Double stars, airy disks and filters

[Stu]

I couldn't find anywhere more appropriate for this, feel free to move it if that makes sense.

After some discussion with and info from Nick about resolution and Dawes/Rayleigh limits, I did a little research and re-discovered this formula which I know but haven't really considered for a while.

x = 2 * 1.22 * Lambda * (f/D)

Where

x = Airy disk diameter in mm

Lambda = frequency of light in nm

f = focal length of scope

D = Diameter of scope

(aperture in mm)

I think there is a factor of 10^6 missing above somewhere to factor in the nm but I managed to make it make sense... I am of course open to corrections in all my comments by those who know far more about this than me!! :-)

This is interesting for a couple of reasons. Firstly airy disk size increases with focal length, and decreases with aperture, so there is a balance to be found between the two. It does imply that scopes with long focal lengths and comparatively smaller apertures (maks/SCT's) will not show particularly small airy disks, whereas an f4 16" for example or even a short focal length smaller frac would show the tightest airy disks.

This does bear out my experience to a large degree, the Mak I had was lovely on a range of objects but it never really delivered on double stars, this may be the reason whereas smaller fracs do.

Actually the Genesis I have comes out very well in this calculation, and that may explain why good examples are able to split doubles at low magnifications?

The second point to my post relates to how the frequency of light affects the airy disk size. If you run some numbers through it, the formula results say that for red light (620 nm), the airy disk is larger than for blue light (450nm) by 38%.

I see a number of comments in the forum about using red filters to help when viewing double stars. Other than pure dimming of glare, is this not missing the opportunity to filter on a frequency of light which gives a smaller disk, and thus better chance of splitting the double? The eye is not very sensitive to blue, so that probably doesn't make sense, but does it make more sense to try a green filter to give an optimum balance between sensitivity and airy disk size?

I've even been considering trying my continuum filter as this has a very tight bandpass at 540nm but quite possibly will dim the view way too much in a small scope, but may work well in a 12" or above?

This is intended to be a post to provoke (in the nicest way) discussion to help me (and others) understand the subject better, so, over to you all.......

Cheers,

Stu

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[Moonshane]

I cannot really comment too much to be honest but can certainly try some filters next time out

[ronin]

The equation is x/f=1.22*λ/D

However x/f is an approximation to Sin(θ).

Where θ is the angle that the ring forms from the axis, and that is where f comes into the equation, it is not a fundimental part of the equation.

The equation starts out as: Sin(θ) = 1.22*λ/D

The main concern is the amount of energy that the airy disk contains. If a lot then the disk is visible and a pain, if too dim to be seen then it has almost no impact visually, more for imaging that summs the collected light.

A short focal length refractor will have the airy disc very close to the image, thus making the image appear bigger and less well defined. Whereas a long focal length will push the airy disk well away from the image. In theory therefore you can push the airy disk outside of the instrument, thereby getting an image with no visible airy disk in the image plane.

[chiltonstar]

Blue  filtration improves the separation of close doubles, red reduces the glare for dissimilar doubles.

Small Airy disk is not the only criterion - dispersion of the two stars across the focal plane is another, hence many would say that a long-focus Mak is excellent for doubles!

Chris

[Stu]

Blue filtration improves the separation of close doubles, red reduces the glare for dissimilar doubles.

Small Airy disk is not the only criterion - dispersion of the two stars across the focal plane is another, hence many would say that a long-focus Mak is excellent for doubles!

Chris

Thanks Chris.

Can you explain what you mean by dispersion of the two stars across the focal plane? What does this mean and how is it a benefit?

I hope it was clear from my post that I was not professing to know these answers but was seeking more explanation and knowledge.

Stu

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[chiltonstar]

I'm not sure I know the answers either but I believe that doubling the focal length for the same mirror or objective diameter doubles the distance between two close stars at the focal plane and doubles the apparent diameter of the Airy disk (the image scale in terms of arcsec/mm halves in other words, dispersing the image formed across 4x the area). However, this effect is cancelled out in practice by using an EP with half the fl.

I think the reasons why a long focal length scope (slow) is usually better than a short focal length one (fast) for close doubles is said to be because the focal plane is flatter, and also optical errors are less in a slow system than in a fast one - less extraneous light gets scattered around in other words which doesn't get packed into the Airy disk and diffraction pattern.

Chris

[HumptyMoo]

Ok, I read this, now I'm done for the night. Will revisit in a couple of years...

[Stu]

The equation is x/f=1.22*λ/D

That equation is the same as the one I quoted, just re-arranged?

Forgetting other factors, the question I guess is if you have two scopes of the same aperture but different focal lengths, does the shorter focal length one produce smaller airy disks (which is how I read the equation), or not?

Stu

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[John]

Forgetting other factors, the question I guess is if you have two scopes of the same aperture but different focal lengths, does the shorter focal length one produce smaller airy disks (which is how I read the equation), or not?

I think the airy disks will be the same size, but I can't get my head around the maths ! 

[Stu]

I think the airy disks will be the same size, but I can't get my head around the maths !

:-), glad it's not just me then John :-)

Stu

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[Merlin66]

Guys,

I think there's some confusion about the size of the Airy disk.....

There are two key measures:

The angular size and the linear size....

The angular size varies with aperture and the linear size with focal length.

A 200mm f8 scope would give a diffraction disk of 11 micron, whereas for a 400mm f4 system you get 5.5 micron.

The 400mm aperture gathers x 4 the light and concentrates it into a much smaller disk - giving x16 the intensity.

[Stu]

Guys,

I think there's some confusion about the size of the Airy disk.....

There are two key measures:

The angular size and the linear size....

The angular size varies with aperture and the linear size with focal length.

A 200mm f8 scope would give a diffraction disk of 11 micron, whereas for a 400mm f4 system you get 5.5 micron.

The 400mm aperture gathers x 4 the light and concentrates it into a much smaller disk - giving x16 the intensity.

Thanks very much Ken. Can you clear this up finally and tell me the same information for 200mm f5 and say a 200mm f10 or f20? Is there a difference due to the focal length?

Cheers?

Stu

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[chiltonstar]

What is behind the question above is a little disturbing, re the long focus Mak not delivering on doubles. I assume this is/was a SW Mak 180 Pro, which I have been looking at seriously as my next scope. All the reviews seem to imply it is "the" scope for doubles, yet you Stu seem disappointed??

Chris

PS yes, I think Ken is correct re angular and linear, but I don't think you can forget optical aberrations in practice.

[Stu]

What is behind the question above is a little disturbing, re the long focus Mak not delivering on doubles. I assume this is/was a SW Mak 180 Pro, which I have been looking at seriously as my next scope. All the reviews seem to imply it is "the" scope for doubles, yet you Stu seem disappointed??

Chris

PS yes, I think Ken is correct re angular and linear, but I don't think you can forget optical aberrations in practice.

Chris,

The question is largely about me trying to understand the equation, but also trying to put some context on both experiences I have had, and read about on the forum.

The scope I am talking about was my OMC200 mak. It was/is a lovely scope and delivered lovely views on planets, globs and planetary nebulae. I never, however, particularly liked or used it for double star work. It may have been an issue of tube currents but the airy disks were not generally as tidy or small as a refractor. It would split the double double say, but needed higher magnification and aesthetically was not as nice.

Running the equation for this scope vs a short focal length 8" newt say says that it has a larger airy disk which surely will have a negative impact on ability to split tight doubles? So is that equation wrong, or does the focal length have an effect? I don't know the answer so am looking for clarification.

The other experiences I've read about are, for instance, reports from Ken of splitting the double double at x49 in a Genesis. Again, if you calculate the airy disk size for this scope, it comes out small relative to other scopes. So, is this a red herring or is it a genuine effect as suggested by the equation? I know that I never got close to this with the mak, but have done with my Genesis, and also other refractors.

Plenty of other effects come in to play, cooling and tube currents probably most significant.

The frequency of light also impacts this size, hence my other question. Some of the links I looked at are below:

http://www.astro-imaging.de/astro/wavelength.html

http://www.astropix.com/HTML/I_ASTROP/FOCUS/DEFS.HTM

There are also photography related sites such as this, obviously focal length in these cases is a result of stopping down the lens, but presumably the same principles apply?

http://www.cambridgeincolour.com/tutorials/diffraction-photography.htm

I produced a small spreadsheet of different scope airy disk size and also the variance with light frequency which I will post up when I get a chance next week. I found it interesting and it has been driving these questions.

Cheers,

Stu

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[Merlin66]

To quote your original post

""

Firstly airy disk size increases with focal length, and decreases with aperture, so there is a balance to be found between the two. It does imply that scopes with long focal lengths and comparatively smaller apertures (maks/SCT's) will not show particularly small airy disks, whereas an f4 16" for example or even a short focal length smaller frac would show the tightest airy disks.

""

OK first of all let's assume we are visually inspecting the Airy disks (this infers using the angular size of the Airy disk as the measure and then the magnification of the eyepiece to "separate" the Airy disks)

The size of the Airy disk will then depend ONLY on the aperture of the telescope. Bigger aperture, smaller Airy disks.

OK, now apply some magnification...the "normal" eye can resolve between 1-2 min of arc. To see the gap between a double star with (very small) Airy disks we have to apply sufficient magnification to "enlarge" the separation to an angular gap of say 2 arc mins.

If the separation was 10 sec arc we'd need (2 x 60)/10 = x12. This would show two "separated" disks.....but what about the disks themselves? well, your 200mm scope would give a typical disk of 2.44*550*10^-6/200 = 0.00000671 radians, approx. 1.3 sec arc - about 1/10 the separation between the star images.......

Does that help?

( A similar but different story exists for imaging but based on pixel size(s) and linear dimensions)