Hello Eddgie,
Thank you for that post. Can you explain me why a 90mm obstruction would mean only 164mm of free aperture? I thought that I would have to compare the free area rather than the free radius (since the obstruction is in the center the area of obstruction is not 35.4% but rather 12,9%).
Thank you,
Arnheim
It is a lazy rule of thumb and it only applies to visual use of the instrument, not imaging and it applies to contrast transfer, not resolution (resolution is almost completely a function of aperture and aperture alone. Even a poor instrument with a large obstruction will have the same resolution (the size of the Airy Disk) as a perfect one.
When there is larger obstruction, you can subtract the diameter of the obstruction from the diameter of the primary, and what is left is what most people call the "free aperture" or "clear aperture" of the system, meaning the amount of aperture that is free or clear of obstruction.
In a refractor, the only diffraction source is the aperture itself, but in a reflector, there is diffraction that comes from the secondary so when the light converges on the primary, it is already very slightly diffracted before it gets to the primary, where it is diffracted a second time.
This is nothing to do with light transmission. It only factors in the diffraction surfaces and does not care about light gathering.
Contrast transfer is different from resolution of a point source. It is really looking at the ability of the system to preserve contrast and the best way to show that is using the contrast retention of line pairs of increasing frequency.
The chart is a model of the Modulation Transfer Function (MTF) of the system. To use the chart, you use the vertical axis as the starting contrast. Most people will use this assuming 100% contrast and is based on alternating black and white sinusoidal line pairs starting at 1 line pair per millimeter, and increasing in frequency linearly until you reach the maximum number of line pairs the system can resolve (this is based on focal ratio). You can make the vertical axis any starting contrast you like. If starting contrast is 100%, then at the point on the vertical axis where the line passes .5 on its way to 0, then the contrast of the lines compared to one another will be only 50%. If the contrast of the lines actually started at 20% (typical of many features on Jupiter) on the vertical axis, then at the point the line where it crosses .5, for that frequency, the contest you would see would be reduced to 10%, which is very near the contast sensitivity of the dark adapted human eye.
Even a perfect system (represented in the chart by the red line) will lose contrast as the frequency goes up. On the left, it will show that virtually all telescopes will resolve 1 line pair per millimeter at the focal plane with 100% contrast transfer. As the frequency goes up though (shown in the horizontal axis) some of the energy is removed from the Airy Disk and directed around it. The horizontal axis shows the change in contrast between the lines in the pair. The white line starts to become less white and wider, while the black line becomes less black and narrower. When the line pairs have both become 50% grey, you have reached the maximum linear resolution of the system.
The left side is the frequency range that would be considered the "Important Visual range" because once the lines get very close together, the human eye lacks the contrast sensitivity to separate them. A camera though, has much better contrast sensitivity (down to 1% or 2%) so it can separate lines of much higher frequency than the eye can.
So, the free or clear aperture means free or clear of obstruction and the bigger the obstruction, the more contrast is lost in the important visual frequencies.
The rule of thumb for free or clear aperture is just a quick way to estimate the contrast transfer of a larger obstructed system as compared to that of a smaller aperture with no obstruction when used visually. The larger aperture will almost always resolve finer detail that is beyond the contrast and resolution of the human eye.
If we assumed that both scopes could resolve 100 line pair, then down to about .33, or 33 lines per millimeter, both modeled systems would show roughly the same contrast for the lines, which would about 36% or so from the 100% starting contrast. Rather than appear as white lines on a black background, they would appear as light grey lines on a dark grey background, making them harder to see. Now if the lines started 20% contrast, at this point they would be shown with only about 7.2% contrast. The white lines would appear as a grey that was only a bit less grey than the black lines, which are now also grey, just a bit darker and at 7.2 contrast, the eye would have difficulty seperating the lines. The perfect instrument would be showing black and white lines starting with 100% contrast as having about 57% contrast, and the lines starting with 20% contrast would show at 11.4% contrast at this same frequency, and in this case, the would appear with enough contrast that would be visible to almost anyone.
That is why contrast transfer is important to understand. It lets you visualize how well to different instruments would show the contrast of different size features starting with different levels of contrast. The shadow of Ganymede crossing Jupiter is a very high contrast feature, but the starting contrast of a festoon is usually maybe 15%. Even though they are fairy large features, the more contrast the instrument loses, the harder they will be able to see visually, even though a camera will easily show them.
Edited by Eddgie, 05 February 2025 - 09:51 AM.
