VI. Radiative equilibrium: the relation between the spectral energy curve of a star and the law of darkening of the disc towards the limb, special reference to the effects of scattering and the solar spectrum
An examination from a theoretical standpoint of the form of the law of darkening of a stellar disc towards the limb needs no apology at the present time. A knowledge of this law is required in two astronomical studies of the first importance: one is the deduction of the orbits and densities of eclipsing binary stars from observations of the light curves, the other is the deduction of the angular diameters of stars from interferometer measurements. In both cases some assumption has to be made as to the distribution of intensity over the disc before the solution becomes precise. Now the sun is the only star for which the intensity-distribution is at present known in any detail; the observations of Abbot, Fowle and Aldrich, as well as those of earlier investigators, have determined the law of darkening both for the integrated radiation and for the separate wave-lengths. As regards other stars, indirect evidence is indeed provided as to the existence of darkening at the limb by the results for eclipsing variables, since in most cases the darkened solution gives a better agreement between the observed and computed light-curves, but as yet numerical precision as to the amount of darkening is hardly to be expected. The interferometer method of measuring angular diameters is theoretically capable of determining the light distribution also from the position of the second minimum of visibility of the fringes; but here again the realisation of this possibility is a matter for the future. In the absence, therefore, of direct observations, it would seem worth while to investigate the types of darkening predicted by theory, for stars of different temperatures and of different atmospheric constitutions, on suitable assumptions; and to examine also the converse problem, namely, that of the deductions it is possible to make as to the state of the star if its law of darkening is given. It seems the more desirable to elucidate the principles underlying the existence of darkening, since certain misconceptions appear to exist on the subject.