DIFFRACTION OF LIGHT 235
breadth b the first minimum in the diffraction image lies, by (79) on page 218, at the angle <p = A. : b* If then two spectral lines are to be separated, their dispersion di must at least be greater than this angle 0, which is half the angular width of the central band in the diffraction image of a spectral line; i.e. by (94) the following must hold:
Hence the resolving power of a prism depends only upon the thickness of the prism at the base, and is independent of the angle of the prism. Thus for the resolution of the two sodium lines a prism of flint glass (n = 1.650, dn = 0.000055, A. — 0.000589 mm.) at least I cm. thick is required. But for the resolution of two lines for which d^- : A = 2 : io6, which may be accomplished with the Michelson echelon or with a grating of half a million lines, the thickness of the prism would need to be e •=. 5 • ioa cm., i.e. 5 m., a thickness which is evidently unattainable because of the great absorption of light by glass of such thickness. A grating device permits, therefore, of higher resolving power than a train of prisms.
21. Limit of Resolution of a Telescope. — If a telescope is focussed upon a fixed star, then, on account of the diffraction at the rim of the objective, the image in the focal plane is a luminous disc which is larger the smaller the diameter of the objective. The diffraction caused by a circular screen of radius h gives rise to concentric dark rings. The first minimum
occurs when the angle of diffraction is such that sin 0 = o.6iT-1'
Assume that a second star would be distinguished from the first if its central image fell upon the first minimum of the first star; then the limiting value of the angle which the two stars
* Since b is large in comparison to A, 0 is substituted for sin <f>.
f For the deduction of this number cf. F. Neumann, Vorles. ii. Optik, p. 89,trical to Q with respect to C, must be the image of Q upon the circle.