6.3. Telescope aberrations and object type
For any given nominal aberration, the effect on image quality perceived by the eye will depend on several factors. The most important are:
(1) detail type
(point-like vs. extended,),
It will be helpful to clarify some basic terminology. While the effect of diffraction is often treated as something different than the effect of aberrations, it is in fact one same phenomenon. What is usually called "diffraction" is the diffraction in aberration-free aperture. However, aberrated apertures also produce diffraction pattern - and so do obstructed apertures - it only has different properties. In general, we can say that optical aberrations worsen diffraction effect, and that it is at its minimum in an aberration-free (still not "perfect") aperture.
How much optical aberrations affect image quality is directly dependant on image magnification. Large aberrations may be entirely invisible at low magnifications and, vice versa, quite small aberrations may noticeably impair image quality at high magnifications. In order to establish some general guidelines in respect to the effect of aberrations, it is necessary to start with the visibility of diffraction effects in a perfect aperture.
The two main object types to consider are: (1) point-like (stellar), and (2) extended. Along with their angular size on the retina, important properties of either type are their brightness and inherent contrast, both having great effect on image appearance in either perfect or aberrated aperture (since diffraction minimum itself acts as an aberration even in a perfect aperture). However, the very basic consideration starts with the determinants set by the eye itself.
For the effect of diffraction to be noticed, retinal image has to be large enough for the brain to create an image of finite dimensions. This requires retinal image extending over at least three cone cells. With the angular size of the smallest cones (in the middle of the retina) being ~1/2 arc minute, it is approximately 1.5 arc minutes in diameter. Angular size of the retinal image needed for the average eye to start recognizing its shape is about twice as large, or ~3 arc minutes for bright, contrasty details. For low-contrast details, it is roughly double, or ~6 arc minutes, on average.
Anything smaller than
~1 arc minute appears point-like to the eye, hence neither diffraction
minimum, nor whatever amount of aberration in might be containing in
addition, is not visible. The larger retinal image, the more apparent
will be the effect of aberration on image quality.
6.3.1. Aberrations and point-like objects
Angular size of a point-like object, such as star, in a perfect aperture depends on its brightness on the retina. It is determined by star's apparent brightness and, given light transmission coefficient, aperture size. In brightest stars, the first bright diffraction ring will appear nearly as bright as central disc of the diffraction pattern, with the disc itself being nearly as large as the Airy disc. Since the diameter of the first bright ring is about 1.8 times the Airy disc diameter, or about 8/D in arc minutes for telescope aperture D in mm (1/3D for D in inches), diffraction pattern of a bright star starts appearing non-stellar at a relative magnification M~0.2D for D in mm, or M~5D for D in inches (which is the same as 5x per inch of aperture). The form starts emerging with magnification about doubled, at ~0.4D for D in mm, and ~10D for D in inches. It becomes clearly defined with magnification increased by another factor of ~2.
Stars of average brightness, somewhere between the brightest and faintest visible will have significantly smaller visible central disc - approximately half the Airy disc diameter - with no noticeable ring structure. Since the ring, due to its faintness, is not visible before the disc itself becomes non-stellar, respective magnifications needed to visually recognize such a star as non-point-like object are nearly four times higher than for a bright star. And stars close to the limit of detection will remain tiny patches of light within the range of usable magnification.
This is how the eye perceives diffraction minimum (i.e. diffraction in aberration-free aperture) for point-like objects. Hence, we may conclude that point-object images in an aberration-free aperture do not appreciably differ from perfect as long as magnification remains below 0.2D for the aperture D in mm, or 5D for D in inches. This limit magnification grows exponentially with the apparent (telescopic) star brightness.
Introduction of wavefront aberrations causes transfer of energy from the disc to the ring area. In order to affect appearance of point-object image as seen in aberration-free aperture, this transfer of energy needs to be sufficiently significant in its extent and intensity to produce enlarged bright portion of the diffraction pattern. Very approximately, this begins to take place at the aberration level of ~0.15 wave RMS, and grows roughly in proportion to the error. While the central diffraction disc does not appreciably change in size at ~0.15 wave RMS error level, the surrounding energy pattern becomes brighter and generally somewhat larger. This, in effect, makes image of a point-object that was near-perfect in aberration-free aperture, now appear slightly more blurry; in other words, magnification that will not show effects of aberration is lower than those for aberration-free aperture. However, this will affect mostly brighter stars; those of lower brightness will not change appreciably, because the energy expanded out of the central disc is still too faint to have appreciable visual effect.
This makes larger apertures in general more sensitive to the effect of wavefront errors on star images, due to their higher visual telescopic brightness.
Since the level of aberrations inherent to amateur telescopes is generally well below 0.15 wave RMS, the effect of aberrations on visual quality of star images vs. that in aberration-free aperture may be noticeable, but not significant. However, there are exceptions to this. One is the size of aberrations farther off-axis, which is often significantly higher than 0.15 wave RMS. Another is wavefront error caused by seeing, which can be higher even in medium-size apertures.
Off-axis aberrations are usually coma and astigmatism, with the latter being especially large in conventional eyepieces. Coma is present in all Newtonian reflectors, with the angular size of sagittal coma in the eyepiece, from Eq.17, given by CSa=CS/ƒe=215tanε/F2, in arc minutes (for small ε simply by CSa=3.75ε/F2), ε being the apparent field radius in zero-distortion eyepiece in degrees (for the field edge, ε=AFOV/2, with AFOV being the apparent field of view diameter in a distortion-free eyepiece) and ƒe the eyepiece focal length in mm. Equating this expression with 3, for the apparent size at which sagittal coma just begins to appear not point-like, gives tanε~3F2/215, for the approximate radius of the "coma-free" apparent eyepiece field, defined as the field within which the geometric sagittal coma is not recognizable as not being point-like. For an ƒ/5 mirror, this gives 19° coma-free field radius, regardless of the eyepiece focal length. It is important to note that this relates to star images; extended objects, in particular brightly illuminated low-contrast type (planets), have more stringent coma-free criterion and, consequently, smaller coma-free field.
As a consequence of the magnification factor, i.e. non-point images appearing point-like if magnified to less than ~3 arc minute, visual coma-free field in the eyepiece is larger than "diffraction-limited" field in the image formed by the objective. Since, substituting h=tanεƒe in Eq.17, sagittal coma in terms of eyepiece apparent field and focal length is CS=tanεƒe/16F2 in mm, and the P-V wavefront error is smaller than sagittal coma by a factor of 3F, i.e. W=CS/3F=tanεƒe/48F3, the P-V wavefront error at the boundary of coma-free field in a 20mm eyepiece in an ƒ/5 Newtonian is about 2 wave P-V for 550nm wavelength. It is larger than "diffraction-limited" coma level (0.42 wave P-V) by a factor of 5, which means that the linear coma free field here is also as much larger than the formal "diffraction-limited" field.
This, of course, is very much dependant on the telescopic star brightness: fainter star will not show deformation farther off, while on the bright ones will be apparent somewhat closer to field center.
With 5mm eyepiece and identical AFOV, coma wavefront error at the edge of coma-free field is four times smaller than in a 20mm eyepiece, or ~1/2 wave P-V, just above the "diffraction-limited" 0.42 wave. Sagittal coma is still ~3 arc minutes, but now, due to higher eyepiece magnification, it is appreciably smaller than the Airy disc diameter (given by 4.6F/ƒe, in arc minutes, for 550nm wavelength). The eyepiece focal length at which the two are about identical is found from tanεƒe/48F3=0.00023 (i.e. for coma at field radius ε being at "diffraction-limited" level), giving ƒe~F3/90.6tanε. For an ƒ/5 mirror, it comes to ƒe~4mm, and for shorter focal length eyepieces the "coma-free" field defined by the angular size of sagittal coma below 3 arc minutes becomes smaller than diffraction-limited field. However, the actual coma-free field, as seen in the eyepiece, is always larger, since it requires sagittal coma appreciably larger than the Airy disc to produce noticeable deformation of the central maxima. Approximately, it takes place with sagittal coma larger than the Airy disc by a factor of 1.5, which corresponds to about 1.2 wave P-V coma wavefront error.
Again, these criteria are valid for the stars not too bright nor too faint. Changes in diffraction pattern due to coma error are less noticeable as telescopic star brightness subsides, and the visual "coma-free" field becomes, in effect, larger. Also, this consideration only includes mirror coma, while for the actual field quality in the Newtonian it is eyepiece astigmatism that is significant or dominant factor. Finally, the visual coma-free field, in the sense of no star deformation visible still has large enough coma in its outer portion to significantly lower its performance level with respect to extended objects like Moon, planets and deep sky.
It is similar, in general, with off-axis astigmatism, only the numbers are somewhat different. As mentioned, it is eyepiece astigmatism that usually dominates in visual observing. A typical conventional eyepiece will have roughly between ƒeε2/60F2 and ƒeε2/30F2 P-V waves of Seidel astigmatism, ε being the eyepiece field angle in degrees. A 20mm eyepiece at ƒ/5 will, therefore, have about 1.6 wave P-V of astigmatism at the approximate 10° boundary of the coma-free visual field radius. This already exceeds coma, which is only ~1 wave P-V this far off. Considering that the smallest geometric astigmatic blur is about 15% larger than geometric sagittal coma for given P-V wavefront error (both, geometric and diffraction, for P-V errors larger than ~, and that the two are roughly proportional to the the actual diffraction blur for errors greater than ~1/2 wave, the coma-free field boundary would have visible blur nearly doubled due to the eyepiece astigmatism. In other words, the actual aberration-free visual field would be somewhat smaller.
Also, since astigmatism
RMS wavefront error for given P-V error is nearly 14% larger than that
for coma, the combined loss of energy is likely beginning to noticeably
degrade contrast of extended objects before the aberration shows visible
deformation of the diffraction pattern of brighter stars. This means
that the actual aberration-free field, as determined by these two
aberrations alone, is yet smaller.