From: "Sidor . Kurt" 
To: 
Cc: 
Reply-To: "Sidor . Kurt" 


Scott and List,

Somebody mentioned that Dawes limit may have been chosen for being able to
distinguish two close Airy discs by the characteristic shape of the
intersection of the two patterns rather than by being able to see the
approx 4% contrast difference between the two brightest peaks.  I agree,
this may be true but lets just say for theoretical argument sake for now
that the contrast change of 4% is perceivable.

John Sherman also wrote:

  Date: Sat, 11 Jan 2003 19:39:32 -0800
  From: "John Sherman" 
  Subject: Re: ATM Ultimate Optical Capability

  Thanks Rich,

  > J.B. Sidgwick's "Amateur Astronomer's Handbook" lists
observations of
  >dark spots against light backgrounds of Dawes/3.  Dark line against light
  >background out to Dawes/15.

From my previous computer model I estimated that a 0.05 arcsecond dark
feature could be identified against a light background for a
"perfect" 6 inch aperture producing approx 6% contrast.  Dawes
limit for this case was 0.77 arcseconds. 0.77/0.05 equals 15.4.  It is in
agreement with the quote from Sidgwick's "Amateur Astronomer's
Handbook" given above.

Scott per your request I re-ran the model for your case of 8 inch aperture,
0.015 arcseconds feature on a 3 arcsecond object.

Lambda = 550 nM (0.00022") wavelength of light D = 8 inches, mirror diameter

Rayleigh: For a 8" diameter 1.22*Labmda/D = 0.69 arcseconds

Dawes: 4.5/D seconds of arc, or in our case = 0.58 arcseconds.


The 0.015 arcsecond wide object produced a contrast difference of 5%. It
looks like it could (theorectically) be seen. The plot can be seen here:

http://images.andale.com/f2/115/106/3663062/1042572983198_RALEIGHDAWES3.JPG

A wider 0.030 arcsecond wide object produced a contrast difference of 10%
.The plot can be seen here:

http://images.andale.com/f2/115/106/3663062/1042572993036_RALEIGHDAWES4.JPG


Then I tried two 0.015 arcsecond wide objects spaced only 0.34 arcseconds
apart.  They each produced a contrast change of 5% but it only got about 1%
brighter between them, I seriously doubt this could be seen if we consider
4% to be Dawes limit.
The plot can be seen here:

http://images.andale.com/f2/115/106/3663062/1042573010906_RALEIGHDAWES5.JPG


I then increased the two 0.015 arcsecond wide objects to be spaced 0.56
arcseconds apart.  They each produced a contrast change of 5% and the space
between them returned to the full white ambient background.  This spacing
of 0.56 arcseconds is in very close agreement with Dawes limit for this
case of being 0.58 arcseconds.  Would this imply that I have (loosely)
proven that Dawes limit in the inverted case of dark against a light
background is equally true as the original case of light objects against a
dark background?
The plot can be seen here:

http://images.andale.com/f2/115/106/3663062/1042573021023_RALEIGHDAWES6.JPG


Mel Bartels also added:

   Kurt, very interesting analysis.  So the key appears to be to make the
   contrasted line big enough to cover enough cones so that the eye can
either
   directly detect this contrast difference or, through its image processing
   filter, deduce the presence of the line.

   Mel Bartels

As for eye cone coverage I think that would be a function of focal length
of the instrument and eyepiece magnification.  The larger the Airy disc and
tiny feature appear to be to the human eye, more solid angle (cones) of the
eye would get used.  My model is aperture dependant, not focal length or
image size dependant. I did notice though that for this last case the 0.015
arcsecond wide features produced a contrast change (gray area)
approximately 0.56 arsceconds wide in my plots, this too is close to the
Dawes limit number.  A very narrow feature produces a "blur" much
larger than itself, this limits any further increase in system
"resolution".  This can be seen in my last two plots, discrete
small features spaced closer than Dawes limit can not be resolved from each
other.

Regards,

Kurt Sidor

--- BBBS/NT v4.00 MP