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▪ CONTENTS ◄ 5.1.1.1. Seeing and aperture ▐ 5.2. Lowlevel turbulence, tube currents... ► 5.1.2. seeing error: limiting resolution, Strehl, otf Due to the random nature of seeinginduced error, the Strehl approximation for nonrandom aberrations, given by Eq. 56 becomes inaccurate for larger errors, considerably more than it is for conventional aberrations (FIG. 51). The effect could be mainly due to the roughness error component, whose increase reduces the average relative size of wavefront irregularity, while at the same time increasing its average RMS error. Similarly to ordinary surface roughness, narrow zones and turned edge, the increase in the PV (and thus RMS) error after a certain point begins to drain less energy out of the Airy disc, with the main effect gradually becoming merely spreading the energy out wider. As a result, the increase in the RMS (surface or wavefront) error is followed by a relatively slower Strehl degradation, as given by the relation derived by Racine (The Telescopic PointSpread Function, 1996; its general form is given on FIG. 83):
e being the natural logarithm base (e~2.718, rounded off to three decimals) and φ2 the variance over the pupil. For longexposure, φ2=1.03(D/r0)5/3, and for short exposure φ2=0.132(D/r0)5/3 (note that the constant value here is slightly different than 0.134 with Mahajan). Variance is directly related to the standard deviation, i.e. the RMS error. If φ2=(2πφL)2 is the phase variance across the pupil, then φL=0.162(D/r0)5/6 amounts to the timeaveraged (longexposure, defined as an exposure significantly longer than the coherence time) RMS wavefront error induced by seeing (Eq. 52.1). So, for D/r0=1, the average RMS wavefront error φL=0.162, giving φ2=1.03, and the seeing Strehl S=0.445; for the nominally identical linear RMS wavefront error ω of nonrandom aberrations the Strehl (from Mahajan's approximation) is 0.357. For short exposure (at the level of coherence time, or shorter), the corresponding RMS wavefront error is φL=0.0582(D/r0)5/6 (Eq. 52.2). A simpler longexposure Strehl approximation, for (D/r0)~1 and larger, is S~1/[1+1.23(D/r0)2], and even simpler, for (D/r0) values over 5, is S~1/(D/r0)2. A quick long exposure seeing Strehl approximation, empirically fitted for (D/r0) around 1 and smaller is S~10.55(D/r0).
The exact Strehl values from the Racine's paper are given in the
following table.
Note that the calculations are based on the standard Kolmogorov turbulence model, which assumes an infinite outer scale of turbulence. While the paper claims good agreement with experimental data, it is to expect that these numbers are generally somewhat pessimistic, due to the error reduction resulting from the finite outer scale of turbulence in the actual field condition. Due to increase of the effect of the tilt error component in visual observing, either with the increase in aperture (for given seeing level), or in the seeing error (for given aperture), the resulting visual Strehl gradually shifts from being near identical to the shortexposure Strehl (for D/r0~1, or smaller, such as in very small apertures in the average or better seeing, or smalltomedium apertures in exceptionally good seeing), to be closer to the longexposure Strehl (for D/r0~5 and larger seeing errors). Plot on FIG. 83 illustrates basic relations between the three main forms of seeinginduced error: long exposure, short exposure and visual.
Standard approach to the seeing error is for longexposure. In it, the size of r0 determines angular resolution of large telescopes as α~λ/r0 in radians, or α~206,265λ/r0 in arc seconds (λ being the wavelength of light), as opposed to the "standardized" diffraction resolution given by ~λ/D. While the size of r0 results from a complex function, it can be expressed much more simply in terms of the resolution limit it imposes to large telescopes. For seeing imposed stellar resolution α in radians, and wavelength λ, it is approximated by: r0 ~ λ/α (54) Hence, an average ~2 arc second seeing at the zenith, for λ=0.00055mm, results from r0~57mm (2 arc seconds in radians is 2/206,265). Note that the averaged turbulence limited stellar resolution is often cited as somewhat inferior to the above: α~1.2λ/r0, or α~1.27λ/r0 (Schroeder). Obviously, if these are taken to represent seeing, the value of r0 corresponding to the given nominal seeing figure will be proportionally larger as well. Since r0 is changing in proportion to λ1.2 and (cosγ)0.6, γ being the zenith angle in degrees, a more complete expression for its size is given by:
Thus, the extracted limiting stellar resolution is:
where r0 is the atmospheric coherence length for λ=0.00055mm corresponding to the zenith angle γ. So, for instance, at 30° from zenith (γ=30°) and the wavelength λ=0.0005mm, atmospheric coherence length r0=100mm for 0.00055mm wavelength results in r0'=(0.0005/0.00055)1.2(cos30)0.6=0.77r0=77mm coherence length for 0.0005mm wavelength. The corresponding limiting stellar resolution ("seeing") at this wavelength is α'~0.0005/77 radians, or 1.34 arc seconds (after multiplying by 206,265).
Following table gives rounded off values of
longexposure r0 and
α
in 2 arc seconds seeing for
several values of zenith angle γ, for
λ=0.00055mm. Values are calculated based on seeinglimited resolution
given as
α~λ/r0
in radians (α~206,265λ/r0
arc seconds); for seeing limited resolution given as
α~1.2λ/r0
or
α~1.27λ/r0,
the atmospheric coherence length is larger by a factor equal to the
numerical constant.
For small amateur telescopes over 60mm in aperture diameter, limiting visual stellar resolution for these r0 values will be generally better due to significantly smaller seeinginduced error level. Linear longexposure seeing diameter  an aspect of seeinglimited telescope resolution that can be important in imaging applications  is FWHML=αƒ=λƒ/r0=(D/r0)λF, with ƒ being the system focal length. In the average 2 arc seconds seeing, with r0~57mm, a 300mm aperture with (D/r0) of ~5, the average longexposure seeing FWHM diameter is about 5λF linearly, twice the Airy disc size, and larger than system's aberrationfree FWHM (~λF) by a 1.2 D/r0 factor or, in this case, five times. For short exposures (generally less than 1/20 of a second, specifically at the level of the coherence time, or smaller), the speckle movement caused by wavefront tilt error ceases, and both seeing blur and its FWHM shrink. If, for the purpose of obtaining a general idea of shortexposure FWHM, the tiltcomponent RMS wavefront error  from Eq. 52.12, rounded to 0.1(D/r0)5/6, in units of the wavelength  is assumed to be ~1/3 of the averaged PV tilt error W on the wavefront, averaged angular displacement of the speckle structure due to wavefront tilt error is αt~Wλ/D~0.3(D/r0)5/6λ/D in radians. The corresponding average linear displacement from the center in the image plane is αtƒ, ƒ being the focal length, or 0.3(D/r0)5/6λF. In other words, for given r0, nominal angular tilt error decreases relatively slowly  in proportion to D5/6/D=1/D1/6  with the increase in aperture, but effectively increases relative to the angular size of diffraction pattern, in proportion to D/D1/6. For instance, nominal angular motion of the image will be about 32% smaller in 100cm compared to 10cm aperture, but in terms of the ten times smaller angular pattern of the former, it will be 6.8 times larger. Nominal linear motion for given λF is proportional to (D/r0)5/6. or, for given r0, in proportion to D5/6. Accordingly, linear diameter of the shortexposure FWHM is approximated by deducting 0.6(D/r0)5/6λ from longexposure FWHM, i.e. FWHMS~(D/r0)λF0.6(D/r0)5/6λF. Alternately, it is smaller than longexposure FWHM by a factor 1[0.3(D/r0)5/6/0.5(D/r0)]=10.6(D/r0)1/6. This is only a crude approximation, producing FWHMS value for D/r0=1 well below λF  in part due to FWHML~(D/r0)λF underestimating FWHML value for D/r0 close to 1  but it does indicate two important features of the seeing FWHM: (1) shortexposure FWHM is nearly identical to the aberrationfree FWHM for the low singledigit D/r0 values, and (2) relative longexposure enlargement of the shortexposure FWHM diminishes with D/r0 increase. The variance of the image motion is given by Tyler as φ02=0.34(λ/r0)2(D/r0)1/3 arcsec2, or as standard (RMS) deviation φ0=0.58(λ/r0)(D/r0)1/6 arc seconds, where λ/r0 is the longexposure seeing in radians (206,265λ/r0 in arc seconds). Gaussian approximation of the intensity distribution within the image (atmospheric PSF) is given by I~exp0.5(θ0/φ0)2. Following graph (based on Hardy, 1998) shows the actual relationship between longexposure and shortexposure FWHM angular size, both normalized to λ/r0, as a function of D/r0. Thus, the diffractionlimited plot (blue) shows the size of diffractionlimited FWHM (~λ/D) in units of λ/r0; since r0~D/x, x being the numerical value for D/r0, λ/r0~xλ/D.
FIGURE 84: The plots are showing that the seeing FWHM is significantly smaller for short vs. long exposure. The difference is largest at D/r0~1 or less, gradually diminishing toward larger D/r0 values. Limit to the long exposure resolution is generally somewhat larger than λ/r0, more in accord with 1.2λ/r0 or 1.27λ/r0 However, the specific value of the numerical portion varies with D/r0, being the highest for D/r0~1 (about 1.8), then quickly decreasing toward D/r0~2 (about 1.4), after which it slowly decreases toward 1 with higher D/r0 values. Note that 1.8λ/r0 long exposure FWHM for D/r0~1 (which equals 1.8λ/D, since r0~D) is significantly larger than in some other sources (Mahajan, Bass/DeCusatis, around 1.4λ/r0, following graph). Discrepancies are fairly common, reflecting complexity of the subject. Since the limiting resolution of ~1.8λ/D cannot be considered diffraction limited, this particular discrepancy may be result of a different modeling of the turbulence and/or exposure. Short exposure image becomes significantly larger than diffraction limit as D/r0 exceeds 2, which also indicates larger relative size compared to, for instance, Bass and DeCusatis, who show no appreciable effect on FWHM for this D/r0 value (following graph). The relative image motion plot shows already mentioned tendency of subsiding with larger D/r0 values. However, it diminishes only in terms of λ/r0. The relative amplitude of image motion, in units of the actual FWHM, λ/D, steadily increases, from about 1 at D/r0=1 to about 8 at D/r0=10, and nearly in proportion to the change in D/r0 for its larger values. It is mainly caused by the increase in the tilt error component. The increased amplitude of image motion causes increased blurring of the visual image, as perceived by the eye. Graph below shows how the averaged intensity distribution (PSF) for long and short exposure seeing PSF deteriorate with the increase in seeing error.
With long exposure, ring structure is already lost at (D/r0)=1. The plot asymptotically approaches horizontal axis, thus can be approximated with simple Gaussian functions for any D/r0 value. After relatively small initial FWHM increase for (D/r0)~1 or smaller, its diameter further increases approximately by a factor of 1+(D/r0)λF, for (D/r0) values up to 10, and beyond. Energy encircled within FWHM slowly diminishes from 50% of the total energy at (D/r0)=0 to 40% at (D/r0)=5. Loss of FWHM energy is less of a factor than its enlargement: at (D/r0)=5 the FWHM diameter is ten times larger than in aberrationfree aperture. This results in as much inferior limiting stellar resolution. For unobstructed aberrationfree aperture, radius of the 80% energy circle ranges from 0.9λF for (D/r0)=0 to 5.6λF for (D/r0)=5.
Aberrationfree aperture with D/2
central obstruction diameter is
clearly inferior at
(D/r0)=0,
with the 80% energy radius of 1.9λF;
as seeing error increases, the difference in the circle radius vs.
unobstructed aperture diminishes, nearly disappearing at
(D/r0)~5
and larger.
Numerical aspect of the changes in FWHM and EE are given in table below.
* estimate based on projecting plots beyond (D/r0)=3 in Optical Imaging and Aberrations 2, Mahajan (p418) Consistent with its general effect, presence of central obstruction causes shift of energy outside of central maxima. Expectedly, the effect on PSF and encircled energy generally diminishes with the increase in seeing error (FIG. 86). In part, it is due to the longexposure Strehl  the measure of wavefront quality, independent of the effect of central obstruction  becoming higher for the obstructed aperture as the size of obstruction and D/r0 ratio increase. The effect of obstruction on the Strehl is nearly neutral at (D/r0)~3, negative for smaller, and positive for larger ones. While seemingly contradictory, it could be rationalized by the aberration contribution from the area blocked by obstruction changing vs. that of the annulus area for with the change in (D/r0). For the lower values, i.e. for coherence length larger relative to the aperture, pupil area blocked by obstruction tends to have lower than average wavefront error, thus the annulus alone has it higher. For the smaller relative coherence lengths, however, pupil area blocked by obstruction begins to contribute more to the overall error (in unobstructed aperture), while the narrower annulus has more of the effect analogous to that of aperture reduction.
Short exposure PSF shows no significant FWHM increase even at (D/r0)=5; its intensity distribution retains some form of secondary maxima well beyond this D/r0 value. Its relative intensity of about 0.3 peak intensity means that it appears nearly half as bright to the eye. Since this intensity distribution is a product of significant wavefront roughness, the ring structure is broken into randomly moving segments, already resembling speckle structure. In general, for achieving nearlimiting resolution and contrast, pixel size shouldn't be larger than FWHM/2. This relative pixel size preserves limit to resolution set by FWHM, with pixel contrast transfer being only moderately lowered. For maximizing limiting magnitude, however, it should be somewhat larger than FWHM. A compromise pixel size is in between, depending on which of the two aspects is more important. Even with pixel size optimized for contrast (i.e. not significantly larger than one half of the seeing FWHM), contrast transfer will be lowered due to the spread of energy caused by seeing error  the larger error, the more so.
A fairly popular concept in imaging is the probability of
"lucky exposure", defined as a very
short exposure (less than half coherence time) capturing moment of
diffraction limited seeing, or better. Diffractionlimited is
defined as 1 radian2
RMS phase variance, or less, which corresponds to 1/2π
RMS, or less. For instantaneous exposure, this error level
corresponds to (D/r0)~3.4.
The original Fried's paper gives the following probabilities (P)
for selected D/r0
values:
The paper also gives a fitted equation approximating P (whose exact value is given by a complex integral) as P~5.6exp[0.1557(D/r0)2], where exp stands for the natural logarithm base e~2.72 under the exponent in brackets. It becomes progressively too optimistic for D/r0~6 and smaller, coming to P=1 for D/r0~3.35. For that reason, the paper states that it is valid for D/r0≥3.5. The probability applies only for the area covered by isoplanatic angle (isoplanatic patch), usually less than 10 arc seconds in diameter. Significantly larger image will have sporadic isoplanatic patches with diffractionlimited phase error over them, but the rest of image will have significantly lower quality. In terms of contrast transfer, the effect of atmospheric error is given by atmospheric OTF (Optical Transfer Function), which is for long exposure given by:
and for short exposure by
where exp(...) is ℯ(...), ℯ being the natural logarithm base, 2.718 rounded to three decimals, and ν is the normalized spatial frequency. It is important to note that the short exposure relation is not accurate for higher frequencies, for v~0.5 and larger. It produces higher contrast transfer in mid to high frequency range. It also produces an upward contrast spike between 0.9 and 1 frequency, becoming exponentially more pronounced as D/r0 value increases beyond ~5 (as Mahajan states, it is the consequence of assuming no correlation between tilt and roughness component of the seeing error, which is not strictly correct). The worst part of this highfrequency error  contrast spike up between 0.9 and 1 frequency  is effectively removed simply by using (1ν1/3) instead of (11.042ν1/3). However, it still leaves in somewhat higher contrast in the mid to high frequency range. It can be nearly corrected by further lowering the constant in front of ν1/3 slightly below 1. Combined atmospheric and telescope contrast transfer is given by the product of their respective OTF's. It is the final OTF in the image plane of the objective. Obstructed apertures will have lower combined OTF due to the contrastlowering effect of central obstruction, but not more. In fact, the larger obstruction, the larger relative size of r0 vs. annulus, resulting in the slower increase in phase variance over annulus area and, consequently, slower decrease in the average Strehl (as a measure of wavefront quality, independent of the effect of central obstruction). This beneficial effect partly offsets the negative diffraction effect of central obstruction. Effect of seeing on contrast transfer of a clear aberrationfree aperture is given by a product the seeing OTF and Eq. 57, as shown below.
The OTF plots show that, unlike classical aberrations, contrast drop caused by relatively significant seeing errors continues to increase toward high frequencies, without partial recovery in mid frequencies. This is a consequence of the expanding central maxima, which is the primary form of energy transfer with the seeing error (as opposed to nonrandom, classical aberrations, which mainly transfer energy to the rings, without significantly affecting size of the central maxima). It also causes loss of resolution sooner than classical aberrations that would result in similar contrast level in the lower range of frequencies. Looking at the short exposure graph, as more representative of the visual seeing error in amateursize apertures, it is evident that, contrastwise, larger apertures tend to remain ahead of the smallest ones. The pattern is similar to that for stellar resolution. A 3 times larger aperture has the cutoff (zero contrast) projected from its midtolow frequencies (dashed line) still about 45% of the full resolving range for this aperture size, and 5 times larger aperture about 22% of the full resolving range for its aperture size. Adjusting for normalization (effectively, by compressing the plot horizontally by a ratio smallertolarger aperture diameter, i.e. smallertolarger D/r0 value), the 3 times larger aperture acts as about 30% larger aperture, contrastwise, than the unit (D/r0=1) aperture, and 5 times larger aperture as about 10% larger (note that both larger apertures do maintain significant contrast well beyond the cutoff frequency of the smaller unit aperture, which are here 0.33 and 0.2, respectively, as well as beyond their projected cutoffs). This directly implies that 3 times larger aperture has better contrast level than either significantly smaller, or significantly larger apertures. However, looking at the approximate visual plots, they are already closer to their respective longexposure plots for D/r~3, and nearly coinciding with them for D/r~5 an larger. It makes improbable that five times larger aperture has any significant advantage in contrast transfer over the one for which D/r~1. Those 2 to 3 times larger are still better, and represent the best aperture range with respect to contrast transfer. The discrepancy between the optimum aperture diameter being about 2r0 for limiting stellar resolution, and 23r0 for contrast transfer should be mainly the consequence of the shortexposure MTF becoming less representative of the actual visual MTF as D/r0 increases to ~2 and beyond, due to the increase in the effective magnitude of the tilt error component. In other words, their actual MTF plots are shifted toward the longexposure plots, the larger D/r0, the more so. Consequently, the optimum D/r0 shifts toward lower value, in this case 2. However, considering that seeing fluctuates, and that it will always be better than the average within appreciable portion of time, an optimum aperture, which would perform best during those times, should be somewhat larger than 2r0. Assuming that, according to the simplistic scheme on FIG. 82, seeing is likely to be 25% or more better than the average in about 1/4 of the time, the optimum aperture would be about 25% larger, or about 2.5r0. Factoring in those relatively frequent nights when the seeing gets better than typical would increase it to about 3r0. Since it is the contrast level that determines resolution limit to extended details, like those on planetary surface, this implies that larger apertures, in general, outperform smallest ones in this respect, despite having a larger seeing error. At the same time, a range of apertures between really large and small apertures offer best performance level. As the magnitude of seeing error increases, the visual seeing error shifts closer to the long exposure mode (FIG. 51). As the graph at left shows, even in the full longexposure mode (unlikely to occur in visual observing), loss of contrast transfer in larger apertures tends to be roughly offset by their size. In other words, contrast level remains similar in wide range of apertures. Since the actual visual seeing is between short and longexposure mode, the implication is that larger apertures tend to retain edge in contrastrelated resolution, with the optimum aperture size being somewhere between small (D/r0~1) and large ones (D/r0~5, or more).
Closer to the observer, it is terrain topography, immediate
surroundings and the very parts of a telescope itself that can
create layers of unsteady air resulting in
wavefront deformations.
