136 Light Waves and Theib Uses
tive. This angle is easily calculated. Thus if I represent the wave length and s is the distance between the two slits, then
the angle is equal to ^• Hence, if we know the length
of the light wave (we can take it as one fifty-thousandth of an inch if we choose), by measuring the distance between our slits when the fringes disapj)ear we have the means of measuring the angular distance between double stars. y/ In the case of a single-slit source we can also get some sort of an idea of the conditions which prevail when the fringes disap{>ear. For we may conceive the slit source to be divided into a number of line sources, parallel and adjacent to each other. Then each line source would form its own set of fringes, and when the angle between the two outside lines, i. ethe edges of the slit, is equal to the angle subtended by the distance of the first dark band from the center, the fringes again overlap in such a way as to disappear. The value of this angle is easily found to be So,
supposing that we had such an object in the heavens as a narrow band of light, we have the means of finding its width. If, instead of a slit, we used a circular opening as a source, there is a little more difficulty in the mathematical analysis. In this
case the coefficient of - , instead of being 1 as in the second
case, or i as in the first case, is found to be 1.22. In observing such an object we measure the distance between our two slits when the interference fringes have just vanished, and compute the angular magnitude of the object by using this coefficient. If we knew the distance to the object, we could calculate also its actual diameter.
The curve representing the clearness of the fringes as the slits approach is rather interesting. It varies with the form
Interference Methods in Astronomy 137
of the object viewed. In the case of a double star it falls very rapidly from its maximum to zero; then it rises again, and if the two slits themselves could possibly be infinitely narrow and the light perfectly homogeneous, it would rise to its original value. But because the slits themselves have a certain width, and because the observation is usually made with white light, this second maximum is usually less than the first.
If the source is a single point of light, then the fringes are equally distinct, no matter what the distance between the slits; whereas, when the source is a disc of appreciable angular width, the fringes fade out as the distance between the slits increases, so that there is no possibility of a doubt as to whether we are looking at a point or a source of appreciable size.
Suppose we are looking at a disc of a given diameter through such a pair of slits which are close together. If we gradually increase the distance between the slits, the visibility becomes smaller and smaller until the fringes disappear entirely. As the distance between the slits increases again, the clearness increases, and so on; i. e., there are subsequent maxima and minima which may be measured, if it be considered desirable. It is necessary, however, to measure this distance between the two slits at the time the fringes first disappear; we may measure this distance at the subsequent disappearances if we choose, but it is not essential, for we are able to find the diameter of the object (the distance between two objects in the case of the double star) if we know the distance between the slits at the first disappearance. If, however, we do not know the shape of the source, we must observe at least one more disappearance.
In Fig. 99 the visibility curves which characterize a slit, a uniformly illuminated disc, and a disc whose intensity is greater at the center, are shown. The full curve corresponds to a slit, the dotted one to a disc, and the dashed