Practically the general reduction of illumination in the fainter stars cuts down the apparent diameters of their discs, and reduces the number of rings visible against the background of the sky.
The scale of the diffraction system determines the resolving power of the telescope. This scale is given in Airy’s original paper (Cambr. Phil. Trans. 1834 p. 283), from which the angle α to any maximum or minimum in the ring system is defined by
sin α = nλ/R
in which λ is numerically the wave length of any light considered and R is the radius of the objective.
We therefore see that the ring system varies in dimension inversely with the aperture of the objective and directly with the wave length considered. Hence the bigger the objective the smaller the disc and its surrounding ring system; and the greater the wave length, i.e. the redder the light, the bigger the diffraction system. Evidently there should be color in the rings but it very seldom shows on account of the faintness of the illumination.
Now the factor n is for the first dark ring 0.61, and for the first bright ring 0.81, as computed from Airy’s general theory, and therefore if we reckon that two stars will be seen as separate when the central disc of one falls on the first dark ring of the other the angular distance will be
Sin α = 0.61 λ/R′
and, taking λ at the brightest part of the spectrum i.e., about 560 μμ, in the yellow green, with α taken for sin α, we can compute this assumed separating power for any aperture. Thus 560 μμ being very nearly 1/45,500 inch, and assuming a 5 inch telescope, the instrument should on this basis show as double two stars whose centres are separated by 1.″1 of arc.
In actual fact one can do somewhat better than this, showing that the visible diameter of the central disc is in effect less than the diameter indicated by the diffraction pattern, owing to the reasons already stated. Evidently the brightness of the star is a factor in the situation since if very bright the disc gains apparent size, and when very faint there is sufficient difficulty in seeing one star, let alone a pair.
The most thorough investigation of this matter of resolving power was made by the Rev. W. R. Dawes many years ago (Mem. R.A.S. 35, 158). His study included years of observation with telescopes of different sizes, and his final result was to establish what has since been known as “Dawes’ Limit.”