From Edward Arnold GRAFTON
® . . . . . . . Date: Thu, 25 Dec 2003 20:14:13 -0600 (CST)
Subject: Re: [Saturn-ALPO] Re: Saturn on December 16th.
At 10:22 AM 12/25/03 +0100, Paolo R. Lazzarotti wrote:
>I attach you a formula by astronomer Paolo Tanga (who's reading us in
>CC) measuring the Tmax for a correct planetary imaging:
>Tmax [sec] = 1150xPxM/D
>Where: P is the rotation's period of a planet [hours]; M is the max
>rotation allowed at the CENTER of the disc [arcsec] and D its apparent
>Assumed a resolution power of 0.43 arcsec for your C11, Nyquist says you
>have to put this separation in 2 pixels; so you would have a 0.21 image
>scale with you.
>But you also well know this is an high value for a fine imaging, so I
>agree with you to use smaller values as you are doing.
>Assumed you didn't resample your final image, you managed a 0.09
>arcsec/pixel, so this is M parameter of the above formula.
>Tmax = 1150x10.5x0.09/20.5 = 53 secs.
For this discussion lets assume that the 2x sampling for max resolution is correct, although I am no longer as convinced of this as I once was. Let me also state that I have not see a rigorous mathematical proof to refute the 2x rule as applied to our application.
The application of this rule DOES NOT extend to the time window of gathering images for later averaging or "stacking". The 2x rule is satisfied by the image scale (sampling) and the TIME WINDOW OF A SINGLE FRAME. All the SAMPLING CRITERIA is completed on the INDIVIDUAL FRAMES. Stacking or averaging the individual frames is a REGISTRATION issue not a sampling issue. During the registration and "stacking" ones hopes to maintain the resolution already captured on the individual frames and improve the S/N. The resolution captured on each frame would be at best .43 arc seconds. If the planet rotated less than .43 it was not resolved by the scope. The window calculated from the formula given by Paolo's formula would then be 253 seconds.
In practice it would take 10/10 seeing for several minutes to achieve the .43 arc seconds resolution, which I have never seen. In fact the seeing might have to be .43 arc seconds/2 to achieve the .43 arc seconds resolution since that is the sampling being performed by the required Nyquest 2x rule!
>Damian Peach wrote:
>I would agree with you that this isnt an appropriate timing to reaching the theoretical limit of the
>telescope, but the fact remains that you are NEVER going to resolve low contrast spot-like features
>better than ~0.5", so the window works (as proved time and time again by Grafton/Myself.)
>You are not taking into account noise is a major limiting factor in resolution of low contrast detail >to this level. Halving the window would not improve resolution, as the image would be so much more
>noisy, resulting in this detail being lost in the noise. Why shorten the window when the one being >used already works regardless of the seeing!?.
This S/N point is a good one. None of the low contrast spots that I have imaged have shown on any one frame. The resolution is there but it is lost in the noise and can only be recovered by improving the S/N by "stacking" many images. The latest dark spot that Damian imaged is a good example. The low contrast .7 arc second spot imaged by Damian would have gone undiscovered if the S/N would have been lower than he achieved, due to a less a less productive imaging strategy having been employed, ie a shorter window.
® . . . . . . .Date: Fri, 26 Dec 2003 09:26:59 -0600 (CST)
Subject: Re: [Saturn-ALPO] Re: Saturn on December 16th.
Paolo R. Lazzarotti wrote:
>What you say is correct; the M parameter of my formula is referred to
>the max blur accepted.
>If you don't want any blur in your final image, that means the first
>frame and the last one have to be captured before the planet would spin
>1 pixel at least at the center of its disk.
>If your image is sampled at 0.4 arcsec/pixel, that means you can accept
>a max blur of 0.4 secs; if you sample at 0.2 arcsec/pixel, you can
>accept a max blur of 0.2 secs and so on.
>I assumed Damian's image scale as 0.09 arcsec/pixel, so my consideration
>But if he was 2x resizing the final image, than the appropriate value to
>consider is 0.18. This is clear!
>>In practice it would take 10/10 seeing for several minutes to achieve the
>>.43 arcseconds resolution, which I have never seen. In fact the seeing might
>>have to be .43 arc seconds/2 to achieve the .43 arcseconds resolution since
>>that is the sampling being performed by the required Nyquest 2x rule!
>Yes, the discussion we're feeding is based on the math.
I would love to see a mathematical solution to this issue but my feeble mind does not corporate! We all know of Rayleigh's criteria for resolution which states that the sin(a)=1.22L/D where L is the wavelength of light and D is aperture. For small angles sin(a)= a (units in radians ).
For Damian's .28 meter scope (11 inch) scope: a=1.22x500x10^-6nm/.28m radians =2.18x10^-3 radians. Given that there are 57.295 degrees/radian and 3600 arc seconds per degree, this works out to .45 arc seconds. The wavelength used above was 500nm. A CCDs sensitivity runs from about 400nm to about 1000nm so according to this criteria the resolution would vary between .36 arc seconds at 400nm to .9 arc seconds at 1000 nm. This is all well and good but as we know the Enke's division is sometimes resolved on Damian's Saturn images and is smaller than the .36 arc second threshold.
This ambiguity calls into question as to the actual resolution obtainable with a given aperture for extended objects. From my experience red or IR images almost always have better resolution that blue light images, no dought to the kindness of the atmosphere at the longer wavelengths. But this further complicates this simple look at the theory as the scope should have better resolution at the blue wavelengths. If one were to only consider the above math it would lead one to a shorter imaging window (due to the .36 arc second resolution) and using blue light. I seriously doubt that anyone would recommend this approach for better resolution especially given the low QE od CCDs at this end of the spectrum which would degrade the S/N (and therefore the resolution) of the image.
Given my uncertainty about the credibility of the 2x sampling rule and the apparent ambiguity of Rayleigh's criteria, I think that the math can only be used a starting point. I would love to see these issues resolved from a mathematical point of view for our application but until then there will remain an element of "Black Magic" in obtaining high resolution images guided by the success of employed techniques.
® . . . . . . .Date: Tue, 30 Dec 2003 10:47:20 -0600 (CST)
Subject: Jupiter Observation December 30th 2003 at 10:42 UT
Hi Jupiter Observers
Here is an observation from December 30th 2003 at 10:42 UT. The transparency was excellent, 9/10 and the seeing was very good, 7/10 with moderate dew. The weather was very cool with the temperatures in the upper 30s F.
Barlow Projection to f/27 aprox .21 arc seconds per pixel.
BLUE + IR rejection, .6 seconds, Edmunds filter
GREEN + IR rejection, .4 seconds, Edmunds filter
RED + IR rejection, .2 seconds, Edmunds filter
Clear filter, No IR .13 seconds, Edmunds filter
Ed GRAFTON (