Walt Morgan...

Sandy, Your Airy disk summary of a couple days ago looks to me like it hits the important considerations. I suppose that the smaller the Airy disk the better, but if it is really small compared to the dimensions of a pixel, will there be an apparent drop to half brightness as the image crosses from one pixel to the next? I question whether it is realistic to keep the disk on a single pixel for a long time (because this is addressing portable gear, with less than perfect alignment), or whether a scope can be re-pointed reliably in increments that will move the disk less than one pixel.

You list three CCD chips currently available. For the specific purpose of making a scope especially sensitive for occultation work, the high-resolution chip would seem to me to be a bad choice – unless it were somehow more sensitive. The larger pixels of the other chips give a better chance of getting the light on a single pixel, and they give ‘enough’ resolution.

In searching the web a bit I found a site that addresses the problem of a central obstruction as related to the Airy disk. It is by David Whysong, a graduate student at UC Santa Barbara, and presumably he is providing good information.

He shows some relationships between disk radius and obstruction size (and his percentages I think are all between obstruction diameter and objective diameter – not areas). If you click on SEE SOME NEWER PLOTS HERE! at the top of the page, you will find more interesting plots. The last one I thought was of particular interest, plotting the integral amount of light versus angle, with a parameter of different obstruction fractions.

He has six lines, but identifies five. My hunch is that the top three are 10%, 20% and 30%, and those are the only ones of particular interest right now. My interpretation for those is that the first Airy minimum is at about 3.3E-6 radians. I can’t make the calculation for a 100-mm aperture come out with that number, but he’s probably right. Regardless, the integral is shown to be constant through that region, so likely the plots show the correct trends even if they should happen to be incorrectly scaled.

At the first minimum, my recollection is that the central disk contains slightly more than 85% of the total light, if unobstructed. With 10% and 20% obstructions he shows about 84% and 83%, respectively. I interpret that to mean that if a central obstruction can be held to 20% or less of the objective diameter, the reduction in total light within the central disk is minimally impacted. That’s nice to know.

To test the significance of that, I measured the diameter of my PC164C, and found it to be about 35 mm. It could be mounted with its ¼-20 thread so that 35 mm would be its maximum diameter, in which case the 20% obstruction criteria could be met using a 175 mm objective – 7 inches. If the CCD were to be mounted in a concentric holder, the obstruction diameter would need to increase, but probably to no more than 42 mm, and the 20% criteria would be met for a 210-mm scope. With an 8-inch objective, the central obstruction would be 20.7%. CONCLUSION: concentrically mounting a CCD camera at prime focus is compatible with retaining a very large fraction of the light within the central Airy disk if objectives of 8-inch are larger are used. (And the penalty increases only very slightly for 6-inch objectives.)

Taking 83% as the integral within the first disk (i.e., using a 20% obstruction), 90% of the light in the central disk will then be about 75% of the total light. Looking at the same integral plot, the 75% line intersects the 20% curve at about 2.2E-6 radians, or at an angle two-thirds as large as that of the first minimum. The same approximate ratio holds for 10% and 30% cases, so presumably for the unobstructed case, too. From that I propose using a figure-of-merit type term for the present purpose:

Significant Airy disk diameter: two-thirds the theoretical Airy disk diameter. (Or SADD = 2/3 TADD) The purpose is to introduce an element of reality into the approximations. This means that I would consider using values 2/3 as large as you list for comparing with dimensions of the CCD pixels.

Unfortunately, as you point out, all of this assumes ideal conditions, and, frankly, it has been a while since I saw anything resembling point images on my video records. Nevertheless, one can aspire. But maybe the honest approach would be to recognize that it is desired to achieve the goal of high sensitivity for faint stars even when atmospherics are less than ideal. If so, perhaps the goal should be for a pixel to capture an image that is double the theoretical diameter:

Reality Airy disk diameter: twice the theoretical Airy disk diameter. (Or RADD = 2 TADD)

Looking at the problem from a totally different perspective, I routinely use my f/3.3 focal reducer for asteroid occultations because of the wider field of view it provides. (With my 8-inch SCT that turns out to be about 15 by 20 arcminutes, and effectively f/4.3.) I have not been able to establish that fainter stars are picked up by using the focal reducer – or if using my f/6.3 focal reducer. The f-o-v is small when the inexpensive CCDs are used, and life is simpler, and results more reliable, when the effective focal length is reduced. A simple, sensitive scope then wants to have a short focal length so that the f-o-v at prime focus is big enough to show more than one very faint target star. I have found faint stars using the 8-inch SCT without a focal reducer, meaning f/9.4 and 6.6x9 arcminutes f-o-v, but that is definitely less reliable in terms of being sure that the correct target is found. For a system more sensitive than mine (magnitude 11 is my common threshold), perhaps a 7.5x10 arcminute f-o-v could be a limiting criterion. The idea is that a target is confirmed when you can relate it geometrically to other stars, and that can be done with a smaller f-o-v if more stars are detected. Your idea of an electronic finder with a zoom lens would go a long way toward making this practical.

Because of the non-ideal state of the atmosphere, I would be inclined to treat all the theory of Airy disk not as an absolute, but as a relative correction. Using as a reference my system: (8-inch SCT with effective focal length of 865 mm, PC164C -6.6x9 arcminute f-o-v) What happens when any parameter is changed? With the same CCD, sensitivity is gained by increasing aperture or eliminating glass surfaces. If one of these is changed, what is the impact on sensitivity? Is the f-o-v still acceptable? Do the changes begin to approach the two magnitude (6.3:1) increment needed to reach magnitude 13?

Since I haven’t found any sensitivity benefit from using a focal reducer, perhaps the better reference point is my 8-inch SCT with effective focal length of 1900 mm. A small improvement could still be obtained by using a smaller obstruction and eliminating the secondary reflection, but loss through the focal reducer drops out of the calculation.

Dave Gault...

I have been thinking about the ideal occultation telescope too and being an amateur telescope maker, I suppose the skies the limit (pun....! where?).

Using my current telescope (10inch f5 Newtonian) and my PC164 camera as a guide, I can reach most mag. 12 stars and some mag. 13 stars depending on spectral colour. With a faster scope, possibly f3 and with the camera at prime focus eliminating the need for a large secondary I'd reckon on at least an extra magnitude sensitivity, possibly two.

So, the features and details for the design I have been thinking about are...

• OPTICS -The focal length needs to be kept fairly short and the aperture as large as practically possible, keeping the overall dimensions within transportability needs. A 10inch f3 would be ideal I reckon but a f3 paraboloid is not a trivial mirror to make, but doable. I don't think the coma inherent in such a fast optic without additional corrective optics would be objectionable due to the small size of the ccd chip.
• ALUMINISING -My current scope has enhanced and overcoated mirrors. I'm unsure if there is a better alternative.
• CAMERA - A PC180 or PC182 uses the same chip sets as the PC164C and PC164EX but has a much smaller body. I'd want to keep the camera interchangeable in normal 1 1/4inch focusers and one day I hope to have a starlight express ccd camera. The diameter could be kept down to 35mm, making for an obstruction of 14% in a 10 inch scope, not bad eh? My only worry is heat plumes from the camera in the optical path.
• TUBE & MOUNT -KISS principle is to apply. (Keep It Simple Stupid) A 300mm dia cardboard tube by about 1 meter long mounted like a normal DOB using plywood for the structure and rollers or bearings instead of teflon on formica would meet the need. All up weight 20kg max.
• FOCUSING -Two methods to choose... move the primary or move the camera. Focusing the camera would lend itself to robo focus. The contortions required to reach into the optical path while twiddling a knob while watching the monitor would not be pretty. Focusing the primary means moving a heavy mirror and allowing for collimation screws, floatation cells and possibly change of balance point. I'd reckon the best bet is a sliding sled focuser. Normally an eyepiece is mounted in the slider and the secondary is mounted on a stalk to the back of the slider and it moves with the eyepiece. We would simply mount the camera mounted on a stalk attached to the slider.
• DRIVE CONTROL -No contest here. A Mel Bartels scope.exe software based system would allow automatic slewing and tracking. It can utilize almost any old PC (laptop) as the controller and a simple motor control box driving surplus stepper motors for each axis. It can be used in open loop mode (without encoders) and closed loop (with encoders). An add on board can be used to control the focuser. For the PC challenged there is a variant called MacDob.
• STAR ALIGNMENT -Mel's software utilizes the usual two star alignment method and doesn't need excessive precision in leveling and NO polar alignment. That is... put the scope on the ground, switch on, point telescope at a known star, centre and acknowledge, repeat and then GOTO. IIRC, Mel's software allows setup and alignment in daylight by using the sun. Center the sun (using a filter...duh) and wait an hour or two and then re-center the sun..... this would allow catching an early event soon after sundown.
• TIMING AND GEOGRAPHICAL DATA -Another no contest- KIWI OSD plus a monitor and a camcorder and you are all set to record events to a precision of +/- 8 milliseconds and latitude and longitude to 15 metres, and height to about 30 metres for something like the Deluo and other non averaging GPS's.
• CAMERA MODIFICATIONS -The cameras noted above have auto gain which if fine for minor planet events where most of scene is black but for total and grazing lunar events a method of manual gain control is required. Further, because the camera is in the optical path, remote control by radio or wire would be desireable. AND if remote radio control is possible, why not a radio data link as well. Look Mum, no wires.... Bluetooth... Hmm. You think I'm joking... well I'm not.
  

  Well, that's my wish list of specs for the scope...

  I have a web page showing progress (or lack of) and it can be found here.

  Dave...

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  Last updated - 12th January 2005

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