The conventional Kolmogorov model of
atmospheric turbulence is strictly valid for an inertial range of
turbulence with the size much smaller than its outer scale - the
large scale air movement at which the turbulence is initiated - and
much larger than its inner scale, at which the turbulence is
dissipated by air viscosity. However, the effect is calculated by
integrating contributions to the refractive index fluctuations - and
to the corresponding phase and wave error - on the scale from zero
to infinity, assuming an infinite outer scale of turbulence (**OST**).
If the telescope aperture is much smaller than OST, this general
approach does not result in a significant error, even if OST is
always finite. However, out in the field this is often not the case,
and the error in applying this general concept may not be negligible
even at quite small apertures. This is particularly the case in good
seeing, when - usually - the lower level turbulence dominates, with
the OST there possibly smaller than 10m.

Integrating from zero instead from some
finite (actual) inner turbulence scale also induces error, but since
it is limited to the smallest-scale contributions to wavefront
disturbance it is much smaller than the error due to assumed
infinite OST, and generally negligible in comparison.

In
1991, D.M. Winker investigated the consequences of using a finite
OST for calculations of seeing-induced error. As a function of the
ratio of the aperture diameter (**D**) to OST (**L**), the
seeing induced long-exposure error, given as phase variance over
Zernike-decomposed wavefront normalized to (D/r0)5/6,
shows significant reduction as the aperture increases relative to
the (finite) OST. The effect is the greatest for the first two
(tilt) Zernike terms. As the top solid line indicates, with all
terms except piston included, not insignificant error reduction of
nearly 20% occurs even in an aperture 1,000 times smaller than OST.
Even in poor seeing, when the upper level turbulence dominates, and
the OST there may reach a few hundred meters, that would be an
aperture of up to 12 inches in diameter, or even larger. While the
tilt error component may not be fully present, even in poor seeing,
in visual observing at these aperture sizes, it is for the better
part, and the effect wouldn't be negligible.

In good seeing, when the OST can be expected to
be in the 50-100m range, this reduction ratio would,
correspondingly, apply to 50-100mm aperture diameter. But at these
aperture sizes the tilt component is not a significant error source
in visual observing. However, a hundred times smaller aperture is
still within the amateurs range. In a 0.5m aperture, with the good
seeing **r**0
of around 10cm, D/r0~5
and the long-exposure phase variance 1.03(D/r0)5/3
being the standard deviation 2πφ**
**squared - **
φ** being the phase
analog of the RMS wavefront error - the variance is 3.94 and the
corresponding RMS wavefront error is 0.32. At this D/r0
level, most of the tilt component is effectively in the visual
seeing error; taking that 2/3 of it is, and knowing that the tilt
component makes 64% of the entire (long-exposure) seeing error gives
that 40% reduction in the tilt error indicated by the plot for
D/L=0.01 results in some 22% smaller RMS wavefront error.
Specifically, 0.2 vs. 0.25.

For the long-exposure error, the reduction
is from 3.94 to 2.36 in the variance, hence from 0.32 to 0.24 in the
RMS (since 2π is a
constant, the RMS changes as the square root of the change in
variance).

The roughness component of the seeing
error, i.e. the error remnant with the tilt removed, is much less
sensitive to the use of a finite OST in calculation. The difference
becomes significant only at D/L~0.1 and larger, and at even larger
D/L values when the first 6 Zernike terms (Noll's scheme) - piston,
tilt (horizontal and vertical), defocus and astigmatism (horizontal
and vertical) are removed.

The effect of a finite OST on the
Kolmogorov-based Zernike wavefront
decomposition is a reduction in the error as a whole, reduction
in the relative contribution of the image motion component (tilt
terms) and increase in the relative contribution of the roughness
error component. Specific ratio of motion vs. roughness component
depends on the aperture-to-OST (D/L) ratio: larger ratio values
result in the relative increase of the roughness component. At
D/L~0.001, the error is reduced by about 20% which, with all of it
coming from the reduction in tilt (image motion) contribution,
increases the relative roughness variance component from 0.134 to
about 0.16 (i.e. the roughness RMS wavefront error component from
36% to about 40%). At D/L~0.01 and 40% error reduction, still
entirely due to the smaller tilt component, the relative roughness
component in the error increases to ~0.22 (~46% of the RMS error),
and so on.

Other authors had also addressed this
issue. An interesting angle is the effect of finite OST on the
seeing FWHM (*On the Difference between Seeing and Image Quality:
When the Turbulence Outer Scale Enters the Game*, Martinez et al. 2010).
As shown on the graph below, a finite OST results in a smaller
seeing FWHM, the smaller OST, the more so. Similarly to Winker's
modeling, limiting the maximum OST size takes out significant
portion of large-scale contributions to wavefront disturbances due
to turbulence, hence most of the resulting error reduction is in the
tip-tilt component, i.e. random motion of image as a whole. The
degree of FWHM reduction vs. Kolmogorov (infinite OST) model due to
it is approximated (Tokovinin, 2002) based on the von Kārmān
finite-OST turbulence model.

The "default" seeing FWHM for the Kolmogorov model, FWHMI=0.976λ/r0
(used in Martinez et al.) is for λ=500nm
and zero zenith angle, is valid under assumption of an aperture
infinitely large vs. **r**0
which, again, is not fulfilled in field conditions (obviously, since
it would imply seeing FWHM for D=r0
slightly better than with zero image motion). For large ground
telescopes, **r**0
is not negligibly small and the resolution (theoretical seeing FWHM)
is somewhat worse. The most frequently quoted figure is 1.2λ/r0,
and the calculation specifically addressing the seeing FWHM in the
proximity D=r0
gives long-exposure FWHM as 1.35λ/r0
radians (i.e. 206,265 as many arc seconds).

The graph shows the reduced FWHM (FWHMF,
with the subscripts "F" and "I" referring to finite and infinite
OST, respectively) vs. "default" Kolmogorov FWHM (blue vs. black),
but it is fair to assume that the relative reduction (top left)
would be similar for other FWHM modes as well. Since OST size
generally increases with turbulence - or, rather, the other way
around, as more energy is injected into the air - the reduced FWHM
is plotted for four arbitrary, partly overlapping ranges with the
averaged OST values from 200m, roughly corresponding to the average
~2 arc second seeing, to 25m, roughly corresponding to 0.5 arc
second seeing (more accurate plot would have been the one
interpolated through the blue graph segments). Graphs indicate that
the nominal FWHM reduction due to finite OST decreases, and its
relative reduction increases with better seeing.

The reduction shown on bottom and top left is for **r**0
values in the visible range of wavelengths (centered at ~550nm).
Since **r**0
varies with the wavelength, as **λ**1.2,
FWHM reduction due to finite OST increases into the infrared, the
smaller OST the more so. For instance, if the relative FWHM
reduction (top right) is for **r**0
values for λ~550nm, going to
λ~1μm would result in about twice larger **r**0,
and the reduction given by the relation for **r**0~200mm
at λ=550nm (i.e. ~400mm at
λ=1μm) and L~25m would have been 0.5,
about 40% greater than at 550nm. Top inset at right (green)
illustrates the range and dynamics of FWHM reduction in a wider
spectral window, from 385 to 1100nm. The reduction is always greater
for longer wavelengths. However, it is near negligible in poor
seeing (top), quite modest in the average seeing (middle), while
becoming significant in excellent seeing (bottom).

Following page addresses in more
details how the effect of atmospheric turbulence relates to the
aperture size.