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5.1.2. Limitations to the standard model       5.1.4. Seeing error: Strehl, resolution, OTF

5.1.3. Seeing and aperture

Since the atmospherie-induced wavefront error - so called seeing error - changes with (D/r0)5/6, it will vary, for given atmospheric coherence length (Fried parameter) r0, with the aperture size D.

Eq. 52.1 and 52.2 give following averaged RMS errors in the average 2 arc seconds seeing (r0~70mm for 550nm), over the span of amateurs' apertures, with the corresponding Strehl values calculated from the seeing Strehl approximations:
 

ERROR/APERTURE (mm)

50

100

150

200

300

400

500

D/r0

0.71

1.43

2.14

2.86

4.29

5.71

7.14

RMS

Short exposure (roughness)

0.044

0.078

0.109

0.139

0.194

0.246

0.3

Long exposure (tilt + roughness)

0.12

0.22

0.3

0.39

0.54

0.69

0.84

Strehl

Short exposure

0.93

0.79

0.64

0.48

0.24

0.11

0.043

Long exposure

0.61

0.28

0.16

0.11

0.05

0.03

0.02

While these figures appear rather pessimistic, one should keep in mind that they are only averaged values. Statistically, one of many possible break-down's for the dismal 0.69 wave RMS long-exposure error at (D/r0)=5.7 (for 400mm aperture in 2" seeing) is 1.38 and 0.34 wave, half the time each. Likewise, its short exposure error splits to 0.5 and 0.12 wave RMS. Placing the visual error level roughly in between the two would give 0.94 and 0.23 wave RMS. This means that half the time the image would be a real mess, but the other half it would be significantly better, with the visual Strehl of around 0.35, comparable to little over 1/2 wave P-V of spherical aberration.

Also, the tilt error effect is not detrimental at very small apertures (smaller than ~4") where, in average seeing, it causes image to move about randomly as a whole, relatively slowly, without significantly  affecting its visual quality. Thus, in this aperture range, the size of seeing error in visual observing is mainly determined by the wavefront roughness component (Eq. 52.2).

And, since these figures are based on the standard Kolmogorov model, which assumes an infinite outer scale of turbulence, the actual error is somewhat smaller for the long-exposure modality due to the smaller tip-tilt error component with a finite actual scale of turbulence.

As the aperture increases - or the seeing worsens - the eye has more difficulty to keep up with the frequency of image motion, as the diffraction pattern gradually breaks into speckle structure, with an increasing number of  pattern components moving randomly within the speckle envelope. The result is progressive blurring and bloating of the star image. The roughness error component too increases with (D/r0)5/6 until, at a sufficiently large aperture - or in sufficiently bad seeing - diffraction pattern disintegrates into a random speckle structure (FIG. 77). The illustrations show pattern structure as a short-exposure snapshot; physical size of the Airy discs is identical (i.e. focal ratio is the same for all apertures) to make the atmospheric effect directly scalable.

FIGURE 77: Illustration of a point source (stellar) image degradation caused by atmospheric turbulence (linear pattern size, identical f-ratio). Left column shows best possible average seeing error in 2 arc seconds seeing (r0~70mm @ 550nm) for four aperture sizes. The errors are generated from Eq. 53-54, with 2" aperture having only the roughness component (Eq. 54), and larger apertures having tilt component added at a rate of 20% for every next level of the aperture size, as a rough approximation of its increasing contribution to the total visual error (the way it is handled by the human eye is pretty much uncharted territory). Columns to the right show the possible range of error fluctuation, between half and double the average error. Best possible average RMS error is approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect would be identical if the aperture was kept constant, and r0 reduced). The 2" aperture is little affected most of the time;  4" is already mainly below "diffraction-limited", while 8" has very little chance of ever reaching it, even for briefly. The 16" is, evidently, affected the most; D/ro ratio for its x2 error level is over 10, resulting in clearly developed speckle structure (magnification shown is over 1000x per inch of aperture, or roughly 10 to 50 times over the practical limits for 2"-16" aperture range, respectively. Also, since the angular blur size is inverse to the aperture size, the x2 blur in the 16" and 2" aperture are roughly of similar size angularly).

The speckles, being formed by the energy coming from the fragments of the broken wavefront over the pupil that meet in phase at the points randomly scattered around Gaussian image point, approximate in size diffraction disc (this apply to the brightest, so called first-order speckles; fainter speckles are smaller in size). The diameter of the 1st-order speckles envelope, the bright central core of the entire star image, roughly analogous to the FWHM (Full-Width-at-Half-Maximum) of the PSF of aberration-free diffraction image of an aperture of diameter equal to r0, is approximated by λ/r0 in radians. This means that it is r0, not the aperture, that sets the resolution limit. The speckle phenomenon is not only characteristic of large professional telescopes. Even medium-size amateur telescopes in bad seeing begin to develop speckle structure.

Probably the simplest model to use for explaining the mechanism of speckle formation is to assume that the aperture of a telescope is, as a result of atmospheric turbulence, effectively broken into sub-apertures, each approximately of the diameter equal to r0, with a different, varying amounts of tilt. The resulting diffraction pattern is a product of superimposed sub-aperture patterns. Similarly to a multi-aperture telescope, the combined pattern shows tendency to segmentation - mostly in the rings area - with the number of first-order radial segments (spikes, lobes, etc.) approximately equaling that of sub-apertures. Unlike multiple-aperture telescope, the seeing-formed sub-apertures are in an optical disarray, poorly defined and in a constant motion. Thus the increase in number of "sub-apertures" quickly causes disintegration of the bright central disc and formation of the random speckle structure which is stationary only within time intervals smaller than ~20 milliseconds (short-exposure image).

Since speckles originate from wavefront roughness, which is proportional to (D/r0)5/6, their number will be larger for larger (D/r0) values, while their average angular size will be smaller in inverse proportion to the aperture size (thus, fully developed speckle pattern remains of similar angular size, regardless of aperture). The number of 1st-order speckles N can be approximated as a ratio of the area of the speckle envelope's FWHM (approximated, angularly, by λ/r0) vs. area of a single speckle (taking 2λ/D for the average angular separation of speckle centroids), or N~[(λ/r0)/(2λ/D)]2=(D/2r0)2; hence the spackle structure begins to form, at least statistically, at (D/r0)~3, and should become apparent at (D/r0)~4 and larger.

Note this approximation may somewhat overestimate the number of speckles, since based on r0, which is the seeing FWHM that includes the tilt error, i.e. random motion of the speckle structure around its central point.

Formation of the dynamic speckle structure changes the realm of a telescope; it voids its resolution limit, making it dependant on the atmosphere, not aperture size. Energy spread caused by it also significantly lowers telescope's contrast level. Since the error increases with aperture, given large enough difference in aperture size, there comes the level of seeing error which may take away the resolution/contrast advantage from the larger aperture and award it to the smaller one (FIG. 78).

FIGURE 78: Effect of seeing on the visual time-averaged seeing PSF for three relative aperture sizes, referred to, from the smallest to the largest as D, 2D and 5D, respectively. The sizes of seeing disc are only approximate, since the effect of tilt error in visual observing varies from small/negligible at (D/r0)<2 to significant/dominant at larger D/r0 values.

Top image (A) shows the approximate visual size of the central diffraction maxima in the three apertures for zero seeing error, and for D/r0=1 in the smallest aperture., D At this level of seeing error, an aperture is visually somewhat above diffraction-limited level (0.80 Strehl), with its central maxima remaining intact and unchanged in size. In 2D aperture, (D/r0)=2, with significantly more energy transferred to the rings area, but with the central maxima also remaining mainly intact and with no significant change in size (it is shown slightly enlarged). The 2D aperture, therefore, retains most of its resolution advantage, and some of its contrast advantage as well.

In the 5D aperture, (D/r0)=5, at which level the speckle structure clearly shows. Consequently, averaged FWHM radius is approximated by D/2r0 in units of λF. That gives 2.5λF, or five times the aberration-free FWHM for this aperture. In other words, averaged FWHM of the 5D aperture is now nearly identical to that of five times smaller aperture, with similar resolution limit and contrast level (the latter depends also on the specifics of intensity distribution, but for simplicity we'll assume it indicated by the FWHM angular size). However, as seeing fluctuates, it is still possible that (D/r0) falls below ~3, regaining near-intact central maxima and good part of its contrast/resolution advantage.

Bottom image (B) shows these three apertures as seeing worsens by a factor of 2. Now (D/r0)=2 for the smallest aperture, which means that it still has its central maxima near intact most of the time. For the twice larger aperture, (D/r0)=4, with its averaged FWHM grown nearly to that in twice smaller aperture. Since it is still likely to drop bellow (D/r0)=3 with moments of better seeing, it does retain its central maxima near intact some of the time, allowing it do realize its aperture advantage during those moments. However, (D/r0)=10 in the 5D aperture means that it has fully developed speckle structure, with very little chance to fall bellow (D/r0)=3 (it would happen only in seeing more than three times better than the average). With fully developed speckle structure, its resolution limit is approximated by λ/r0; since r0 is 1/10 of its aperture, its resolution now is also 1/10 of its maximum theoretical resolution λ/D, or only about half of the resolution power of five times smaller aperture.

Further doubling the seeing error brings the smallest aperture to (D/r0)=4, and twice larger aperture to (D/r0)=8. The latter is now at a similar level as the 5D aperture in twice better seeing, while the smallest aperture still have realistic chance of falling bellow (D/r0)=3, in which case it out-resolves, and out-contrasts both larger apertures. Yet another doubling of the seeing error takes this advantage of the smallest aperture away - but now an aperture twice smaller than this one will out-resolve and out-contrast the three larger apertures.

This general scenario would play out with, say, 150mm, 300mm and 750mm apertures in 0.8, 1.6 and 3.2 arc seconds seeing (i.e. r0 of 144, 72 and 36mm, respectively).

Image below illustrates how angular size of diffraction pattern and its appearance change with the decrease in atmospheric coherence length r0 (top) and increase in aperture D (bottom).



FIGURE 79: TOP: At fixed aperture size, continuous increase in seeing error causes diffraction pattern to deform, break into speckle structure and greatly expand. BOTTOM: Effect of seeing on the angular diffraction pattern tends to equalize very small and very large apertures in terms of the effective point-image size (i.e. point source resolution), leaving a mid-size aperture with the relatively smallest central maxima.

Decrease in r0 for given aperture results in the progressive break up and enlargement of the diffraction pattern. For given r0, angular pattern size for (D/r0)>3 will be nearly constant, approximated by λ/r0. As D/r0 decreases below 3, the central maxima gradually regains its integrity, and limiting resolution shifts toward λ/D. Despite the central maxima becoming larger than speckles in larger apertures, its diameter becomes smaller than that of their primary speckle structure. Consequently, for 1.5<(D/r0)<2.5, the smaller aperture has better resolution than two or more times larger apertures. At (D/r0)~1, resolution level in the smaller and much larger (five or more times) apertures are similar; further decrease in aperture results in decreased resolution in smaller vs. larger apertures.

If we take for the smallest aperture D=50mm, and r0~100mm (~1.5 arc second seeing), then the bottom patterns, from left to right, correspond to 50, 100, 175, 300, 600mm and 1.2m apertures. Keep in mind that these represent averaged, statistical errors. In actual field conditions, as seeing fluctuates, 175mm aperture will have its D/r0 in the 1.5-2.5 range during periods of better seeing, out-resolving both smaller and larger apertures. Even 600mm aperture is likely to have moments when D/r0 drops to ~2.5, or somewhat bellow; that is, however, very unlikely for 1.2m aperture, which remains inferior to 150-300mm apertures (approximately) resolution-wise.

The degree of pattern integrity directly determines the maximum usable point-source magnification, defined as one needed to achieve limiting visual resolution allowed by seeing. Limiting magnification for extended object is more complex, varying with detail's inherent contrast, brightness and shape. In general, contrast of extended objects is affected more than point-source resolution - and particularly those with low inherent contrast - thus seeing-limited usable magnification for this type of objects should be somewhat lower. An indication of the effect of seeing on contrast and resolution in smaller vs. larger aperture is best presented with MTF graph (FIG. 80).

FIGURE 80: TOP: Taking the near optimum D/r0 stellar-resolution-wise of ~2 and assuming that at this error level eye still filters out most of the tilt error component, would place the corresponding visual MTF closer to that for the short-exposure MTF. Contrast over the range of resolvable bright low-contrast detail would be at the level of a 40% smaller aperture, (dashed gray extending to the horizontal scale) while its threshold for these details would be reduced by about 1/3. Its stellar resolution would  be near its theoretical limit.

On the other hand, a 2.5 times larger aperture at D/r0~5, would have its visual MTF closer to the long-exposure MTF for that D/r0 level. Its low-contrast detail transfer would have been at the level of five times smaller perfect aperture, with its bright low-contrast detail threshold reduced by about 2/3. Its stellar resolution would be about 1/3 of its theoretical, which means that it could be inferior to 2.5 times smaller aperture, which has 40% of its limiting resolution. The approximate mid-frequency contrast levels, at the level of 40% and 80% smaller aperture, indicate that after applying the 2.5 aperture differential factor the larger aperture would still have an edge in bright low contrast detail resolution. It is confirmed by reducing the small aperture visual MTF plot's width 2.5 times (dotted orange), making it directly comparable to the larger aperture's visual MTF. It shows that the smaller aperture has better contrast transfer over all frequencies with non-zero contrast in the larger aperture, becoming significant for frequencies larger than ~0.4. It also has better limiting resolution for high-contrast details (MTF cutoff frequency), but somewhat inferior bright low-contrast detail resolution threshold. That seems illogical at first, but it is a consequence of the threshold line - which represents the minimum contrast required by eye at given normalized frequency - being steeper in the smaller aperture when scaled relative to the larger aperture, since any normalized frequency ν in the latter corresponds to 2.5 times higher frequency in the former, and a higher spatial frequency requires higher minimum contrast.
 
BOTTOM: We shouldn't forget that seeing error varies in time, and that the averaged error is not telling the whole story. MTF plots show the the effect of seeing error on visual contrast transfer for an aperture in (D/r0)~1 seeing (approximately diffraction limited visually) vs. two and four times larger aperture in the same seeing, in what could be considered a common r0 ((i.e. seeing error) fluctuation. Due to the larger average error, the contrast transfer variation range is significantly wider in the larger apertures. Also, for given r0 fluctuation, the larger apertures deviation from the average is wider toward better contrast level, while opposite in the smaller aperture. With the entire fluctuation range of the smaller aperture below the larger aperture average, it can be said that the latter retain some contrast/resolution visual advantage in moderately bad seeing (below D/r0~5), as long as the seeing is the only error source.

     This crude approximation using MTF model indicates that the smaller aperture would probably have relatively small advantage in both, planetary and stellar resolution, when at the seeing error level of D/r0~2. The advantage is small enough that could not materialize if the assumptions are somewhat biased toward smaller aperture.

     However, considering that most other factors - inherent optical quality, thermal equilibrium, absence of central obstruction, better collimation - generally favor smaller aperture, it makes likely the scenario suggested by the size of seeing-affected central maxima, i.e. that an aperture at D/r0 seeing level would outperform both, significantly smaller and significantly larger apertures (plots for twice smaller aperture would fit between D/r0=0 and D/r0=2, thus it is obvious that it would be outperformed by both larger apertures).

     Main difficulty in accessing more accurately effect of seeing on visual performance in apertures of different sizes is caused by the gradual, unspecified transition from seeing error effectively including only the roughness component (short-exposure error level) at small aperture sizes (i.e. at low seeing error levels, approximately D/r0<2) to also fully including the tilt error component at large seeing error level (approximately D/r0>5, long exposure error level). The dynamics of this transition can be only hinted at from the seeing MTF for both error levels over a wide range of seeing-induced error (FIG. 81).



FIGURE 81
: MTF plots for short and long-exposure seeing error, for
D/r0  range 0-10. Points marked at the horizontal scale indicate approximate relative aperture size corresponding to the cutoff frequency for bright low-contrast detail. Note that these MTF are somewhat pessimistic - more so for long than short exposure, but for the latter as well - since based on the standard Kolmogorov model assuming an infinite outer scale of turbulence. In the actual field condition, outer scale of turbulence is always finite, resulting in a reduced seeing error - primarily in the image motion (tilt) component  - and the difference vs. standard model grows larger as the seeing improves.

The very nature of seeing - its constant fluctuation around the average value and, in an extended time frame, fluctuation of the average itself, benefit larger aperture more when swinging above the average. For given seeing level, D/r0 value most of the time remains within 50%, and ~25% deviations can represent appreciable portion of the time. With the seeing average following similar pattern - it can be more, or less at any given night, but this should be reasonable assumption, within the range of a rough average - r0 could reach more than two times larger size, and could be nearly twice larger over relatively significant portion of the time (on the other hand, it would be also more than twice smaller, and nearly twice smaller over nearly as much of the time).

FIGURE 82: Simplistic scheme of the variation in r0 around its averaged value. D/r0 changes in inverse proportion to it, and seeing error in proportion to (D/r0)5/6. For the maximum amplitude of deviation from the average A, the portion of time within which r0 will differ from A by a, or more, is at least roughly approximated with t~0.5-(a/2A),
where a is the deviation from the averaged 
r0 value. For instance, for the likely amplitude of change in r0 of A=50%, the time within which r0 will be 25% or more larger - and the seeing error smaller by a factor of 1/1.255/6, or more - is t~0.5-(25/100)~0.25, or about one quarter of the time. 

In this particular case, with the larger-to-smaller aperture ratio 2.5, a nearly doubled r0 would put the larger aperture close to D/r0~2.5 (now closer to the short-exposure level), and the smaller one close to D/r0~1, nearly coinciding with the short-exposure level. MTF plots above indicate that it would give to the smaller aperture little in improved stellar resolution, since it already was close to 100%, while its planetary contrast would vary from that of about 1/3 smaller aperture for large details to that of nearly 90% smaller perfect aperture for planetary details near resolution threshold. The larger aperture, on the other hand, would regain most of its stellar resolution (near-equal doubles), with its planetary detail contrast at the level of about 2 times smaller perfect aperture, with its planetary detail threshold also comparable to that in a two times smaller perfect aperture. Hence, it would easily outresolve the smaller aperture on the near-equal brightness stars, while also regaining an edge in planetary contrast and resolution (effective relative aperture ~1.25D vs. 0.8D and 0.9D for planetary contrast and resolution, respectively, in the smaller aperture).

What all this implies is that it is possible for a significantly smaller aperture to outperform the larger one, but it requires them to be, and remain at a specific seeing error level, generally involving compromised seeing conditions. It is only possible when D/r0 in the larger aperture is ~4, or larger, in which case a smaller aperture with D/r0~2 will have - all else equal - better resolution and contrast transfer. In other words, smaller aperture could perform better while the seeing error is large enough, but if it lessens sufficiently due to seeing fluctuations, larger aperture would rebound and perform better. Typically, seeing fluctuations are wide enough for that to happen.

This is, of course, very simplified concept, omitting the role of other induced errors (thermals, collimation), inherent optics errors, as well as observer's individual limitations and experience, but it is necessary to assess the effect of seeing alone. Other factors can also come into play, such as luminosity of the telescopic image while observing low-contrast details. Apertures smaller than about 6 inches in diameter will provide less than optimum image brightness at high magnifications, which negatively affects eye's low-contrast detail resolution.

Following page addresses other important aspects of the seeing error: specifics of the seeing PSF, Strehl, and OTF.


5.1.2. Limitations to the standard model       5.1.4. Seeing error: Strehl, resolution, OTF

 

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