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Obstructed optics and rings around stars

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#1 sharkmelley

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Posted 29 September 2020 - 03:16 AM

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:

 

asi6200_ha_1.jpg
ZWO ASI6200 camera on AG Optical 10" iDK with Chroma 3nm filter

 

Here's an example from a less recent thread:

 

SharpstarHNT150.jpg

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:

 

HST_ConcentricRings.jpg

 

 

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):

 

AiryRingsCO47_2.jpg

 

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:

 

AiryRingsCO47.jpg

 

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:

 

AiryRingsCO55.jpg

 

Mark


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#2 freestar8n

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Posted 29 September 2020 - 03:41 AM

Nice work.  I guess we tend to see this in images that have been heavily stretched so the rapid fade out may not matter as much.

 

I have occasionally seen this in my own sct images and thought it was odd - but at the same time didn't have an idea where they came from since it was slower than diffraction rings should be.

 

I have images at f/10 with 3.75um pixels and 700nm filters and I will see if I can see anything.

 

But if this is the mechanism then we a) should not see the effect with a refractor and b) they should not be visible far from the star unless the image is heavily stretched - because they die out quickly.  Is that correct?

 

Frank



#3 sharkmelley

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Posted 29 September 2020 - 05:04 AM

But if this is the mechanism then we a) should not see the effect with a refractor and b) they should not be visible far from the star unless the image is heavily stretched - because they die out quickly.  Is that correct?

Agreed.

 

However regarding refractors, simulation exhibits some weird effects when the pixel size is close to an integer multiple of the Airy ring spacing.  A beat frequency is set up which leads to interesting Moire patterns.  Here's an example:

 

RefractorMoire.jpg

 

Do such refractor patterns occur in real life or is it simply an artefact of the simulation?

 

Mark



#4 Jerry Lodriguss

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Posted 19 October 2020 - 01:42 PM

Agreed.

 

However regarding refractors, simulation exhibits some weird effects when the pixel size is close to an integer multiple of the Airy ring spacing.  A beat frequency is set up which leads to interesting Moire patterns.  Here's an example:

 

attachicon.gifRefractorMoire.jpg

 

Do such refractor patterns occur in real life or is it simply an artefact of the simulation?

Hi Mark,

 

Philosophical question: if you can't reproduce this pattern in a refractor in the real world, does it call into question the theory for an obstructed scope?

 

Jerry



#5 Jerry Lodriguss

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Posted 19 October 2020 - 01:56 PM

Mark,  I ran across this paper on using software called Tiny Tim to model the point spread function of the Hubble.

 

https://www.stsci.ed...inytim_spie.pdf

 

It's got some info that may be of interest.

 

Jerry



#6 sharkmelley

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Posted 19 October 2020 - 04:09 PM

Hi Mark,

 

Philosophical question: if you can't reproduce this pattern in a refractor in the real world, does it call into question the theory for an obstructed scope?

 

Jerry

An interesting question.  As I said earlier it is possible the Moire pattern is simply an artefact of the simulation.  On the other hand, the large scale structure in the rings of an obstructed scope is not an artefact but can be proved mathematically.

 

Here's an example of Airy rings from an unobstructed aperture vs a 50% obstructed aperture - the brightness has been gamma scaled to make them easily visible:

AiryRingExample.jpg

 

Large pixels won't have the resolution to sample the individual rings but they are more likely to successfully sample the larger scale structure of the 50% obstruction.

 

Mark



#7 Jerry Lodriguss

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Posted 19 October 2020 - 04:56 PM

An interesting question.  As I said earlier it is possible the Moire pattern is simply an artefact of the simulation.  On the other hand, the large scale structure in the rings of an obstructed scope is not an artefact but can be proved mathematically.

 

Here's an example of Airy rings from an unobstructed aperture vs a 50% obstructed aperture - the brightness has been gamma scaled to make them easily visible:

attachicon.gifAiryRingExample.jpg

 

Large pixels won't have the resolution to sample the individual rings but they are more likely to successfully sample the larger scale structure of the 50% obstruction.

Hi Mark,

 

Well, between proving it mathematically, and seeing an example from the real world, like the obstructed images and Hubble, I will always side with the experimentalists over the theorists, just like in quantum relativity. :-)

 

Jerry




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