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# Resolution, magnification and seeing.

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### #1 Javier1978

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Posted 23 January 2013 - 04:36 PM

Hi everyone,

I´m trying to understand some concepts about this topics, and I have several doubts. Lets see:

1.- Suppose I have a 6" scope and a 10" scope in a perfect seeing night. Given the same magnification, will the larger scope give more resolution or the resolution is something that you will achieve gradually when pushing up the power of the scope?

2.- How does the seeing affects the resolution in each scope at the same magnification?

3.- Is there any way to calculate the resolution considering the aperture and magnification. I know x aperture will give x resolution, but my eye will only be able to see some details by having a big image with high magnification. So, can I say, in a 8" scope in a perfect seeing night an eye will be able to see x arc sec details at 140x?

BTW, I know contrast might be another variant here in terms of planetary resolution, but I´m interested in the theorical resolution of the scopes.

Thanks in advance,

Javier.

### #2 Tony Flanders

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Posted 23 January 2013 - 04:44 PM

1.- Suppose I have a 6" scope and a 10" scope in a perfect seeing night. Given the same magnification, will the larger scope give more resolution?

That depends. Up to magnifications around 20X per inch of aperture, the limitations of the scope don't kick in. So at 120X both scopes will have the same resolution. At 30X per inch or higher, diffraction effects definitely kick in. So while a 6-inch scope will resolve more at 180X than at 120X, it will resolve less than a 10-inch scope at 180X.

The numbers cited are fairly arbitrary, and may very depending on the individual.

How does the seeing affects the resolution in each scope at the same magnification?

That is a subject of much debate. But to a good first approximation, it will affect both the same at the same magnification.

Is there any way to calculate the resolution considering the aperture and magnification.

It really depends on your eyes. People with superb vision can typically see all the available detail at 25X per inch. People (like me) with worse vision may need higher powers.

### #3 Javier1978

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Posted 23 January 2013 - 05:22 PM

Thank you Tony, a very clear answer!

### #4 GlennLeDrew

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Posted 23 January 2013 - 06:15 PM

To further complicate matters, object brightness, image surface brightness and contrast are variables which must be considered.

To provide just one example. Consider an object of low or moderate surface brightness, oh, say, M57, the Ring Nebula. You have on hand two telescopes of differing aperture but working at the same magnification. The bigger scope will thus operate with a larger exit pupil, thereby providing an image having higher surface brightness. The eye 'likes' to have a goodly amount of light to work with so that it can operate at higher resolution. A dim view causes our visual system to work at lower resolution, by grouping together into larger units bunches of light receptors. And so the brighter image, even though of the same size on the retina, will present more detail, or higher resolution.

And even for objects rather brighter than your tylocal DSO, such as planets, a larger aperture at given magnification often presents a more detailed view. Even when neither the resolution limit of the smaller scope nor the atmospheric seeing limit has been reached. Up to a certain point, at any rate, the eye 'prefers' a brighter image.

### #5 Javier1978

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Posted 23 January 2013 - 06:44 PM

Hi Glenn,

That was another doubt that I had. So, even though you are not reaching the theorical limit of either scope, the larger exit pupil will help in terms of resolution because the eye works better with a brighter image.

Thank you!

### #6 Jon Isaacs

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Posted 23 January 2013 - 07:15 PM

2.- How does the seeing affects the resolution in each scope at the same magnification?

Javier:

This is how I think about it:

Regardless of the magnification, the resolved image exists at the focal plane of the telescope. And, assuming that the eyepiece is of good quality, it also exists in the exit pupil of the eyepiece. In terms of the performance of the optics, the image can be considered to be resolved at all magnifications.

You question has to do with the ability of the observers eye to resolve the magnified image. This varies with the individual as well as the object and a dozen other things.

Tony and Glenn have addressed this aspect. Even for planetary viewing, I find that at 180x, the increased brightness of Jupiter in a 16 inch (2.25mm exit pupil) to provide more colorful, more detailed views over what I see in a say a 10 inch scope (1.4mm exit pupil).

Jon

### #7 Javier1978

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Posted 23 January 2013 - 10:20 PM

Jon,

I understand the concept of a resolved image at the focal plane and that in terms of optics that image could be resolved at any magnification, but isn´t that more a theorical concept?

Despite of the abilities of the different persons, the human eye probably has a standar resolving power given a magnified image formed at the focal plane. I mean, I guess there might be some standars of the average human vision as an optical element, and in this case, as the last optical element of the chain.

### #8 Jon Isaacs

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Posted 24 January 2013 - 05:42 AM

Jon,

I understand the concept of a resolved image at the focal plane and that in terms of optics that image could be resolved at any magnification, but isn´t that more a theorical concept?

Despite of the abilities of the different persons, the human eye probably has a standar resolving power given a magnified image formed at the focal plane. I mean, I guess there might be some standars of the average human vision as an optical element, and in this case, as the last optical element of the chain.

Javier:

In my mind, it's not a theoretical concept, there is a real image at the focal plane of the telescope. From that point on, all you are doing with an eyepiece is magnifying that image so that your eye can resolve it.

So questions are not really about the telescope or eyepiece optics but the ability of the eye to resolve the image. There are standards for the eye, you can back calculate them from the ability of the eye to resolve double stars. In general, it depends on the exit pupil... The 25x/inch to 50x/inch rule of thumb is based on the resolving power of the eye and the size of the Airy disk.

To use a simple case, a 1 inch aperture produces an airy disk that is 5.45 arc-seconds in diameter, 50 times that is a 4.5 arcminutes, 25 times that is about 2.3 arc-minutes. This about what the eye can resolve when viewing high contrast targets.

Consider the double-double. It's separation is 2.3 arc-seconds. At what magnification do you begin to resolve the pair, at what magnification do you easily resolve the pair? To my eye, I can convince myself that I can see something at 60x, it is generally well resolved at 100x though it is better at 120x...

2.3arcseconds x 60 = 2.3 arc-minutes, 2.3 arc-seconds x 120 = 4.6 arc-minutes, the 25x-50x rule...

Notice this is independent of the telescope, the assumption is the telescope is able to easily resolve the pair but it is the eye that is the limitation...

Jon

### #9 Tony Flanders

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Posted 24 January 2013 - 05:43 AM

So, even though you are not reaching the theorical limit of either scope, the larger exit pupil will help in terms of resolution because the eye works better with a brighter image.

Up to a point. When the target is really bright, like some double stars, extra brightness may actually be counterproductive.

But for planets, the extra brightness is usually helpful. Moreover, using a larger scope allows you to use bigger exit pupils, which helps people with visual defects such as floaters.

### #10 Tony Flanders

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Posted 24 January 2013 - 05:46 AM

I understand the concept of a resolved image at the focal plane and that in terms of optics that image could be resolved at any magnification, but isn´t that more a theoretical concept?

Not at all! You can place your digital camera at the focal plane and resolve that image directly. That's because the pixels on a digital-camera sensor are much, much smaller than the cells in your retina.

### #11 Javier1978

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Posted 24 January 2013 - 07:40 AM

Jon,

I understand the concept of a resolved image at the focal plane and that in terms of optics that image could be resolved at any magnification, but isn´t that more a theorical concept?

Despite of the abilities of the different persons, the human eye probably has a standar resolving power given a magnified image formed at the focal plane. I mean, I guess there might be some standars of the average human vision as an optical element, and in this case, as the last optical element of the chain.

Javier:

In my mind, it's not a theoretical concept, there is a real image at the focal plane of the telescope. From that point on, all you are doing with an eyepiece is magnifying that image so that your eye can resolve it.

So questions are not really about the telescope or eyepiece optics but the ability of the eye to resolve the image. There are standards for the eye, you can back calculate them from the ability of the eye to resolve double stars. In general, it depends on the exit pupil... The 25x/inch to 50x/inch rule of thumb is based on the resolving power of the eye and the size of the Airy disk.

To use a simple case, a 1 inch aperture produces an airy disk that is 5.45 arc-seconds in diameter, 50 times that is a 4.5 arcminutes, 25 times that is about 2.3 arc-minutes. This about what the eye can resolve when viewing high contrast targets.

Consider the double-double. It's separation is 2.3 arc-seconds. At what magnification do you begin to resolve the pair, at what magnification do you easily resolve the pair? To my eye, I can convince myself that I can see something at 60x, it is generally well resolved at 100x though it is better at 120x...

2.3arcseconds x 60 = 2.3 arc-minutes, 2.3 arc-seconds x 120 = 4.6 arc-minutes, the 25x-50x rule...

Notice this is independent of the telescope, the assumption is the telescope is able to easily resolve the pair but it is the eye that is the limitation...

Jon

You are right Jon, my question is about the eye resolving power ability, but I don´t see how this can be taken apart from magnificaction and scope resolution in terms of practical observation. Well, I think your explanation shows this thought very well, doesn´t it?

Thanks for the given information.

### #12 Javier1978

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Posted 24 January 2013 - 07:44 AM

I understand the concept of a resolved image at the focal plane and that in terms of optics that image could be resolved at any magnification, but isn´t that more a theoretical concept?

Not at all! You can place your digital camera at the focal plane and resolve that image directly. That's because the pixels on a digital-camera sensor are much, much smaller than the cells in your retina.

Well, but if you place a pixel big enough you won´t reach the scope theorical resolution, I´m right? So, the image resolved at the focal plane sounds OK but you will need a last element (eyepiece-eye or a sensor) that will determine your final resolution.

### #13 Asbytec

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Posted 24 January 2013 - 12:10 PM

Here's a cool theoretical explanation. Find the linear size of the Airy disc on the focal plane, then calculate the eyepiece focal length that will enlarge the Airy disc so the eye can resolve it at about 120" arc apparent angular size.

Turns out to be about 13x per inch of aperture and /consistent/ with Jon's 25x-50x rule of thumb.

Linear size in millimeters = k*lambda*Focal ratio. In my scope, that's 2.44 * 0.00055 * 13 = 0.017mm for the Airy disc on the focal plane.

Magnification required to expand 0.017 mm on the focal plane to 120" arc apparent size is eyepiece fl * 120" arc/206265. So, 30mm * 120" arc/206265 = 0.017 mm. That's about 65x at 1950 mm FL or ~ 11x per inch (150mm) minimum, according to the math, to see the Airy disc (which includes some dark inter-space between the bright central disc and the first ring.) To see the central disc, add a little more magnification ~ 13x/inch. I think that's where that magical number comes from.

Of course, at higher magnifications, image brightness and scale, exit pupil, etc., begin to have effects. The eye is complex and not standardized between observers. But, for the minimum power, I thought this was an interesting calculation. I hope I got that right, at least it works out to be 13x per inch in my own scope.

It should work in all scopes. Faster focal ratios have smaller Airy patterns, shorter focal lengths, and require shorter eyepieces to achieve 13x per inch and resolve their Airy discs. A similar thing happens with larger scopes and their Smaller Airy patterns, they need higher magnification but still ~ 13x per inch.

http://www.handprint...html#resolution

### #14 GeneT

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Posted 24 January 2013 - 04:43 PM

Javier,
You asked some excellent questions. And, there are some excellent answers here!

### #15 Javier1978

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Posted 24 January 2013 - 06:16 PM

Thank you Norme, I did not understand anything of that!

Gene, I think I have asked some questions that are beyond my understanding at this time. I think in basic concepts, but then the a little more sophisticated concepts or maths appears and I get lost. Even I got lost with Jon´s didactical explanation.

My english doesn´t help either. Anyway, I did learn a couple of useful things in this thread.

### #16 derangedhermit

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Posted 24 January 2013 - 08:43 PM

That's because the pixels on a digital-camera sensor are much, much smaller than the cells in your retina.

I read that rod cell density in the human eye is 80,000-160,000 / square mm.

I read that digital camera pixel density is from 15,000-1,000,000 / square mm. (1 to 8 microns per pixel)

If these are correct, then rod cell density is in the middle of the range of digital sensor pixel density.

Maybe I am making a mistake.

### #17 derangedhermit

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Posted 28 January 2013 - 07:30 PM

That's because the pixels on a digital-camera sensor are much, much smaller than the cells in your retina.

Rod cell density is in the middle of the range of digital sensor pixel density.

So folks, which of these statements (if either) is correct?

### #18 Tony Flanders

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Posted 29 January 2013 - 06:14 AM

That's because the pixels on a digital-camera sensor are much, much smaller than the cells in your retina.

Rod cell density is in the middle of the range of digital sensor pixel density.

So folks, which of these statements (if either) is correct?

Depends on the digital camera -- they're really all over the place. Pixels in webcams, inexpensive point-and-shoots, and cell phones are just a few times the wavelength of green light. With these, plus maybe a 2X Barlow, you really can image all the available detail directly.

The cost is very poor sensitivity, because a small pixel gathers few photons.

At the opposite extreme we have low-light video cameras, with pixels literally thousands of times larger (by area). Poor resolution and high sensitivity.

DSLRs are somewhere in between.

### #19 derangedhermit

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Posted 29 January 2013 - 08:25 PM

Compact digital cameras and cell phone cameras mostly have pixels between 1 and 2 microns.

Rod cells in the eye are about 2 microns in diameter.

Low-light security cameras have fewer but larger pixels than compact cameras, typically 3-4 microns in diameter, although there is more variation here than in the other categories. Low-light cameras often get much of their capability by using an IR LED as a light source, since the sensors are sensitive to near-IR.

DSLRs have the largest pixels in this little list, ranging from just under 4 microns to over 8 microns.

It's important to remember that the light-gathering capability of the pixel or cell is related to the square of the diameter, not the diameter itself.

### #20 Starman1

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Posted 02 February 2013 - 01:09 AM

I was taught the maximum resolution of a scope was two points separated by Dawes Limit, or 4.5/aperture of the scope in inches.

Hence, a 4.5" scope can resolve a 1arc second distance, and a 9" scope 0.5arc second.
I was also taught that, for "comfortable" viewing, the apparent separation in the eye for those two points should be 8' of arc. Since 8' is 480", that means a magnification of 480X would have to be used on that 4.5" scope to view the 1" separation "comfortably".
No way. That's 107x/inch.

Better to use 4' as a practical apparent separation. 4' is larger than the maximum resolution of the eye with good acuity (usually 2-3'), and would prove difficult for only a fairly small percentage of observers.

That means 240X for the 4.5" scope, which is much more practical, at 53X/inch, which is about where most books say the maximum magnification should be.

But could the 9" scope handle 480X to see the maximum resolution of that scope? Depends on the seeing conditions. Maybe yes, but probably not. Few places have good enough seeing to use 480X successfully.

### #21 Jon Isaacs

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Posted 02 February 2013 - 08:03 AM

Hence, a 4.5" scope can resolve a 1arc second distance, and a 9" scope 0.5arc second.
I was also taught that, for "comfortable" viewing, the apparent separation in the eye for those two points should be 8' of arc. Since 8' is 480", that means a magnification of 480X would have to be used on that 4.5" scope to view the 1" separation "comfortably".
No way. That's 107x/inch.

Better to use 4' as a practical apparent separation. 4' is larger than the maximum resolution of the eye with good acuity (usually 2-3'), and would prove difficult for only a fairly small percentage of observers

Don:

I consider myself a dedicated double star observer. Here's a few thoughts and experiences.

- To split a 0.5 arc-second double requires 0.5 arc-second seeing. The magnification necessary to make the split is immaterial, that depends on the eye.

- The double double is about 2.3 arc-seconds. 4 arc-minutes at the retina corresponds to about 104x. This is certainly a comfortable magnification if one is using a 3 or 4 inch scope where the airy disks are considerably smaller than the separation but even then higher magnifications can show the separation better. If the seeing is not optimal, higher magnifications may provide a cleaner image.

- The Rayleigh criteria is the separation when the two Airy disks are just touching, this is given by 5.45 inches/aperture. To see whether they are actually touching or not requires finer resolution of the eye than just the separation.

- If one is working at the Dawes limit (4.56inches/Aperture), the Airy disks are overlapping, and the observer is trying to see a 5% drop in brightness that appears as a thin dark line. This is a low contrast object and very difficult to see, quite different from the double-double in a 3 or 4 inch scope. In my experience, magnifications on the order of 80X/inch are necessary to see that thin line and "split" a Dawes limit double. A couple of years ago, when Zeta Bootes was right at the Dawes limit for an 10 inch scope, several observers in the double star forum were able to make the split but all used magnifications on the order of 800x or even more.

Sidgwick in his 1950s discussion of maximum magnification derives the 25x/50x rule but says that double stars may require significantly higher magnifications.

- Generally floaters are a problem for me but for some reason they are not so problematic splitting doubles at very high magnifications.

Just some stuff to think about.

Jon

### #22 Starman1

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Posted 02 February 2013 - 11:08 AM

Epsilon Lyrae makes an interesting test of your eye's resolution.
Can you see it as double with the naked eye (separation about 3')?
How low a magnification in the scope shows all 4 stars?

I've done the test:
1) I can *just* make out epsilon as double with my naked eye.
2) Using the 3' separation figure for my eye's resolution, that translates to a magnification of 78X to see all 4 stars in the scope.
3) I have eyepieces at 59X and 87X. I find the 4 stars easily split at 87X, which goes along with the naked eye test. I find the stars also split at 59X, which means that, theoretically, I can resolve objects at a naked eye separation of 2.26'.
4) Actually, I can't see epsilon as a separated double star with the naked eye--I see it as an elongated image.[My driving glasses separate the star into 2 points, but since they provide a half diopter of magnification, that doesn't count] What allows the separation to be seen at 59X is not primarily the magnification, since this image's exit pupil matches that of my naked eye, nor the brightness of the images in the scope. What does it is the star images are round tiny points in the scope and slightly flared, out of focus, images to the naked eye.

I wonder if there is another test of the eye's resolution in a naked eye double. Any ideas?

### #23 GlennLeDrew

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Posted 02 February 2013 - 11:34 AM

Don,
If your eye is the weak link, through the scope at an exit pupil equal to or larger than your eye's pupil the image should be equally bad. The telescopic image at best provides an image of fidelity approaching the direct view. In other words, even if the scope delivers to your eye a perfect wavefront, your eye will 'mangle' it just as badly as when no instrument is placed before it.

Epsilon Lyrae is not the best test for this purpose, as each naked eye component is a not-bright 5th magnitude. Were they a bit brighter, the resolution as a pair would be easier. And the apparent brightness of the naked eye test subject should be the same as the telescopic one. Through even a 60mm aperture, any one of the four components is rather brighter in appearance than the whole lot together with the unaided eye.

This in another example of how subject brightness goes most directly to the matter of resolution.

### #24 Starman1

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Posted 02 February 2013 - 11:56 AM

Don,
Epsilon Lyrae is not the best test for this purpose, as each naked eye component is a not-bright 5th magnitude. Were they a bit brighter, the resolution as a pair would be easier. And the apparent brightness of the naked eye test subject should be the same as the telescopic one. Through even a 60mm aperture, any one of the four components is rather brighter in appearance than the whole lot together with the unaided eye.

This in another example of how subject brightness goes most directly to the matter of resolution.

That's what I think, too.
I've seen what the limit of the eye's resolution can be--a 10 year old boy spotted the directions the "horns" of Venus pointed when Venus was nearing inferior conjunction. A few of us confirmed his sighting with binoculars that showed he had the orientation of the crescent just right.
Since the tips were, at that time, about 1' apart, he was obviously able to see the tips as separated. Incredible acuity. And no shortage of brightness, either. [Not to mention excellent seeing conditions]

Glenn, do you know any other separations that would be a better test for the naked eye?

### #25 GlennLeDrew

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Posted 02 February 2013 - 03:08 PM

Don,
The only test subject which comes to mind--and it's a severe one at that--is the 1 arcminute, equal pair, nu Draconis.

The ideal is if course equal or near-equal brightness pairs, the components having a brightness somewhere about 3rd magnitude (which will be a short list), going to as faint as 5th. A routine to search the Hipparcos catalog, looking for such pairings in the 1-4 arcminute separation, would make short work of the task. In essence, each star of suitable brightness is searched for neighbors not exceeding a defined magnitude delta and which have a calculated separation in the range sought. Later, on-line images could be inspected so as to ascertain no other neighbors which could compromise or 'contaminate' the view; an example would be a suitable pair found in a bright cluster or crowded association.

Or there might already exist a list compiled for just this purpose.

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