Naturally, if the photoelectric method is used to observe occultations, detection of much closer pairs should be possible. For high-speed photometry with millisecond (ms) resolution, separations of a few milliarcseconds (mas) should be detectable, far exceeding the diffraction limit of the telescope being used. However, such observations are not straightforward to analyse, since diffraction and stellar diameter effects dominate the high-resolution light curves. During an occultation event a series of alternating bright and dark fringes, the Fresnel zones, are generated and sweep across the observer during an interval of some 40 ms. The first zone, across which the intensity of the light drops smoothly to zero from a value 1.4 times its pre-occultation level, is about 13 m wide on the surface of the Earth and subtends an angle of about 8 mas at the distance of the Moon. Stars with apparent angular diameter less than about 1 mas will generate a diffraction pattern close to that expected from a point source. Those with diameters significantly greater than this will create patterns that can be considered as the sum of a series of point source diffraction patterns displaced in time relative to each other.5 Thus for high-speed measurement of an occultation event where the diffraction pattern is sampled say at a resolution of 1ms, the characteristics of the resulting light curve will depend upon the diameter of the star. This effect is illustrated in
Figure 18.3 where theoretical light curves are computed for a point source and for a star of angular diameter 6 mas. The light curves illustrated here have been further modified from the purely theoretical ones to take account of a realistic bandwidth of the detector system and non-zero telescope aperture (modelled as 50 cm).
The variation apparent in Figure 18.3 of the shape of the light curve as a function of stellar angular diameter can of course be exploited in the analysis of observed light curves; both the precise time of occultation and the stellar diameter may be estimated by non-linear least squares methods. An initial estimate of the diameter is made, perhaps from previous observations or from theoretical considerations based upon the star's spectral characteristic6 and used to compute an approximate light curve. This is then compared point-by-point with the observed light curve and the differences used to solve for corrections to the initial estimate. The process is repeated until convergence is reached and depending upon the quality and signal-to-noise ratio of the data, precisions of better than 1 mas may be achieved. In practice several other parameters are solved simultaneously with stellar diameter, such
Theoretical light curves for occultation of a point source (dashed curve) and for a star of angular diameter 6 mas (solid curve).
as an estimate of the brightness of the star, the background noise and rate of motion of the lunar limb. A large number of stellar diameter measurements has been obtained by this method and published in the astronomical literature.
The method can readily be used for the analysis and discovery of close double stars. If evidence of duplicity is suspected in an observed light curve, the modelling process is extended in order to compute a theoretical curve by summing two such curves displaced in time and amplitude by the initial estimates of component separation and brightness and lunar limb-rate. The fitting process is identical to the single-star case, except that now two diameters may be estimated along with the parameters of the double star system. The results of such analyses are of course the same as for the visual observation method, in the sense that only the component of the double star separation in the direction of motion of the lunar limb is determined from a single observation. However, separations as small as a few mas are detectable.
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