7.1.1. TELESCOPE CENTRAL OBSTRUCTION: SIZE CRITERIA

Setting I=0.80 puts the maximum acceptable CO size at ~0.32D according to Eq. 61, and at ~0.35D according to Eq. 61.1. However, it assumes perfect optics. For an actual optical set of the Strehl ratio S higher than 0.80, the minimum acceptable obstruction size for the combined ~0.80 Strehl level for low-to-mid MTF frequencies would be obtained from SI=0.80, with the peak intensity factor for the obstruction, as mentioned, I=(1-ο2)2=1-2ο2+ο4. In this concept, linear obstruction has to be  smaller than 0.35D, so the ο4 factor can be neglected, and the maximum acceptable CO size for the combined ~0.80 Strehl level is:

οmax~[0.5-(0.4/S)]1/2  or, adjusted for better contrast transfer,   
οmax~[0.6-(0.48/S)]1/2                                          (62)

with ο being, as before, the relative obstruction diameter in units of the aperture.

    For the entire range of MTF frequencies, οmax~[1-(0.80/S)]1/2, with the disc brightness becoming non-factor. Following table gives the corresponding c.obstruction sizes for selected optics Strehl values (in units of the aperture diameter).
 

- spherical aberration: ωo = ω(1-ο2)2
- defocus: ωo = ω(1-ο2)
     - coma: ωo = ω[(1-ο2)(1+4ο2+ο4)1/2/(1+ο2)1/2]
     - astigmatism: ωo = ω(1+ο2+ο4)1/2

where ωo and ω are the RMS wavefront error in obstructed and clear aperture, respectively.

Note that these RMS values are with respect to a new reference sphere, best fitted to the portion of wavefront within annulus area (the P-V error also changes, but that is not directly related to image quality). The combined peak diffraction intensity in the presence of aberrations is given by a product of the peak diffraction intensity of aberration-free obstructed aperture, and that corresponding to the RMS wavefront error ω over annulus area, in units of the wavelength, or:

I = (1-o2)2e-(2πω)2       (63)

e being, as before, the natural logarithm base e~2.72. Thus, for instance, a system with 0.37D c. obstruction and 0.074 waves RMS of spherical aberration (0.25 wave P-V) over full surface area of its optics, has the RMS error within annulus reduced to 0.055 waves RMS, resulting in the combined peak intensity of 0.66. That is noticeably better than 0.60 peak intensity that would result from using the unadjusted wavefront RMS.

In effect, in the presence of spherical aberration, CO partly compensates for its damaging effect by reducing the wavefront error. When both CO and inherent wavefront error are large enough, obstructed system can even perform better. For instance, a system with 1/2 wave P-V of lower-order spherical aberration performs better with 50% obstruction than without it (peak diffraction intensity 0.426 vs. 0.395, respectively).

Evidently, these factors may have importance with larger obstruction sizes and wavefront error levels at which the relative change induced by obstruction has appreciable effect (i.e. not too small, and not to large aberration). The effect on defocus error can be significant, the consequence of axial elongation of the central maxima in the presence of obstruction. It makes an aberration-free obstructed telescope less sensitive to defocus by a (1-ο2) factor; hence, from Eq. 25, defocus error in an obstructed telescope, given as P-V wavefront error at best focus, becomes Wd = -(1-ο2)Ld/8F2, with Ld being the longitudinal defocus and F the telescope focal ratio.

The question of the CO size at which its effect becomes insignificant can be answered in a similar manner. For perfect optics, with S=1, it is determined by any chosen Strehl figure SN considered to be the level of negligible image deterioration. Since here SI=SN=I=(1-ο2)2,

οmax~(0.5-0.5SN)1/2     or     οmax~(0.6-0.6SN)1/2        (64)

the latter adjusted for the better contrast transfer efficiency. So, if the desired effective Strehl for resolvable low-contrast details is SN=0.9, the corresponding maximum c. obstruction size (adjusted for better contrast transfer) with aberration-free aperture is οmax=0.24.

For imperfect optics, with the Strehl S<1, but presumably better than S*, it would be determined from

οmax~[0.6-(0.6SN/S)]1/2            (64.1)

also adjusted for better contrast transfer efficiency. If, for instance, the optics Strehl is S=0.95, and the desired Strehl level for resolvable low-contrast details is SN=0.9, the corresponding c. obstruction size is οmax=0.18 (also adjusted for better contrast transfer due to its relatively brighter central maxima vs. that in aberrated aperture).

And for an aberrated optics set with S<SN, valid criterion would be how much of an additional contrast loss of extended details τE, expressed as a ratio number, is found to be either negligible or acceptable. According to it,

 
οmax~(0.6TE)1/2            (64.2)

Taking 5% additional average contrast loss (τE=0.05) on low-contrast details as a reasonable level of hard to notice contrast change, we arrive at the size of obstruction likely to produce negligible effect for most people as ο~0.17 of the aperture diameter. Of course, this applies as well to aberration-free apertures.

As  mentioned, the above consideration is for the left side of the MTF graph, i.e. resolvable low-contrast details. For the entire range of MTF frequencies, the tolerable size of CO is significantly larger, as obtained by replacing (1-ο2)2 factor by (1-ο2). In terms of the additional general contrast loss τG, over the entire range of MTF frequencies, the corresponding relative obstruction size is given by:

οmax~ TG1/2            (64.3)

Thus, while the CO size producing ~20% contrast loss for extended details (τE) is 0.35D, it is as much as 0.45D for the identical drop in general contrast level TG. However, practical importance of the right half of MTF graph for general observing is considerably less than 50%; it mainly limits to splitting near-equal in brightness double stars, and resolving high-contrast line-like features near or beyond diffraction limit (Cassini division, Moon rills). Consequently, extended-detail contrast transfer τE is more relevant indicator of the overall performance level of an obstructed aperture.

More detailed insight into the change of intensity distribution and contrast loss caused by CO is given by the PSF and MTF, respectively (FIG. 67).

FIGURE 67B: LEFT: Contrast transfer in 0.33D obstructed aperture vs. 0.67D unobstructed aperture. When the smaller aperture is placed in moderate-to-good seeing, simulated with 0.26 wave P-V (0.075 wave RMS, the diffraction-limited level) of defocus, with the larger aperture having correspondingly larger seeing error (0.36 wave P-V), larger aperture retains its bright low-contrast (BLC) detail resolution advantage (3 vs. 5), but to a significantly smaller extent than without seeing error (2 vs. 4). The obstructed aperture also loses its edge on contrast transfer toward resolving limit. This implies that the popular concept postulating that the effective contrast/resolution of an obstructed aperture compares to that of unobstructed smaller by the size of linear c, obstruction, expresses - at least at this obstruction level - combined effect of aperture size, obstruction and seeing. Resolution advantage in the larger aperture for the averaged seeing error, becomes more pronounced during the moments of better seeing, but diminishes when seeing worsens.  MIDDLE/RIGHT: The effect of c. obstruction in different aperture sizes. At left, smaller aperture with and without 0.26 wave P-V of defocus to simulate the averaged seeing error, and further deterioration of contrast transfer with the addition of 0.25D c.obstruction. To the right, 2.3 times larger (by diameter) aperture with the same relative obstruction size and doubled seeing error. The effect of c. obstruction on contrast transfer is greater in the smaller aperture (A vs. B), which is to expect considering that the energy transfer induced by obstruction is relatively larger vs. its seeing error level. Interpolating perfect MTF curves to coincide with the lower-to-md frequency contrast transfer sections of unobstructed and obstructed MTF plots indicates that the addition of 0.25D c.obstruction lowers contrast level for extended details over most of the range of resolution from about 0.75D to 0.61D (19% linear aperture reduction) and from 0.43D to 0.37D level (14% linear aperture reduction) in the smaller and larger aperture, respectively. Higher frequency contrast transfer in both apertures is nearly entirely determined by seeing, with little effect due to added obstruction. Loss in the limiting BLC resolution in the averaged seeing is, however, quite small, and slightly higher in the larger aperture (2 vs. 3, approx. 10% vs. 5%), which has already lost significantly more of its theoretical potential due to its larger seeing error. Seeing combined with the obstruction in the larger aperture decrease its extended object contrast level and BLC resolution to that of approximately three times smaller perfect clear aperture. Smaller aperture is only degraded to the level of about twice smaller perfect unobstructed aperture. Hence, they now compare as 2.3D/3 and D/2, or 0.75D vs. 0.5D, with the larger aperture keeping its relative contrast/resolution advantage. This, however, will diminish with the increase in seeing error.