Fourier analysis of the light curves of eclipsing variables,XI

The aim of the present paper will be to introduce a new definition of the loss of light suffered by mutual eclipses of the components of close binary systems: namely, as across-correlation of two apertures representing the eclipsing and eclipsed discs.The advantages of such a strategy over the more conventional (geometrical) approach are (a) greater symmetry of the respective expressions; (b) greater affinity of expressions arising from distortion with those expressing the light changes due to eclipses of spherical stars; and (c) greater freedom in dealing with the effects of particular distribution of brightness over the disc of the star undergoing eclipse (generalized limb-darkening), as well as of possible semi-transparency of the eclipsing component (Wolf-Rayet stars!). In point of fact, none of these tasks could be handled with equal ease by any other technique; nor could the corresponding loss of light be so automated by any other approach.In Section 2 which follows brief introductory remarks we shall evaluate the loss of light arising from distribution of brightness within the aperture undergoing eclipse, and appropriate opacity of the occulting disc. In Section 3 we shall take advantage of these new forms of our results to deduce a number of new properties of the eclipse functions — both algebraic and differential — which have so far escaped attention and which are of considerable practical interest. Lastly, in Section 4 we shall generalize the same concepts to the modification of the light changes caused by the departures of the respective apertures from circular forms.It will be shown that all these phenomena can be most naturally described in terms of Hankel transforms of the products of two Bessel functions with orders depending on the physical characteristics (distribution of brightness; opacity) of the two components; while the geometry of the system (i.e., the fractional radiir _1,2 of the two stars; or the inclinationi of their orbit) enter only through their arguments. Such formulation of our problem should bring a theory of the light changes of eclipsing variables in much closer contact with the adjacent parts of physical optics.