I hope to answer a question that has bugged the amateur astro-community for some time - ring patterns seen around stars in diffraction limited telescopes with central obstructions.
Here's an example from a recent thread:
Here's an example from a less recent thread:
ZWO ASI183MM-Pro camera on Sharpstar 150 Hyperbolic Newtonian (f/2.8) with Optolong 6.5nm Ha filter
It also appears in the Hubble space telescope images:
I will use the Sharpstar HNT150 image as a worked example to explain what I think is going on. Inevitably any explanation will be quite technical but I'll do my best to make it understandable.
The Sharpstar HNT150 (f/2.8) has a central obstruction of 70mm which makes it 47% (70/150)
The formula giving the intensity of Airy rings for centrally obstructed optics can be found in the "Obscured Airy Pattern" section of the Wiki Airy Disk article. It uses a first order Bessel function of the first kind. This is the function BesselJ in Microsoft Excel which means anyone can plot it. So here is a plot of a slice through the circular Airy pattern of a star with no central obstruction (CO) and a star with a 47% CO (e.g. the HNT150 telescope):
The effect of the CO is to narrow the width of the central peak and to throw more energy into the surrounding rings. Everyone is already familiar with this. But the amplitude of successive rings dies down quite quickly which means that few people are familiar with what happens in those outer rings. So here's a plot where I have applied a scaling function to the rings, to make the effect visible:
There is an obvious periodic modulation to the amplitude of the CO rings, with a maximum brightness approximately every 3.75 rings.
My argument is that the individual rings are too finely spaced to be sampled by the sensor pixels but these bright groups of adjacent rings form a regular coarsely spaced pattern that is easily sampled.
Let's do the arithmetic on the Sharpstar HNT150 with ZWO ASI183MM-Pro combination:
Given the focal ratio of f/2.8 and using the well known formula for unobstructed optics, the radius of the first zero in the Airy pattern is:
radius = 1.22 x wavelength x focal_ratio
Successive rings have an asymptotic spacing of:
spacing = 1.0 x wavelength x focal_ratio
This gives a ring spacing of 1.83 microns for the H-alpha wavelength (656nm) and f/2.8 optics. But the pixel pitch of the ZWO ASI183MM-Pro camera is 2.4 microns, so the rings of unobstructed f/2.8 optics are too closely spaced to be resolved by the sensor.
However my plot above shows that there is an amplitude modulation that occurs in the case of a 47% CO with a period of approx 3.75 rings i.e. 6.88microns (1.83x3.75). This is easily sampled by the camera and my conjecture is that this is exactly what we are seeing in the above image. The spacing of the ring pattern in the image would be 2.9 pixels (6.88/2.4)
A further interesting point is that the pattern imposed on the rings varies with the size of the central obstruction. For instance a 55% CO has a pattern with a period of approx 4.5 rings: