[ATM] Quantifying visual seeing error

["vladimir" <svla@isp.com>]

Here's another attempt to figure out how to roughly quantify 
effects of seeing from the appearance of diffraction pattern 
in a telescope. In the absence of professional attention to the
effect of seeing on visual observing, we're left to wonder
where in between time-averaged (long exposure) and 
instantaneous seeing error fits the visual seeing error.

According to Mahajan (Aberration theory made simple, 
p143), the averaged seeing RMS error for long exposure 
(from a few seconds up for large telescopes) is given by:

w=0.162(D/r0)^(5/6)

with D=the aperture and r0=Fried atmospheric coherence
length

Most of the error is caused by wavefront tilt. With the 
exposure reduced sufficiently (to a few milliseconds or less)
to elliminate tilt error, the remaining RMS error is given by:

w=0.058(D/r0)^(5/6)

While the eye is capable of registering movement down
to a few milliseconds, it is much less capable of isolating
it from a continuous series of pattern change in forming
what we see as an image. It constructs what we see as
a pattern image from an arbitrary number of such
flash sequences, likely spanning considerably longer time
frame. This may still be a fraction of a second, but 
inevitably opens up the door for certain amount of
the tilt error. The only question is: how much?

Connection point between the visual pattern and Fried 
parameter should be the blur size. Apparent blur size
for the worst seeing at the Pickering scale (1) is about six
Airy disc diameters. This roughly correspondes to long-exposure 
blur at (D/r0)~5, as given by Mahajan.
Following the dinamics of blur reduction, as given in the
scale for the first three levels, reduction factor for (D/r0)
with every successive scale level can be approximated as ~0.75. 
This would give (D/r0) as 1.6, 1.05, 0.62 and 0.44,
and the instantaneous RMS error as 0.086, 0.06, 0.039 and 0.029 
for Pickering's 5, 6/7, high 8 and 9/10 levels. 

Incidently or not, it is fairly close to the dinamics of
the visual RMS seeing error calculated by Suiter (Star
testing astronomical telescopes, p137-8), who comes
to 0.15, 0.1, 0.075 and 0.05, respectively. With Suiter's
RMS values greater by approx. a factor of 1.8, the visual
RMS seeing error could be approximated by

w~0.1(D/r0)^(5/6)

However, according to Mahajan, this (random aberration)
RMS error has different relation to the (random aberration)
Strehl. For instance, while the conventional relationship
implies 0.37 Strehl for (D/r0)=1 (w/0.16 long exposure RMS error), 
the exact Strehl value is 0.445. The random
aberration (seeing) Strehl is better approximated by 

S~1/[1+1.23(D/r0)^2]

And the "appropriate" conventional RMS error is
 
w=0.241sq.rt(-logS)

Although seems to be having a decent fit, the above rambling is 
a pretty lose one. I'm sure comments can be 
made which would help put missing parts in their place and
give better foundation for quantifying a visual seeing error.

Vlad 








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