How to See the Diffraction Limit of any Telescope

Seeing is in some respect an art, which must be learnt

(William Herschel, 1782)

The Airy diffraction pattern is not easy to observe astronomically in its full and perfect glory - practically, never in anything other than a small telescope (less than about 5 or 6 inches in aperture, the comments below referring primarily to larger instruments) under virtually perfect seeing conditions. Otherwise the best one can hope for is a partial, flickering view which it may take long experience as a telescope user to recognise as "diffraction" rather than seeing blur: it took this author over 20 years with the same 12.5-inch mirror. The rings, in particular, are incredibly sensitive to atmospheric distortion, incomparably more so than the diffraction disk itself, and simply vanish without legible trace in Newtonians of typical amateur size, the moment the seeing falls below I or II (Antoniadi). It is therefore of great value to have a means of displaying these and related effects at the level of the telescope's limiting resolution much more clearly, and so to train the eye to see structure at this level.

The first stage is to learn just what the resolution limit of one's telescope actually looks like, just how tiny this really is, how very much smaller than the usual star-image as seen on 90% of nights. It is very easy to go on using a telescope for years, especially if only using powers up to 20 or 25 per inch of aperture, firmly under the impression that the "splodge" one sees a star as at best focus on typical nights is the diffraction disk and that, even if not, there will be no finer level of structure visible in the image. This is wrong even as a rough approximation, but may be a difficult lesson to unlearn and require a change of observing habits. The agitated "fried egg" which one sees in apertures over 6 inches on all except the very finest nights is nothing whatever to do with the true diffraction image, either as to size or structure. Nevertheless, on all except the worst nights, the true limiting-resolution star disk is visible, buried in the heart of the obvious image,

Figure 11.5. The

Rayleigh limit aperture mask.

Figure 11.5. The

Rayleigh limit aperture mask.

quite accessible (at least in "flashes") to a trained and sufficiently agile eye, perhaps a factor of five smaller than the "splodge". However, no amount of general stargazing will bring about this training of the eye, for which specific exercises are required.

A great aid to this first step of adjusting the eye to the scale of the true diffraction image is a simple aperture mask, of the form shown in Figure 11.5, cut from a sheet of any stiff, opaque material and placed over the aperture (diameter D) of the telescope. Each of the segments symmetrically cut out of the mask is bounded by a circular arc of diameter D struck from a centre P where OP = 0.820 D. A fabrication accuracy of ± 1/16 inch is perfectly adequate.

With this applied to the telescope, one has a Michelson stellar interferometer specifically designed to produce interference fringes having a spacing exactly equal to the Rayleigh diffraction limit 1.22^/D for that telescope. Observe a first magnitude star (not a close double!) with this at a power of at least 40 per inch of aperture (40D), focussing carefully. This time, it is not necessary to wait for a night of first-class seeing, as the interference fringes "punch through the seeing" to an extraordinary degree, a surprising and rather curious fact commented on by many users of the interferometer since Michelson himself in 1891. What you will see is an enlarged and elongated diffraction disk divided into extremely fine bright fringes, perhaps as many as 10 or 11 in all (see Figure 11.6). Unless you have done something like this before, you will probably be surprised at how small this scale of image structure is: in all probability a lot smaller than the star images

Figure 11.6. The spacing of these fringes equals the resolution limit of the telescope at full aperture, according to the Rayleigh criterion.

usually seen in the same telescope. The magnification required to separate these fringes clearly will depend on your visual acuity and this observation provides an interesting opportunity to test the question of so-called "resolving magnification". The majority of observers will almost certainly find that the commonly alleged figure of 13D to 15D is hopelessly inadequate and some may need 50D or more.

Having accustomed the eye to the appropriate scale of image structure, the next stage is to become thoroughly familiar with the Airy diffraction pattern itself. This is made much easier if the pattern is enlarged relative to the scale of the seeing by use of a series of circular aperture stops reducing the telescope's entry pupil to D/4, D/2, and 3D/4. It is advisable when doing this with any reflector having a central obstruction to make both the D/4 and D/2 stops off-axis in order to keep vignetting by that obstruction to a minimum. On a night of seeing I or II (Antoniadi) focus the telescope on a second or third magnitude star with a power of at least 50D (the author's standard working power for this type of observation is 66D = x825) and keep this same magnification on throughout, while examining the image successively with apertures of D/4, D/2, 3D/4 and D. If the telescope is of good quality and properly collimated, you should have no difficulty at all in seeing a nearly perfect "textbook" Airy pattern with the smallest stop: a big, round central disk (not in the least point-like at this power of 200 or more per inch of aperture used), sharply defined, and surrounded by several concentric diffraction rings, extremely fine even on this power, nicely circular and separated by perfectly dark sky.

Figure 11.6. The spacing of these fringes equals the resolution limit of the telescope at full aperture, according to the Rayleigh criterion.

On running successively through the larger apertures D/2, 3D/4 and D this Airy pattern will shrink dramatically and, unless the seeing and the collimation of the telescope are perfect, it will also suffer a progressive deterioration. The result on full aperture is unlikely to bear much resemblance to the ideal image shown by D/4, even ignoring the difference of scale, partly due to the much greater sensitivity of the larger aperture to atmospherics and "seeing", and partly to the almost inevitable residual coma arising from incomplete colli-mation. Note that equation (11.1) implies that coma at full aperture D will be 16 times that at D/4 for the same offset Q, so that an asymmetry like that shown in Figure 11.2, or worse, will now make its appearance even where none was visible at D/4. Nevertheless, if the night is sufficiently fine, it should be possible with persistence to recognise some trace of the pattern of disk and rings even on full aperture. Now is the moment to return to the business of "hyperfine" collimation discussed earlier, completion of which should result in a perfectly round Airy disk, at least, even though the rings at full aperture are unlikely ever to be as clean as those seen at D/4. The telescope will resolve to the Dawes limit if and only if this state is achieved; if the Airy disk absolutely refuses to come round as a button the instrument is defective and consideration will need to be given to the possible causes of image distortion discussed in the previous section or, in the worst-case scenario, to the imperfections of the main mirror itself.

The final stage of this ocular training programme is to learn to cope with the seeing on more typical nights when the diffraction rings will be so fragmented and perpetually on the jitter as to be completely unrecognisable. Here I refer to seeing down to about III (Antoniadi), the worst at which high-resolution astronomy is possible. But it is not in the end the rings with which we are primarily concerned and the emphasis on them here has been purely for their great sensitivity as a diagnostic tool, for identifying and curing removable coma in the telescope. The real image is the disk and the fundamental point about that is that it is often still there even on second-rate nights when the outer envelope of the seeing blur may reach several times the Dawes limit. Though then quite invisible to an observer not specifically trained to work at the diffraction limit, the Airy disk will time and again reveal itself to a trained eye as an intense nucleus buried in the heart of that seeing blur. The object of the exercises suggested in this section is that it should now be possible, with some further practice on these more typical nights, to do what the untrained eye never could - to pick out the true disk and ignore the atmospheric "noise".

This last stage is perhaps the most difficult, though it should not present great problems if the earlier exercises have been successfully completed, and the requirement now is practice on nights of less than perfect seeing: practice, practice, and more practice. In fact these ocular gymnastics soon become quite easy and instinctive. It is probably in part the lack of such training and consequent failure to distinguish the seeing blur (the gross image outline) from the still visible Airy nucleus which is responsible for the persistent myth that seeing limits ground-level resolution to 1'' at best, and is certainly the origin of some of the more spectacularly absurd figures one sees quoted for alleged image size. This author's experience of typical conditions at a very typical lowland site may be of some interest in this context: using a 12.5-inch Newtonian at 400 feet elevation (130 m) in central England, an equal -'' pair (such as n CrB in May 2000) is steadily separated by a clear space of dark sky at x238 in seeing of only III-II (Antoniadi), while PA measures of pairs at 1.8'' and below are frequently within 2° or so of subsequently verified definitive values even when the seeing is III (e.g. X 138 Psc January 2000 and £ UMa, April 2000, both at x238). These observations prove that the mean angular size even of the gross outline of the image as seen under such very middling conditions is no more than about 0.6'', in the centre of which the smaller Airy nucleus is still fitfully visible. When the seeing improves to I or II this accuracy of PA measures extends down to pairs at 1'' or even slightly below, and this is using the most primitive of home-made micrometers on an undriven alt-azimuth telescope.

When described minutely like this, the business of fine-tuning the capabilities of instrument and observer is perhaps likely to appear a rather arduous road. In fact, this could scarcely be further from the truth, as the training of the eye is essentially once-and-for-all, while one soon drops into a virtually unconscious habit of the collimation procedures described earlier, which then take merely a few minutes at the start of each observing session. While it must be emphasised in the strongest terms that, as Herschel put it, "you must not expect to see at sight", there is no obvious reason why a new observer, starting from scratch and following the programme outlined in this section, should have any difficulty in attaining a fully trained eye within a few months of commencing observations. I believe the value of the approach outlined in this section lies entirely in making that possible - it is certainly not necessary for the process to take the 20 years it took this author (with the same telescope) in the absence of any such detailed guidance!

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  • alicia
    How to find the diffreaction limit on telescope?
    6 months ago

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