Central Obstructions and their effect on image contrast
Rodger W. Gordon
The central obstruction (c - o) of a reflecting telescope and its effect on astronomical images is a never ending topic of conversation. The basics of the problem are well known and in general arguments revolve around just how large a c - o can be tolerated in lunar and planetary viewing.
The following table shows the change in the diffraction patterns (Airy disc, diffraction rings) caused by various c - o ratios (secondary dia. to clear aperture).
|C - O||Airy Disc||Rings|
A perfect optical system with a c - o of 33% causes a similar change in the distribution of energy in the spurious disc as 1/4 l (p - v) spherical aberration at the image plane of a perfect unobstructed system. There are subtle differences; the 33% c - o leads to a slightly different distribution of energy within the ring system.
The experimental investigations by Dall and others clearly showed that the c - o ratio should be kept within 1/5 (20%). However, in many instances this is impracticable. Various forms of Cassegrain need 30% to 35% c - o ratios, to elliminate sky flooding. Dall employed a transfer lens system to reduce the size of the c - o in his Dall-Kirkham Cassegrain below 20%, but this is an added complication and no commercial Cassegrain of any type is manufactured in such a fashion to my knowledge.
Theodore Dunham pointed out that the combination of a 33% c - o plus 1/4 l spherical aberration effectively altered the distribution of energy in the spurious disc to the equivalent of a 1/2 l system. In effect such a telescope would not meet the Rayleigh criterion for diffraction limited performance. Commercial Newtonian reflectors are typically fitted with a 20% to 25% c - o ratio flat, and 1/4 l optics. Despite claims that they are diffraction limited, they cannot be. In all but a few exceptional cases there are no commercial Cassegrains (catadioptrics or otherwise) that can be called diffraction limited either, due to the combined effects of the c - o and residual wavefront errors. However many commercial Schmidt-Newtonian or Maksutov-Newtonian telescopes have relatively small c - o's, typically 18% to 23%. If the optics are well corrected, then they will be diffraction limited systems.
W.H. Pickering demonstrated that the brighter diffraction rings caused by c - o's were more susceptible to the effects of mediocre seeing where the entire spurious disc expanded in size compared to an unobstructed system. This resulted in a much more pronounced reduction in image contrast than could be explained by theory alone.
Skilled observers using telescopes of like aperture and optical quality in good seeing, having 10% and 20% c - o's can detect the subtle change in contrast on planetary images. Below 10% c - o, the change is too small to be of any significance, even though the eye may distinguish contrast differences less than 5%.
Image contrast and detail definition are quantified using what is known as the modulation transfer function. The MTF curve below shows that the contrast is decreased by the c-o but only at low to medium spatial frequencies. There is a slight enhancement at higher frequencies. The limiting spatial frequency of 1.0 equates approximately to Dawes' limit. These results are valid only if the instrument is optically good and properly collimated. If it is not the case, the MTF curve is squashed, the resolution limit is lowered and the loss of resolution affects all the objects, including high contrast objects like the Moon.
An empirical rule deduced from these results is that an instrument of diameter D and obstruction d, in the low frequencies, is equivalent to an unobstructed instrument whose effective diameter Deff is:
Deff = D - d
(this rule is slightly pessimistic for small obstructions).
The changes in a lunar or planetary image caused by a c - o are mostly to low contrast features. High contrast detail, e.g. Cassini's division, Galilean satellite shadow transits, sunspot umbras or black shadows in lunar craters, are not noticeably effected. Detail that is effected typically includes faint Jovian festoons and "white" spots, belts and zones ; sub-divisions in Saturn's rings, quasi-linear features on Mars, sunspot penumbras and the terminator shading and cusp-cap collars in the Venus atmosphere.
Many tyros are puzzled to learn that the resolving power of a telescope increases as the c - o ratio increases. For example a 33% c - o reduces the diameter of the Airy disc by roughly 10%. However this enhancement in the resolving power is only realised on equal or nearly equal doubles. Light removed from the Airy disc is dumped into the ring system, and causes a reduction in the contrast of detail in extended images, and a concommitent reduction in the resolving power of the same telescope on markedly unequal doubles. (Dall 1937 - Paul 1965).
To the c - o's effects must be added the effect of the 3 or 4 vane spider that supports the secondary. The brightness of the diffraction spikes produced by the spider are directly proportional to the ratio of the spider vane thickness and the clear aperture, and their length inversely so. The smaller the vane thickness to aperture ratio, the fainter the spikes are, but the longer they are. A spider, although it creates diffraction spikes on bright stars, has no visible influence on the contrast of the lunar and planetary images. For high resolution, it is therefore not necessary to use tricks like curved arms. For reflectors with a 20% c - o and a four vaned spider, the reduction in image contrast is comparable to that caused by the secondary spectrum of a mid-range f/12 to f/16 achromatic doublet. However minus violet filtration can effectively clean up this residual chromatic error.
For very large apertures (by ATM standards) say 24-inches and greater, the effects of a c - o of a given ratio are to some extent mitigated by the tighter spurious disc size, which is almost always smaller than the seeing disc. In the case of the 200-inch Hale reflector the seeing disc is typically 0".3 to 0".5 arc, but the spurious disc is only 0".027 arc. This illustrates how the effects of a c - o manifest themselves to greatest extent in diffraction as opposed to seeing limited apertures.
Since seeing is so variable, it is instructive to set up test charts in controlled conditions and note the alteration in resolving power, comparing results using both a high and low contrast chart (0.9 [b/w] & 0.17 [grey] contrast index). Using a 3 1/2-inch Questar (c - o 30%), the theoretical resolving power was obtained using the 0.9 ci chart but a reduction of 0".2 arc was noted using the 0.17 ci chart.
Planetary markings often have contrast differences below 20% and upon such markings the deliterious effects of a c - o are most felt. The effects of c - o's of various sizes have been photographed by Horace E. Dall. To examine these images one should consult either Horace E. Dall, "Diffraction Effects due to Axial Obstructions in Telescopes", Journal of the British Astron. Assoc., Vol. 48, No.4 p163, or "Telescopes for Skygazing" by Henry E. Paul - Amphoto 1965. There is a useful article posted on the APM website.
As webmaster of this site I would like to add some pertinent comments to Rodger W. Gordon's sage advice on central obstructions and their deliterious effects on image contrast in extended objects.
Whereas it is perfectly correct that the effect of a 20% c - o is comparable with the secondary spectrum of a mid-range f/12 to f/16 achromatic doublet, one has to bear in mind that such a doublet will only achieve a 1/4 l chromatic correction between the C & F lines for apertures below 5-inches. The rule of thumb for this correction (an f/ratio 3 times per inch of aperture) becomes impractical beyond 6-inches. From my own personal experiences in using a 6 1/2-inch f/13.5 Cooke refractor, my Quantum 6 Maksutov with a 33% c - o but otherwise excellent optics, and my 10-inch f/10.6 Calver which has a 2-inch minor axis flat (20% c - o), I would say that my Calver outperforms both the Cooke and the Quantum in terms of image contrast. However my Q6 easily out performs the Cooke refractor both in terms of raw resolving power, and image contrast.
I have also used on many occasions the 8-inch f/15 Cooke refractor at the Jeremiah Horrocks observatory in Preston, and the 15-inch f/12 Grubb refractor at the Wilfred Hall observatory near Longridge in Lancashire. In my estimation my 10-inch Calver outperforms either of these refractors, both in terms of resolution and image contrast. The problems with achromatic refractors much larger than 6-inches aperture are twofold. Unless one is prepared to adopt a folded system, they become unwieldy, and if the focal ratio is deliberately reduced to make them less so, secondary spectrum becomes a problem, and ruins image contrast. Secondly, large object glasses such as the 15-inch f/12 Grubb referred to, rarely exhibit less than 1/4 l residual spherical aberration, and of course absorb light, and scatter it. None of these refractors of course had coated OG's, which may have made a difference. However unless one is prepared to go the expense and trouble of making either a triplet object glass or a large unobstructed reflector, the long focal ratio Newtonian in my opinion is the optimum telescope beyond 8-inches aperture.
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