The flexure of telescope mirror-discs arising from their weight, and its influence upon resolving power

Abstract

The resolving power of a reflecting telescope is proportional to its aperture, the mirror being supposed accurately a paraboloid of revolution, so that a bundle of parallel rays pass through a geometrieal focus after reflection. Regarded from this point of view alone it is of advantage, in the construction of a reflecting telescope, to make the mirror of as large a size as the mechanical difficulties incident to the construction of large mirrors will allow, difficulties, that is to say, such as that of obtaining the glass of the necessary homogeneity throughout, in order to avoid distortion owing to changes of temperature, and the difficulties of grinding, shaping, and polishing. There is, however, another factor to be considered which affects the problem. The mirror, even if perfect in other respects, will, when partially supported, be distorted by its weight, to a greater or less extent according to the nature of the support. Ibis distortion will cause a diminution in the resolving power, the effect of which will evidently increase with the size of the mirror, and so will tend to counteract the advantage accruing from the larger aperture. It is conceivable even that there will, for any given method of support, be a critical aperture, an increase of the size of the mirror beyond which will actually produce a decrease instead of an increase in the resolving power. The object of the present paper is, firstly, to calculate the nature and amount of the distortion which is produced by the weight for various methods of support, and, secondly, to investigate to what extent tins distortion will affect the resolving power of the instrument, and whether any limitation is thereby placed upon the size of the mirrors which are likely to be practically attainable. The types of support which are considered are necessarily comparatively simple and somewhat ideal: the difficulties of the mathematical analysis impose these limitations, but it will be seen that they are sufficient to enable us to give a definite answer to the problem as to whether the critical size of aperture is one which is likely to be reached in the construction of large mirrors, or as to whether this critical size is so large as to be of no practical significance. Since the equations of elastic equilibrium are linear it is sufficient to consider the flexture of the disc for the two cases in which it is horizontal and vertical. The case in which it is inclined at any angle can then be obtained by a combination of these. The notation used throughout will be that used by Prof. Love in his 'Mathematical Theory of Elasticity'.