Re: [ATM] Quantifying visual seeing error

[Michael Peck <mpeck1@ix.netcom.com>]

At 19:15 10/3/05, vladimir wrote:

>and slightly less for balanced coma. The approximation puts it at 0.38.
>This doesn't seem to be explaining the
>0.445 "seeing" Strehl for the same nominal RMS error.

Well, I was no more than half right with my initial response. Here 
<http://home.earthlink.net/~mlpeck54/astro/temppix/strehlvsrmskt.png> 
is a plot of numerically estimated Strehl ratios vs. RMS wavefront 
error for 100 randomly generated "atmospheric" wavefronts with 
D/r0=1. This is for the long exposure case, with tilt included. The 
green line is Mahajan's approximation, which actually errs on the 
optimistic side, contrary to the result I got with randomly generated 
wavefronts with different statistical characteristics. What's driving 
the result is that the RMS wavefront error is itself varying randomly 
over a considerable range, and more than half the time it's actually 
better than it's RMS value of about 0.16 waves.

Here 
<http://home.earthlink.net/~mlpeck54/astro/temppix/histrmsle.png> is 
a histogram of RMS errors for 10,000 random atmospheric wavefronts, 
again with D/r0=1. I think Fried defined the parameter that was named 
after him so that when D/r0=1 the mean value of the mean-squared 
phase error is about 1 rad^2. In fact the rms value of the RMS 
wavefront error is right around 0.16 waves - that's the long-dashed 
vertical green line in the histogram.  The mean RMS error is somewhat 
lower at 0.148 waves, and the median is lower yet at 0.139 waves. So, 
because of the skewed distribution of outcomes most of the time 
you're a little better than average although you're occasionally much 
worse than average.

In the short exposure limit 
<http://home.earthlink.net/~mlpeck54/astro/temppix/histrmsse.png> the 
histogram has the same basic shape but shifted to the left and 
compressed. Most (~88%) of the time the short exposure wavefront 
qualifies as diffraction limited by Marechal's criterion. If one were 
to create a movie of simulated PSFs produced by these simulated 
wavefronts what it would show is a well defined centrally 
concentrated PSF (almost always) that dances around randomly.



>Still, the theory seems to make seeing harsher on optical quality
>than what it appears to me. With r0~2.2 inch in a
>typical 2 arcsec seeing, everything bigger than 60mm
>in aperture would be - on average - less than diffraction
>limited due to seeing alone (low 8 and worse on the scale).

That may be too pessimistic. I think at visual wavelengths a Fried 
parameter r0~10-15 cm. would be typical for poor to average seeing, 
with r0~20-25 cm for good to excellent seeing. If the eye-brain's 
response time is closer to the short than long exposure limit you'd 
actually perceive diffraction limited conditions most of the time 
with an r0 sized telescope. Of course with, say, a 3r0 sized 
telescope you'd almost never experience truly diffraction limited 
conditions. You would, however be diffraction limited by Fried's 
standard of RMS < 1/(2 pi) waves about 75% of the time.

Mike


_________________

Michael Peck
email mpeck1@ix.netcom.com
Wildlife photoblog! http://wildlife-pix.com
Amateur telescope making http://home.earthlink.net/~mlpeck54/astro/astro.html 

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