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The periodic variation in the photon counts induced by the precision Ronchi ruling among the target and field stars imaged in the focal plane of the Thaw refractor has been and continues to be analyzed for phase differences in a purely numerical way. Efforts to develop a theoretical analytical output model for the MAP have so far been thwarted by the lack of an available, readily integrable mathematical function accurately representing the observed extended wing profiles of star images. However, it is shown that such a function exists in the form of a modified Bessel function. A complete theory of the instrument can therefore in principle be constructed.
The derived time dependent output function has one given parameter (the ruling constant) and four adjustable parameters: FWHM image diameter (seeing diameter), semi-amplitude of the star's photon count, the cycle period and the time of zero phase. For each star first order approximations to these adjustable parameters (same for all cycles in a given run) are then improved by the method of differential corrections by solving the linearized equations of condition in a standard least square solution. The least square adjustments may extend over a few cycles or longer, yielding times of zero phase for each star and ultimately mean phase differences between all stars for a given run.
Because the analytical model is capable of fitting the digital output of the MAP with great fidelity and is very flexible (it can accommodate a very wide variation in seeing and sky transparency) it is expected that its application to ongoing astrometric studies will bring about a further increase in the precision of astrometric observations at Allegheny Observatory.
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