A paraboloidal reflector is the perfect single mirror for light originating at astronomical distances. However, if the source is at a distance of only twice the focal length, a sphere is the perfect surface. At an intermediate distance, the ideal mirror is a prolate spheroid. Thus, three different forms do the best imaging at three different distances. A telescope could perform adequately when star tested with a nearby source yet fail when directed at the distant sky. Even worse, a fine telescope could be unfairly misjudged by failing the test on a source that is much too close. How much is the test disturbed if the source is placed nearby?
The largest effect of using an artificial source close at hand is to induce spherical aberration in the system. In other words, when a perfect paraboloid is forced to peer myopically at a nearby source, it shows spherical overcorrection that is not found when directed skyward. W.T. Welford says the star test should be conducted with an artificial star placed more than 20 focal lengths away, but he also warns his readers to do accurate ray tracing of the optical system before it is used at questionably close source distances (Welford 1987). We can see below that Welford's suggestion of 20 times the focal length is a very good one for normal apertures and focal ratios, but it fails rather badly for fast mirrors.
In the May, 1991, issue of Sky & Telescope, Roger Sinnott traced rays through paraboloidal mirrors and saw how close the source could be placed before unacceptable spherical overcorrection was noticed. This empirically-derived formula, rewritten for the notation used here, is
where F is the focal ratio and D, the aperture diameter. This equation is rewritten to calculate the multiplier of the focal length:
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