The Rayleigh Criterion

The theoretical diffraction image, or Airy pattern, of a star, seen in the focal plane of a perfect refracting telescope of aperture D cm, is given by the pattern in Figure 10.1. If a second star, equally bright, and close to the first is also present then two Airy disks and sets of rings are visible.

The Rayleigh criterion is defined as the separation at which the peak of one Airy disk corresponds exactly to

Radius (arcsecond)

the centre of the first dark ring of the other profile. At this point the intensity in the dip between the two profile peaks drops to 73% of the intensity of either peak. In terms of the angular separation of the two stars this is given by 1.22 X/D in radians. In terms of the resolving angular separation, 0res, of the two stars

Because 0res is a small angle, tan 0res « 0res> so a = 122A

res D

but remember that 0res is measured in radians. To convert to seconds of arc, multiply by 206,265.

The power of an objective to separate double stars therefore nominally depends on both the wavelength and the diameter of the objective. For the normal eye

Figure 10.1. The diffraction limited image of a star in a perfect refractor.

the wavelength is that of the peak response, which is usually at 550 nm. So replacing X in the last expression and converting from radians to arcseconds gives the Rayleigh criterion of 13.8/D where D is in cm.

Thus, for a 10-cm refractor, the Rayleigh criterion is 1.38''. This corresponds to a drop in intensity of 27% in the centre of the combined profile, between the two maxima. However, it is possible to see double stars still resolved even if they are closer than this limit. This was first demonstrated, for small telescopes at least, by the Reverend William Rutter Dawes (1799-1868). Dawes says: "I examined with a great variety of apertures a vast number of double stars, whose distances seemed to be well determined, and not liable to rapid change, in order to ascertain the separating power of these apertures, as expressed in inches of aperture and seconds of distance. I thus determined as a constant, that a one-inch aperture would just separate a double star consisting of two stars of the sixth magnitude, if their central distance was 4'.'56; the atmospheric circumstances being moderately favourable."2

Aitken3 points out that it is generally accepted that resolving power rests partly upon a theoretical and partly on an empirical basis. This can be seen in Figures 10.2 and 10.3. In the first, the Rayleigh criterion for a 20-cm refractor is shown with the intensity between the two peaks dropping to 73% of the maximum when the peak of one profile is 0.69'' from the centre of the second profile. Figure 10.3 shows the situation with the Dawes limit demonstrated (the stars are 0.58'' apart in this case). The dip between the peaks is only 3% in this case. The resolution of a double star can therefore depend on the brightness of the stars as it is easier to see a small dip in a bright image than in a faint image.

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