, which will show even more forcibly how the phenomenon is connected with the motion of the earth in its orbit. As the earth pursues its annual course around the sun, its movement at any moment may be regarded as directed towards a certain point of the ecliptic. From day to day, and even from hour to hour, the point gradually moves along the ecliptic, so as to complete the circuit in a year. At each moment, however, there is always a certain point in the heavens towards which the earth's motion is directed. It is, in fact, the point on the celestial sphere towards which the earth would travel continuously if, at the moment, the attraction of the sun could be annihilated. It is found that this point is intimately connected with the phenomenon of aberration. In fact, the aberration is really equivalent to drawing each star from its mean place towards the Apex of the Earth's Way, as the point is sometimes termed. It can also be shown by observation that the amount of aberration depends upon the distance from the apex. A star which happened to lie on the ecliptic will not be at all deranged by aberration from its mean place when it happens that the apex coincides with the star. All the stars 10° from the apex will be displaced each by the same amount, and all directly in towards the apex. A star 20° from the apex will undergo a larger degree of displacement, though still in the same direction, exactly towards the apex; and all stars at the same distance will be displaced by the same amount. Proceeding thus from the apex, we come to stars at a distance of 90° therefrom. Here the amount of displacement will be a maximum. Each one will be about twenty seconds from its average place; but in every case the imperative law will be obeyed, that the displacement of the star from its mean place lies towards the apex of the earth's way. We have thus given two distinct descriptions of the phenomenon of aberration. In the first we find it convenient to speak of a star as describing a minute circular path; in the other we have regarded aberration as merely amounting to a derangement of the star from its mean place in accordance with specified laws. These descriptions are not inconsistent: they are, in fact, geometrically equivalent; but the latter is rather the more perfect, inasmuch as it assigns completely the direction and extent of the derangement caused by aberration in any particular star at any particular moment.

The question has now been narrowed to a very definite form. What is it which makes each star seem to close in towards the point towards which the earth is travelling? The answer will be found when we make a minute enquiry into the circumstances in which we view a star in the telescope.

The beam of rays from a star falls on the object-glass of a telescope; those rays are parallel, and after they pass through the object-glass they converge to a focus near the eye end of the instrument. Let us first suppose that the telescope is at rest; then if the telescope be pointed directly towards the star, the rays will converge to a point at the centre of the field of view where a pair of cross wires are placed, whose intersection defines the axis of the telescope. The case will, however, be altered if the telescope be moved after the light has passed through the objective; the rays of light in the interior of the tube will pursue a direct path, as before, and will proceed to a focus at the same precise point as before. As, however, the telescope has moved, it will, of course, have carried with it the pair of cross wires; they will no longer be at the same point as at first, and consequently the image of the star will not now coincide with their intersection.

The movement of the telescope arises from its connection with the earth: for as the earth hurries along at a speed of eighteen miles a second, the telescope is necessarily displaced with this velocity. It might at first be thought, that in the incredibly small fraction of time necessary for light to pass from the object-glass to the eye-piece, the change in the position of the telescope must be too minute to be appreciable. Let us suppose, for instance, that the star is situated near the pole of the ecliptic, then the telescope will be conveyed by the earth's motion in a direction perpendicular to its length. If the tube of the instrument be about twenty feet long, it can be readily demonstrated that during the time the light travels down the tube the movement of the earth will convey the telescope through a distance of about one-fortieth of an inch.[42] This is a quantity very distinctly measurable with the magnifying power of the eye-piece, and hence this derangement of the star's place is very appreciable. It therefore follows that if we wish the star to be shown at the centre of the instrument, the telescope is not to be pointed directly at the star, as it would have to be were the earth at rest, but the telescope must be pointed a little in advance of the star's true position; and as we determine the apparent place of the star by the direction in which the telescope is pointed, it follows that the apparent place of the star is altered by the motion of the earth.

Every circumstance of the change in the star's place admits of complete explanation in this manner. Take, for instance, the small circular path which each star appears to describe. We shall, for simplicity, refer only to a star at the pole of the ecliptic. Suppose that the telescope is pointed truly to the place of the star, then, as we have shown, the image of the star will be at a distance of one-fortieth of an inch from the cross wires. This distance will remain constant, but each night the direction of the star from the cross wires will change, so that in the course of the year it completes a circle, and returns to its original position. We shall not pursue the calculations relative to other stars; suffice it here to say that the movement of the earth has been found adequate to account for the phenomena, and thus the doctrine of the aberration of light is demonstrated.

It remains to allude to one point of the utmost interest and importance. We have seen that the magnitude of the aberration can be measured by astronomical observation. The amount of this aberration depends upon the velocity of light, and on the velocity with which the earth's motion is performed. We can measure the velocity of light by independent measurements, in the manner already explained in [Chapter XII]. We are thus enabled to calculate what the velocity of the earth must be, for there is only one particular velocity for the earth which, when combined with the measured velocity of light, will give the measured value of aberration. The velocity of the earth being thus ascertained, and the length of the year being known, it is easy to find the circumference of the earth's path, and therefore its radius; that is, the distance from the earth to the sun.