Stargazing: Past and Present

by Sir Norman Lockyer

But whatever curves he chooses he goes to work so that the spherical aberration of the compound lens shall be eliminated as far as possible, and the chromatism in one lens shall be corrected by the other, or in other words, that what is called the secondary spectrum shall be as small as possible; and it is to be feared that this will never be abolished.[^[6]]

Fig. 64.—Images of planet produced by short and long focus lenses of the same aperture giving images of different size, but with the same amount of colour round the edges.

This matter requires a somewhat detailed treatment in order that it may be seen how the four surfaces to which reference has been made are determined.

The chromatic dispersion, in the case of the object-glass, may be roughly stated to be measured by about one fiftieth of the aperture. Suppose for instance the discs, Fig. [64], to represent the image of any object, say the planet Jupiter. Then round that planet we should have a coloured fringe, and the dimensions of that coloured fringe, that is, the joint thickness of colour at A and D, will be found by dividing the diameter of the object-glass used by fifty. Now this is absolutely independent of the focal length of the telescope; therefore one way of getting rid of it is to increase the focal length of telescopes; and as the size of the image depends on focal length, and has nothing whatever to do with aperture, we may imagine that with the same sized object-glass, instead of having a little Jupiter as on the left of Fig. [64], we may have a very large Jupiter, due to the increased focal length of the telescope. Then, it may be asked, how about the chromatic aberration? It will not be disturbed. The aperture of the object-glass remains unaltered, and there is no more chromatic aberration here than in the first case; so that the relation between the visible planet Jupiter and the colour round it is changed by altering the focal length. But as we have seen, we are able by means of a combination of flint and crown glass to counteract this dispersion to a very great extent. How then about spherical aberration?

Up to the present we have assumed that all rays falling on a convex lens are brought to a point or focus, but this is not strictly true, for the edges of a lens turn the rays rather too much out of their course, so that they will not come to a point; just as the rays reflected from a spherical mirror do not form a single focus. The marginal rays will be spread over a certain circular surface, just as the colour due to chromatic aberration covered a surface surrounding the focus. It was explained that for the same diameter of lens the circle of colour remained the same, irrespective of focal length, but in the case of spherical aberration this is not so; it diminishes as the square of the focal length increases; that is to say, if we double the focal length we shall not only halve, but half-halve, or quarter the aberration. Newton calculated the size of the circle of aberration in comparison with that due to colour, and he found that in the case of a lens of four inches diameter and ten feet focus, the spherical aberration was eighty-one and a half times less than that of colour. It is found that by altering the relative curvatures of the surfaces of the lens, this aberration can be corrected without altering the focal length; for any number of lenses can be made of different curvatures on each side but of the same thickness in the middle, so that they have all the same focal length, but the one, having one surface about three times more convex than the other, will have least aberration, so that it is the adaptation of the surfaces of lenses to each other that exercises the art of the optician.

So far we have got rid of this aberration in a single lens; it can also be done in the case of achromatic lenses. The foci of the two lenses in an achromatic combination must bear a certain relation to each other, and the curvatures of the surfaces must also have a certain relation for spherical aberration. In the achromatic lens there are four surfaces, two of which can be altered for one aberration and two for the other. For instance, in the case of the lens, Fig. [45], where the interior surfaces of the lenses are cemented together, although shown separate for clearness, we find that if the exterior surface of the crown double convex lens be of a curvature struck by a radius 672 units in length, and the exterior surface of the flint glass lens to a curvature due to a radius of 1,420 units, the lens will be corrected for spherical aberration, and these conditions leave the interior surfaces to be altered so that the relation between the powers of the lenses is such as to give achromatism.

The flint is as useful in correcting the spherical aberration as the chromatic aberration; for although the relative thicknesses of the flint and crown are fixed in order to get achromatism, still we have by altering both the curvatures of each lens equally, and keeping the same foci, the power of altering the extent of spherical aberration; and it is in the applications of these two conditions that much of the higher art of our opticians is exercised. We have now therefore practically got rid of both aberrations in the modern object-glass, and hence it is that lenses of the large diameter of twenty-five and twenty-six inches are possible.

The nearest approach to achromatism is known to be made when looking at a star of the first or second magnitude, the eyepiece being pushed out of focus towards the object-glass, the expanded disc has its margin of a claret colour. When the eyepiece is pushed beyond the focus outwards the margin of the expanded disc is of a light green colour.

If the object-glass is well corrected for spherical aberration, the expanded discs both within and without the focus will be constituted of a series of rings equally dense with regard to light throughout, with the exception of the marginal ring, which will be a little stronger than the rest.