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What Does It Take To See Width In The Cassini Division?




This interpretation is based on my understanding and application of published data and measurement criteria with specific notation from the following sources. Noted paragraphs are not necessarily quoted but may be summarized directly from the source stated. All information if not noted from a specific source is mine. I have attempted to be as accurate as possible in my presentation of all the data and applications put forth here. Although this article utilizes much information from the sources noted, it represents my opinion and my understanding of optical theory. Any errors are mine. Comments and discussion are welcome.

Clear Skies, and if not, Cloudy Nights.
Edz March 2003,

UPDATED with corrections November 2003.

The original “Cassini” article was published in March 2003. Soon afterwards, I recognized there were errors related to my presentation of resolution and the Airy disk formulae. While all of the errors were mine, some were based on taking for granted that the printed works I referenced were accurate, which all were not. I immediately began the research needed to fully understand those topics and correct the information.

It has taken this long to gain a full understanding of all the aspects of the topic and clearly express the information in my own words. Many hours of reading and note-taking and pages of correspondence has led to a full article encompassing all the research, published here on CN as “Understanding Resolution”. All the information used here to correct this article is contained in that full article on resolution.

Edz Nov. 2003.

Introduction

All lenses or mirrors cause diffraction of light. Assuming a circular aperture, the image of a point source formed by the lens shows a small disk of light surrounded by a number of alternating dark and bright rings. This is known as the diffraction pattern or the Airy pattern. At the center of this pattern is the Airy disk. As the diameter of the aperture increases, the size of the Airy disk decreases. The aperture and the size of the Airy disk determine the limits of the scope to resolve two close point sources. Although there are many other factors involved, if no other aberrations limit the ability of the lens, then the lens is said to be diffraction limited, meaning limited in resolution only by the diffraction pattern it produces.

Resolution in optical instruments is dependant on the aperture of the lens or mirror and the wavelength of the light observed. Resolution is independent of focal length or magnification, however it is dependant on the magnitude and color of the objects observed. In telescopes we use the angular value for resolution, usually reported in arc seconds.

The Airy disk radius is measured from the midpoint of the central diffraction disk to the minimum of the first diffraction interspace. The central diffraction disk, sometimes confusingly referred to as the Airy disk, is somewhat smaller than the true Airy disk. There are numerous texts that claim the Airy disk is the central disk and that the formula results in the radius of the central disk. This is not correct. Keep in mind that the correct formula for the Airy disk gives the radius to the center of the first minima within the first dark interspace, so the angular dimension of the bright central disk itself is a bit smaller. Sidgwick gives a very good explanation of the Airy disk formula.

Knowledge of point source diffraction is necessary to understand the application of resolution to extended objects. While there are some simple conditions where point source diffraction may seem to explain resolution in extended objects, many extended object conditions are much more complex and the criteria explaining point source resolution is altered in some way. Contrast becomes a very important attribute affecting resolution in extended objects. In most cases resolution in extended objects is dramatically affected by our abilities of visual perception.


Saturn 10-23-03 - Wesley Higgins - photo used by permission


What must the observer accomplish?

Most observers would use their best optical system to achieve optimal planetary performance. They would be attempting to observe dimensional width in the Cassini division at the most obvious point that would provide success, that is where the division is at its widest at the ansae. These questions arise:

  1. Why can we see the Cassini division with even small scopes?
  2. What does it take to perceive it as having dimensional width?

Known information that is given:

  • Saturn's disk is 75,000 miles in diameter.
  • Cassini division is 2,800 miles across.
  • Currently (Nov./Dec. 2003) Saturn is 8+ A.U. from Earth
  • Currently (Nov./Dec. 2003) the disk appears 20 arcseconds across
  • The rings are 45 arcseconds across at the widest point.
  • Maximum tilt from the edge-on plane is 27°. In Nov./Dec. 2003 tilt will be 26°.
  • Based on these figures, at the ansae, at the very widest point visually

In Nov./Dec. 2003, Cassini's 2800-mile wide division would appear 0.75 arc-seconds wide.

The geometry of the problem

The discussion begins with the geometric presentation of the rings and the affect the tilt has on our observation. The full circle of the rings, due to the observed tilt, is presented as an ellipse. Without yet applying the affects of diffraction, the maximum and minimum width that we see can be determined based on trigonometry.

Draw a simple XY grid. Construct a line to represent the entire width of the planet’'s ring system along the X-axis but at an angle representative of the tilt (currently 26°). An observer at the origin sees the features as if they were projected on the end of the triangle that is the Y component. The dimensions of these features can be calculated by sine 26° x hypotenuse (true dimension). Any feature that is parallel to the X-axis is seen at its full dimension. However, any feature that is perpendicular to the X-axis is seen at sine 26° x dimension. There are many more points on the ellipse that are observed nearer to the minimum than to the maximum. The determination at all other points between max and min is quite a bit more difficult.

During the 2003 apparition, even though the rings are tilted ~26° to us right now, at the ansae we still see the full width of the rings. The width of the division at the ansae has not been reduced due to tilt. With a maximum tilt of ~27° (sine 27° = 0.45), the 2800 mile wide division, as seen by us directly in front of and behind the planetary disk appears as if it is ~1300 miles wide. If the Cassini Division measures 0.75 arcseconds at the widest point, it measures only 0.35 arcseconds at maximum at the narrowest point.

In future apparitions, when the tilt becomes more extreme, as will happen over the next 8 years or so, we will lose sight of the division where it is perpendicular to our line of sight in front of and behind the disk. Examples: when tilt = 14.5°, then sine 14.5° = 0.25, then 2800 miles x 0.25 = 700 mile visual width (0.2 arcseconds) when tilt is 15°; when tilt = 10°, then sine 10° = 0.17, then 2800 miles x 0.17 = ~500 mile visual width (0.125 arcseconds) when tilt is 10°.

Even when the rings become tilted to only a few degrees from edge-on and seem to be disappearing from our sight, at the ansae we will still see the full width of the rings. We will always see the full 2800-mile width of the division at the ansae, right up until the rings are edge on and we can no longer see the rings at all. We will however lose a great deal of contrast, which helps to see them.

Cassini division is an extended object.

Planets are considered extended objects. Although extended objects may be bright, they are generally considered as having low contrast differential across adjacent points.

Kitchin: “The definition of extended sources can include any source whose image is larger than the size of the elements of the detector. It can even include stars at high magnification or stars in poor seeing conditions.”

The Cassini division varies considerably from the general perception of an extended object, potentially moving it away from the classical definition of an extended object and closer to that of a point source, although something more like many point sources all lined up in a neat row.

Kitchin: For point source images that are smaller than the detecting element of the detector, the rod and cone cells of the eye, the telescope gives a brighter image. The situation for extended sources, images larger than the detecting element, is dependant on how spread out the light of the extended image becomes and the size of the exit pupil as compared to the size of the eye pupil. An image will be at its brightest that it can be observed when exit pupil = eye pupil, and this may be less than 7mm.

Although the rings easily fit the extended object criteria of images larger than the detecting element, the width of the Cassini division does not. The width is small, even smaller than the limit of resolution of many small scopes. With exception to its linear dimension, Cassini is a small, very high contrast image requiring fine resolution that fits the definition of a point source.

Can point source criteria be applied to extended objects?

I have found in my understanding that various point source criteria are applicable to extended objects. There are criteria when used in conjunction with each other that together make up a useful explanation for how we view extended objects.

Price: Planetary observers will have difficulty in applying point source criteria to planetary objects. For this, contrast becomes critically important.

Price makes several uses of applying point source criteria to extended objects. I could find no clear-cut criteria, other than light gathering and surface brightness, developed specifically for extended objects. Light gathering and surface brightness are not appropriate to judge our ability to see the brightly bordered, high-contrast image of the Cassini division. It seems that contrast and resolution are the most important criteria used to judge the ability to see the Cassini division as an extended object.

Porcellino: “The resolving power of a telescope is applied most often to double stars, but that is not the only area where it is important. It dictates such things as the sharpness of detail visible on a planetary disk …”.

Cassini is presented to us at extreme high contrast, making it as easy to see as a close double. In addition, Cassini is like hundreds of doubles lined up in a row taking on linear dimension. Due to this added dimension and the high contrast, Cassini becomes easier to see and may be an example of an object that exceeds normal criteria measurement limits.

Price: The extended images of the planets can be considered as being made up of an infinite number of Airy disks and diffraction ring images of every point on the planet. High resolution is necessary to observe and contrast will improve the visibility.

Sidwick is his “Amateur Astronomers Handbook”, provides excellent examples for understanding the diffraction pattern for a linear presentation of an infinite number of point sources.

Resolution & Contrast

In telescopes, the most important component is aperture. Aperture controls maximum resolution. No amount of magnification will enable you to resolve a double star that is beyond the limit of resolution provided by your aperture. Maximum potential resolution determines the ability to resolve both point sources and fine detail in extended objects.

In addition to maximum resolution, the ability to split close objects, we need to discuss resolution of fine detail. Price has the best explanation of the importance of contrast for the resolution of fine detail.

Price: the separation of two almost touching bright diffraction disks on a black background is different from the resolution of fine irregular detail of varying contrast and brightness…” on planets. “Although usually defined as ability to resolve fine detail, the resolving power and aperture of a telescope are alone not sufficient to determine a telescope’'s ability to resolve detail. Of greater importance is contrast and this has nothing to do with resolving power.” “…the contrast between fine detail and its immediate neighborhood…affects the resolvability of the detail.”

We must rely on superior contrast to provide us with the optimum utilization of the system being used for viewing. In some cases, higher magnification provides greater ability to see, for instance when splitting close doubles. But better contrast and resolution, provided by higher quality optics, at a magnification lower than maximum may result in seeing more. The combination of high quality optics, proper baffling and improved coatings provides increased light to the eye, less scatter, higher contrast and overall higher resolution.

Extended Object Contrast

If the light gradient is not perceptibly different from the adjacent area, such as in extended objects, it is more difficult to see the features in the object. It's easier to see drastically changing features than it is to see slightly changing features. The ability to resolve these differences to any extent is primarily dependant on contrast. There needs to be sufficient contrast between the adjacent areas within the object, or between the object and the background, in order to see it. Higher contrast makes things easier to see.

Cassini provides us with very high contrast. The light within the rings is far brighter than one might think. Price recognizes the rings as the brightest feature by which all others should be judged. The rings have very sharp edges. There is no diffusion of light at the edges of the rings. There is a sharp break between the light and dark areas of the A and B rings and the division.

Price: The width of the Cassini Division is 4500 Km or 2800 miles. The B ring is the brightest feature of the Saturnian system and should be used as the baseline for magnitude estimates of all other features.

Porcellino: Cassini’'s division separates the brightest portion of the B ring from the brightest portion of the A ring. The division is bounded by the brightest features available within the ring system.

What criteria can be applied to the division?

Replies posted by one forum participant stated,

[1] “"A 0.7" double - where a pair of stars is separated by 0.7" -has, due to diffraction, much less than 0.7" separation between the disc edges. How much, depends on both, magnitude (assumed near equal) and aperture.”

[2] “ "If we take a "standard" white 6mag double in 6" aperture, visible diffraction disc will be somewhat less than half the Airy disc diameter (1.8"); that's what allows us to see it just resolved. If you'd have diffraction disc of that size formed by point sources at the two bright edges enclosing the division, you simply wouldn't see… it - not even the widest segment.”

[3]” This obviously shows that stellar diffraction limit can not be applied to a resolution of extended objects. It's due to the fact that any light point source on Saturn's ring is much fainter than a 6mag star in a 6" aperture - consequently, with much smaller (likely invisible) diffraction pattern.”

[1] The image of the Airy disk is not a point but a disk of measurable dimension. Therefore due to diffraction the space separating two stars is less than the space between their centers. Although Airy disk size is not dependant on magnitude, the extent of the bright central disk is dependant on magnitude. The separation for any double is a measurement between the centers. Any double will have much less space separating the edges of the diffraction disks. The separation stated, which is between the centers is not an indication of the space between the bright central disks. And in fact if we are observing near the Rayleigh Limit and if the objects are bright, there may be no space at all between the objects.

[2] The diffraction disk size will help us understand the difference between seeing the Cassini division and resolving it. There are special extended object criteria that explain seeing features much finer than our scopes are capable of resolving. As an example, linear features such as a black line on a light background can be seen at R/3.5 to R/5, where R is the value for point source resolution. It is important to note here, although we can see the linear feature, a small scope is still not capable of resolving the feature if the thickness of the line is less than the smallest image dot size the scope can produce. The smallest feature a scope can resolve must exceed the resolution of the scope. It will be shown that scopes can display certain features they cannot resolve.

[3] Several authors note the use of point source criteria as applied to extended sources. In addition, the rings provide light source that is far brighter than one might think, with extreme high contrast. A much more thorough explanation of diffraction in extended features is to be found in the reference books listed. However, it can be stated here, every source of light has a diffraction pattern. Features such as the continuous bright lines of the rings can be considered as a row of an infinite number of points, each with its own diffraction pattern. The extent to which they overlap, the shape and size of the features and the contrast present in the image is what affects our ability to see it. The applications of Rayleigh Limit and extended object resolution together are needed to understand how we can see features smaller than our scopes can resolve and what it would take to resolve such features.

[4] One respondent asked, “"How is that explanation consistent with the fact that the Cassini division can be seen with a four inch refractor? Could it be that the analogy of splitting a double of 0.7 arc seconds separation doesn't hold, due to the fact that these really aren't point sources like a distant star?”"

[4] Certainly splitting Cassini is not exactly analogous to splitting a double star. I’'ve noted above the special criteria for the resolution of extended linear features as a ratio of point source resolution. However, the criteria by which we measure the splitting of doubles is the same by which we measure the resolution of our optics. Although we can see an extended feature, we need to consider the resolution capabilities of the instruments to determine at what size we can resolve the feature.

[5] Another respondent posted, Then, the 6" scope ought to see it… assuming excellent seeing of course… and 8" should be able to show it as a wide gap like when pumping magnification on double stars after they are split.

[5] The explanations above and the work that follows will help to explain why we can see the Cassini division in a 6” scope or even a 3” scope, even though it is beyond the resolution capabilities of that size scope. Also it will be explained, at least in my opinion, what is required to resolve the division. As I stated in [2] above the smallest feature a scope can resolve must exceed the resolution of the scope. The resolution capabilities, based on the Airy disk formula or Rayleigh criteria, play an important role in just what the scope is able of resolving.

Common Diffraction Limit Criteria

Porcellino: “No amount of magnification will allow you to resolve a star that is beyond your telescope's limit.”

Resolution as used in astronomy is commonly measured by angular measure in units of arc seconds. The Airy disk has the same linear size (more or less) for all scopes of a given focal ratio. The Airy disk has the same angular size for all scopes of a given aperture.

In telescopes, the most important component is aperture. Aperture controls maximum resolution. Maximum potential resolution determines the ability to resolve both point sources and fine detail in extended objects.

The two most commonly used criteria for measuring limits of resolution are Rayleigh limit and Dawes limit.

Rayleigh Limit = 5.45/D inches = 138/Dmm.

Rayleigh Limit is a measure defining the limit at which two components can be clearly identified as separate components. It defines the distance between the centers of two Airy disks where the maximum of one is placed over the minimum of the other.

The Rayleigh limit of a 6”/150mm telescope is 5.45/6” or 138/150 = 0.91 arcseconds.

Rayleigh Limit is a measure that correlates to the wave nature of light. The correlation will be shown later.

Kitchin: The angular resolution is equal to the Rayleigh limit, where separation between two stars is considered as achieved when the stars are just touching.

When observing stars at the theoretical limit, for the condition “just touching” to be possible, the visible disk must be no more than 50% of the diameter of the Airy disk. This would be true only for moderately faint stars. In this case, the distance between the centers of the two Airy disks is equal to the diameter of each of the central bright disks and it has the radii of the two visible disks just touching. This is not to be misunderstood as a measure of a space between the bright edges of the two Airy disks.

Based on a strict interpretation of the Rayleigh limit, two point sources with a separation of more than 0.91 arcsec could not be separated with a 6” scope. This is not always true in the strictest sense, as some equipment may be capable of exceeding standard limitation criteria. In addition, there are many other factors to consider, not the least of which is atmospheric turbulence.

Dawes Limit = 4.56/D inches = 116/Dmm.

Dawes Limit is the first point at which a double star is elongated enough to suspect the presence of two stars. Like Rayleigh, it is not a measure of separation required to see a black space between components.

Dawes limit for a 6” / 150mm telescope is 4.56/6 or 116/150 = 0.77 arcseconds.

Dawes is not a measure dependant on the wave nature of light. It was empirically determined to represent a point of minimum separation where a double can be noticed as two components. It is the first point at which a noticeable notch allows a determination that two components exist.

Dawes limit is not used to indicate an achievable black space between two point sources. It is however a reliable determination of a limit at which two moderately bright close stars can be noticed as being two separate components. It is not a criteria that can be used or applied to resolution in extended objects.

The Airy Disk

In my research of the Airy disk formula, I noted numerous well-regarded astronomy references and several websites with published astronomy related formulae that refer to a shortened form of the formula for the Airy disk size. That shortened form in all cases was given as A = 1.22/D where it is stated that A = arcsec and D = scope dia. in meters. This formula is incorrect. Further, some, not all, of the same books and websites imply that the Airy disk and Dawes Limit are even equivalent. There are in print a number of incorrect associations, wrong units used in formulae or a general lack of explanation to provide a full understanding for the presentation of the Airy disk formula.

Based on the definitions of Dawes Limit and Rayleigh Limit, it is clear that Dawes Limit and Airy disk size are not equivalent, but that Rayleigh Limit and Airy disk size are related and equivalent.

Do not confuse the definition of the Airy disk as the bright central dot in the diffraction pattern. This is really not correct and this term is very often confused in much of the literature in print. Diffraction produces a pattern called the Airy pattern, dominated by the Airy disk and then surrounded by a number of less bright diffraction rings. The Airy disk is measured out to the minimum of the first diffraction interspace. The central bright disk is correctly referred to as the spurious disk. There is no true measurement for the spurious disk itself.


I will refer to the spurious disk or central disk as the visible disk.


The measurement 5.45/D (see derivation below) based on the wavelength of light (and specific only to yellow light at 550 nanometers, the light to which we are most sensitive) is measured out to the first minima. As we move from the center of the visible disk out into the first dark interspace the minima occurs. The edges of the visible disk usually cannot be seen as the light falls off to zero towards the first minima. The dimension of the Airy disk varies with the wavelength of light, being larger for red light and smaller for blue light. Therefore, given equal magnitudes, it will be slightly easier to split two blue stars than two yellow stars and both are easier than two red stars.

Sidgwick: Resolving power is dependant on wavelength of the light observed and the diameter of the objective. The radius of the Airy disk is also referred to as the resolving power of the telescope.

Resolving power is not completely independent of the magnitude of the light observed and this will be explained later.

Beiser: The angular radius of the Airy disk out to the first minima is represented as:

A = 1.22ª / D, where

A in radians = 1.22 λ (Lambda) / D (Aperture)

A is the angular radius of the Airy disk measured in radians.

Lambda is the wavelength of light = 550 μm or 550 nanometers = 550 x 10^-9 meters.

Visible light is between 420 μm and 650 μm. We use 550 μm, the wavelength of yellow light.

D is the diameter of the aperture in meters. For a 150mm scope D = 0.15meters.

Then A = 1.22 x 550 x 10^-9 meters / 0.15meters = 0.0000045 radians

Converting radians to arcseconds,

then 0.0000045 radians x 360/2pi x 60 x 60 = 0.92 arcseconds.

The angular radius of the Airy disk for a point source resolved with a 150mm telescope observing yellow light at a wavelength of 550um is 0.92 arc seconds.

Alternatively, Setting D equal to 1 inch then

A = 1.22 x 550 x 10^-9 meters / .0254meters = 0.0000264 radians

Converting radians to arcseconds,

then 0.0000264 radians x 360/2pi x 60 x 60 = 5.45 arcseconds.

The angular radius of the Airy disk for a point source resolved with a 1 inch telescope observing yellow light at a wavelength of 550um is 5.45 arc seconds.

When D = 1 inch, A arcsec = 5.45/D. Therefore for any D, A arcsec = 5.45/D.

Kitchin: the angular resolution is equal to the Rayleigh limit, where separation between two stars is considered as achieved when the stars are just touching.

As mentioned before, for the condition “just touching” to be possible, the visible disk must be no more than 50% of the diameter of the Airy disk. This would be true only for moderately faint stars. Bright stars put greater light into the visible disk and very faint stars obviously put less light into the visible disk. However the size of the Airy disk remains constant for a given scope.

Repeated here, Rayleigh Limit resolution = 5.45/D inches = 138/Dmm. For a 6” scope, the Rayleigh limit is 5.45/6” = 0.91 arcseconds. This compares closely to the resolution calculated above using the formula for the wave nature of light. It varies only because the value 6” is nominal and does not correlate exactly to 150mm. 150mm/25.4 = 5.9 inches. 5.45/5.9 = 0.92 arcseconds.

Understanding Rayleigh Limit

Rayleigh Limit = 5.45 / D inches (or 138 / Dmm) is a measure of the ability of the scope aperture to split a double star.

Likewise, Dawes Limit 4.56 / D inches (or 116 / Dmm) is another measure.

Rayleigh Limit states you should be able to tell that a double is two stars if the centers of the diffraction disks of the two stars (commonly referred to as the Airy Disks, but see below) are separated by a dimension equal to the radius of the first diffraction interspace. That’'s the distance from the center of the Airy disk to the minimum of the space between the disk and the first diffraction ring. (This is important. I will refer to this a little further down). This calculation is directly tied to optics theory and the ability of a lens to resolve detail based on the wave nature of light. The limit of a lens to resolve is determined by the diameter of the lens and the wavelength of light. Take note that this limit, which has the centers of two disks separated by the radius of a disk may not provide for any black space between the two components.

Dawes limit was determined by actual field-testing of many and varied double stars. Close doubles observed at the Dawes limit might be recorded as elongated or notched, but usually not separated. Dawes is not a criteria that can be usefully explored for relation to extended object resolution.

It is true that you can tell there are two components to a double before you have reached a point where they are completely split with a black space between them. When observing doubles I keep my notes for the various eyepieces, using terms something like elongated, elongated pointed, notched, barely touching, thin black line, clear black space.

Generally, it is held that Rayleigh (and Dawes) should only be applied to equal 6th magnitude doubles. It has been shown that resolving limits are not independent of magnitude. Some very good telescopes are capable of exceeding both of these limits. Conversely, some lesser quality scopes will not be able to even reach these limits. But these are good indications of what a good telescope should be able to see.


Seeing a Black Space Between Components

Now back to the passage I referred to as important. Let me further explain. First I will repeat what I said earlier. Rayleigh Limit states you should be able to tell a double is two stars if the centers of the diffraction disks of the two stars are separated by a dimension equal to the radius of the first diffraction interspace. That’'s the radius from the center of the Airy disk to the minimum light within the space between the disk and the first diffraction ring.

What if I want to see an object with at least a thin black space between the components? What are my limits? What should I expect of my scope?

Based on the definition above, Rayleigh limit is a measure of a radius. It is the measure from the center of the bright central dot, or the visible disk, out to the minimum of the “black space” between the visible disk and the first bright diffraction ring that surrounds the Airy disk. If you want to see two objects as completely separated with a thin black space between them it is necessary for the centers of the Airy disks of the two components to be separated so this “black space” overlaps and becomes visible between them. That separation dimension for bright stars is approximately equal to two radii or the diameter of the Airy disk.

The visible disk itself is slightly smaller than the Airy disk dimension since the Airy disk is measured out to the minima of the first dark space. The visible disk varies with magnitude, but it’'s about 85% of the Airy disk dimension for a bright star. For a faint star less than 50% of the diameter of the Airy disk is occupied by the central bright disk.

A separation of something more than the Rayleigh Limit is needed to have a black space between two objects. If the objects with separation at the Rayleigh limit were faint and had a visible disk just a hair less than 50% the diameter of the Airy disk, you would be able to see a thin line of separation. Using a reasonable 60% bright central disk assumption, it requires objects separated by Rayleigh Limit times 1.2 (60/50) before you can see a split. For very bright objects with a central disk 85% the diameter of the Airy disk, it would require separation of Rayleigh Limit times 1.7 (85/50) before you could see a split. Very bright and very faint objects will begin to impose their own sets of limitations on resolvability due to magnitude.

Resolving Power

It has already been explained, the image of point sources are not points, they are disks. They can be so close that they overlap to the extent they cannot be visually perceived as two points. Magnifying may be sufficient to separate these if their images were points, but since they are disks all it will do is magnify the overlapping disks. In this case, resolution can only be improved by increasing D. This will have the effect of making the disks smaller allowing for less overlap in the image.

Derivation shows as equivalent the results of the formula for calculating the radius of the Airy disk and the results of the formula for Rayleigh limit. Therefore the Rayleigh limit can be used for a given telescope to calculate the Airy disk radius produced by that telescope. Any feature that does not have angular dimension greater than the limit of the telescope’'s ability to resolve cannot be observed as anything larger than the Airy disk. Since this is the smallest image spot size that can be achieved by a given telescope, no matter how large that image is magnified, it cannot be construed as having width. You would not be magnifying a resolved feature. You would simply be magnifying the smallest image spot size.

Based on Rayleigh limit criteria, 5.45/D inches or 138/Dmm:

Airy disk radii for telescopes of diameter D inches: 4” = 1.36arcseconds, 5” = 1.09arc”, 6” = 0.91arc”, 7” = 0.78arc”, 8” = 0.69arc”, 9” = 0.61arc”, 10” = 0.55arc”. The diameter of the Airy disk is twice these values, however the visible disk, depending on the magnitude of the object observed, can range from about 85% to less than 50% of the full diameter of the Airy disk. To confirm the ability of your telescope to achieve these results, it would be necessary to observe and record various results for doubles near and beyond the projected limits for your scope.

Conversely, any telescope that does not have the ability to produce a resolution finer than the size of the feature will never see that feature as having dimension. A feature may be seen by smaller scopes, especially as we have to consider features other than point sources such as craters on the moon or the Cassini division as having length and width, but the feature is not seen resolved with the width dimension exceeding image spot size unless the telescope resolution is sufficient.

Resolving Power in Extended Objects

Knowledge of point source diffraction is necessary to understand the application of resolution to extended objects. While there are some simple conditions where point source diffraction may seem to explain resolution in extended objects, many extended object conditions are much more complex and the criteria explaining point source resolution is altered in some way. Contrast becomes a very important attribute affecting resolution in extended objects. In most cases resolution in extended objects is dramatically affected by our abilities of visual perception.

Stellar diffraction limits must be understood to apply diffraction to the resolution of extended objects. This is supported by more than one author as noted previously, and again noted here. A good example may be the Cassini division in Saturn’'s rings.

Porcellino: “The resolving power of a telescope is applied most often to double stars, but that is not the only area where it is important. It dictates such things as the sharpness of detail visible on a planetary disk …or the moon.

The dimension of the Cassini division has been measured and mapped by close satellite imagery. Therefore, consider it to be a 2800-mile wide dark band bounded immediately on either side by the bright light of the A and B rings. If two points of light were considered as being on directly opposite sides of the division, our perception of the view would be of two point sources with a separation of 0.75 arc-seconds between their centers. We would not see a 0.75” space due to the affects of diffraction.

In order to perceive width in the Cassini division, what we are talking about is seeing across the 0.75 arcsecond gap, the point source separation dimension, and seeing it as having width, not just as a line. If the gap were beyond the resolving ability of your scope, you may see the gap but the gap would be no more than a line. It would not have dimensional width.

Sidgwick: “A bright line of negligible width, crossing a dark area, may be regarded as consisting of a very large number of contiguous points. Each of these will produce its own diffraction pattern, with the result that the image of the line will be thickened by a fringe on either side.”

In Rayleigh limit we have a criteria that provides for a qualified calculation of two objects just touching with a perceivable black space between them. A 9” telescope has a Rayleigh limit of 0.61 arcseconds. Assuming 60% visible diffraction disks, at 0.61 x 1.2 = 0.73” it could just separate objects to appear with a black space between them. Since they are resolved by the scope, this would indicate that any single isolated points bounding a 0.75” separation as viewed through a 9” scope might just be perceived as having dimension in the resolved image of the black separation between them. Other criteria will be shown that account for the fact the division can be seen with smaller scopes, but this may be the smallest scope that actually will allow the observer to perceive width in the division.

A larger aperture will provide a smaller diffraction disk. High contrast will improve the ability to see the division. And finally, an added linear dimension to the feature helps make the image easier to see. All these things will have an affect on the ability to see the image of the division. We do know this; the Cassini division can be seen by scopes with considerably less than 9” diameter.

Why can we see Cassini in small scopes?

Cassini obviously is viewed by much smaller scopes than point source diffraction would dictate and I offer this as an explanation:

I believe the primary reason we all see Cassini with small scopes is due to its linear dimension. This linear feature extends the high contrast image to an appearance of length or an appearance that might be described as width between two bright points stretched out in a linear dimension.

Remember, I've said in my opinion, because it has linear dimension it becomes much easier to see than a point source. Although the dimension across the division would be similar to the resolution limit, the fact that the observed image has a second dimension, a significantly observable length, we get to observe the object in more than one dimension. This allows our eyes to perceive much greater detail. The eyes function more acutely when viewing very high contrast objects and even more-so when viewing linear objects.

Look at the following example.

Type a single line like this ( - ) on a page. Imagine for a moment that the single line is the black space between two point sources. Now step 10-15 feet away and look for the line. You probably cannot see it, I couldn't.

Now type a continuous line on a page like this ______________________

The single line dash represents the space between a double star. Obviously, the long line represents the Cassini division. Imagine that the continuous line represents an infinite number of spaces equal in width to the short line above but continuous, such as the Cassini division. Step 20-30 feet away and you can still see it. It is no thicker than the first line, but because it has much greater linear dimension there is much more of it to see and we can more easily see it.

I recorded the performance of several small scopes in attempts to find the smallest scope/magnification combo that allowed seeing the Cassini division. The significant results are included here:

  • 150mm/1200mm ref - 6mmUO ortho=200x –- seen almost 100% around best view
  • 150mm/1200mm ref - 10mm Radian=120x –- easy view but not enough contrast
  • 125mm/1350mm sct - 7mmUO ortho=190x –- seen nearly 100% except where disk in front
  • 125mm/1350mm sct - 9mmUO ortho=150x –- sharpest 5” view seen about 70% around
  • 90mm/500mm ref - 7mmUO in 2xTV=140x –- distinct about 50% more-so following edge
  • 90mm/500mm ref - 5mmUO ortho=100x –- good contrast, Cassini 30-40%, low limit 90mm
  • 90mm/500mm ref - 6mmUO ortho=80x –- Cassini only suspected
  • 85mm/600mm apo - 5mmUO ortho=120x –- seen except in front and back about 60%
  • 85mm/600mm apo - 6mmUO ortho=100x –- long arcs seen 40-50% around
  • 85mm/600mm apo - 7mmUO ortho= 85x –- long arcs constant at both ansae low limit 85mm
  • 85mm/600mm apo - 8mmTV plossl= 75x –- only glimpsed long arcs at ansae
  • 78mm/480mm ref - 7mmUO in 2xUltima=140x –- difficult view but seen
  • 78mm/480mm ref - 4mmUO ortho=120x –- barely seen maybe 30%, low limit 78mm
  • 78mm/480mm ref - 5mmUO ortho=100x –- only momentarily suspected on following ansae
  • 76mm/600mm newtcas - 5mmCel Vix ortho=120x –- rings very distinct, but No Cassini
  • 76mm/600mm newtcas – 8mmTV x2Vix = 150x –- rings wide, 2 moons, but No Cassini
  • 60mm/415mm ref - 5mmCel Vix ortho = 83x –- nice view of rings, but No Cassini


Extended Object Resolution Criteria

Cassini obviously is viewed by much smaller scopes than point source diffraction would dictate. I believe the primary reason we all see Cassini with smaller scopes is due to its linear dimension.

Sidgwick: “A linear object may stimulate a sufficient number of cones to produce sight even though its width is 20 or 30 times less than the threshold diameter of a spot.”

This above statement refers to the ability to perceive the feature. It does not however indicate that the dimension of the feature can be resolved.

Remember, I've said in my opinion, because it has linear dimension it becomes much easier to see than a point source. Although the dimension across the division would be similar to the resolution limit, the fact that the observed image has a second dimension, a significantly observable length, we get to observe the object in more than one dimension. This allows our eyes to perceive much greater detail. The eye functions more acutely when viewing very high contrast objects and even more-so when viewing linear objects. Visual perception has a great deal to do with explaining why we can see a black line on an apparent white background with an aperture smaller than point source resolution dictates.

Sidgwick: “Diffraction at the edge of a bright area of the image results in blurring of that edge. Encroachment of light from the bright into the dark area causes a dimming of a narrow strip (of the bright area) and the graying of an equal strip (of the dark area).”

This linear feature extends the high contrast image to an appearance of length or an appearance that might be described as width between two bright points stretched out in a linear dimension. The blurring at the edge condition may have the result of making the division appear wider than it is in reality.

Sidgwick gives examples in his book for visual resolution of various stellar and extended conditions. Point source resolution, a white pinpoint on a black background, is explained by the Airy disk formula. In the following example, R represents 5.45/D, the Airy disk diameter or Rayleigh point source resolution.

For a black spot on a white background, several experiments show results that indicate results or R/2.3 and R/3.

For a single dark line on a light background, similar to the Cassini division, the same experimenters show results of R/3.5 and R/5. Two other specific tests provided results of R/14 and R/15.

The above conditions are found to be easier to resolve than point sources. The next example is found to be more difficult.

For parallel dark lines on a light background, several results indicate a limit of 1.1R to 1.4R.

Diffraction Fringes Interfere with Resolution

In a perfect system, a larger diameter objective imparts a smaller diffraction fringe around every point of light in an image and the result is we can see finer detail. Contrast is the quality that allows seeing more in the detail provided by the rest of the optical system. Observing at magnification below optimum results in a diffraction fringe surrounding each point of light broader than when using an optical system at optimum resolution, even in extended objects.

In the Cassini division, the light gradient and therefore contrast across the boundary of the division is at a maximum. Planetary observers are using their best optical system to observe at optimum resolution therefore the individual observer, thru the use of optimum equipment, by default holds diffraction fringes within the system in use to a minimum.

High contrast boundaries seen with an optical system operating near maximum potential resolution provides us with a view of an object that does not seem to fit the standard low contrast light gradient criteria of the typical extended object. The Cassini division seems to be the perfect example of an extended object that is as far away as we can get from the classical low contrast criteria of an extended object.

Magnification helps you to see

The final achievement that needs to be accomplished is to make the image large enough for the eye to perceive. It is not enough to simply have a scope with a large enough diameter to resolve the object. The limiting factor is the eye’'s resolution and sufficient magnification must be employed to raise the image to a size that the eye can perceive.

As I stated earlier, no amount of magnification will enable you to resolve a feature that is beyond your aperture’'s limit. Likewise, no objective lens will reach the limits of its maximum resolution unless sufficient magnification is employed. With increased image scale higher magnification utilizes more of the ability of the available aperture.

Kitchin: The resolution of the eye is 3 to 5 minutes of arc; therefore a minimum magnification must be utilized to enlarge the image sufficiently to exceed the eye’'s resolution. Therefore minimum magnification of about 1300Dmeters or 30Dinches is needed to realize the potential limiting resolution of the optics.

I did encounter a great deal of text and practical discussion relating empirical data gathered over decades that show a significant amount of magnification is required to show all the resolution the telescope is capable of delivering to the eye. In the same practical use discussions, it is readily accepted as magnification increases, contrast of the image thru the eyepiece, as compared to the sky background, is increased. In fact it is this increase in contrast that sometimes allows some objects to even be seen at all.

Liller: The higher the magnification, the more efficient the eye, at least up to a point. A 150mm objective at 100x will see stars more than a magnitude fainter than the same objective at 30x.

Certain objects take magnification better than others. Objects sometimes need the maximum magnification you can get from your instrument in order to resolve them. Sometimes higher magnification is called for to make the most of the telescope’'s resolution ability. Double stars and planets are examples that can take a lot of magnification.

High magnification is sometimes required to achieve a telescope’'s resolution limits. The visibility of the detail provided to the eye depends on a many factors, such as the kind of detail (point source, linear or circular), the amount of light in the detail (brightest and faintest are more difficult), the relative contrast of that detail (white on black, black on white, faint diffuse or colored), and the visual acuity of the eye at the telescope (varies widely).

Sidgwick: What is the low limit of magnification needed to resolve two points within the capabilities of the objective? If they are not resolved in the focal plane, no magnification will allow the eye to see separation.

Though two points may be resolved in the focal plane, they will not be seen as resolved unless a high enough magnification is employed to allow the eye to perceive the separation. This will vary with the individual user. Point images need to subtend an angle of at least 1 arcmin or the eye cannot see two points. This would yield a magnification of 13Dinches. In the case of stellar images, the value is nearer to 2 or 3arcmin and these yield 24D to 36Dinches. With lower magnifications than these, there is still potential resolved detail in the focal plane image that has not been developed to the point of visibility.

A decrease in the exit pupil to less than 0.85mm leads to a progressively more negative effect on vision. This imposes an upper limit on useful magnification. The resultant 30D provides the maximum useful magnification that can be employed. At this limit, all the gain in resolution has been reached. Of course this upper limit can be exceeded by twice for some objects, such as close doubles stars and planets.

Acuity

The resolution of the eyes in practical astronomical observation IS NOT 60 arc seconds. Many measurements relative to visual acuity have been analyzed and results can be found at various sites including google (search on visual acuity), C. R. Kitchin’'s book “Through The Telescope”, Jeff Medkeff’'s website and my own un-scientific testing (search CN Forums for acuity). The resolution of the eye for astronomical uses is more generally found to be in the range of 150 to 200 arc-seconds, while the occasional individual, under the best of circumstances, can obtain results down as low as 120 to 150 arc-seconds. It is a very rare occurrence to obtain results better than that. Sidgwick includes a very clear discussion of the thresholds of human acuity for dark-adapted observing.

The formula 1300Dmeters provides a result of the lowest magnification needed to reach the resolution of the telescope. It provides a result achievable by the most optimistic condition. It also assumes the eye is capable of seeing the resolved image at that magnification, which may not always be enough magnification.

Assuming the eye is in the most optimistic of the range and it can achieve 3 minutes of resolution or 180 arc seconds, then an image that has a 0.75” separation of the Airy disks must be magnified by 180” / 0.75” or 240x to be seen. Generally features at the limit of resolution require greater magnification as acuity is found to be closer to 240 arcseconds (and not the 180arcseconds used in this example). This will vary with the acuity of the observer.

This formula produces a result that shows optimum resolution is achieved at an exit pupil of just less than 1mm, about 0.75mm. Based on field experience, I find the best planetary resolution with my 5” scope is achieved at magnification of 150x to 180x and with my 6” scope around 180x to 220x, exit pupils between 0.83mm and 0.68mm.

Summary / Conclusions

While there are several criteria by which we may measure limits of resolution, not all provide for a clear separation between objects. An understanding of the Airy disk formula, Rayleigh limit, and the affects of wavelength and magnitude of light will provide you with the means to determine limits of separation.

Resolution as used in astronomy is commonly measured by angular measure in units of arc seconds. The Airy disk has the same linear size (more or less) for all scopes of a given focal ratio. The Airy disk has the same angular size for all scopes of a given aperture.

Dawes Limit = 4.56/D inches = 116/Dmm. Dawes Limit is the first point at which a double star is elongated enough to suspect the presence of two stars.

Rayleigh Limit = 5.45/D inches = 138/Dmm. Rayleigh Limit is a measure defining the limit at which two components can be clearly identified as separate components. It defines the distance between the centers of two Airy disks where the maximum of one is placed over the minimum of the other. Like Dawes, it is not a measure of separation required to see a black space between components.

The angular radius of the Airy disk out to the first minima is represented as:

A = 1.22 λ / D, where A in radians = 1.22 λ (Lambda) / D (Aperture)

A radians = 1.22 λ / D is equivalent to Rayleigh Limit = A arcsec = 5.45/D inches

Utilizing point source limiting criteria for determination of potential to see an extended feature is a difficult application, but I believe it is a valid approach.

Although the rings easily fit the extended object criteria of images larger than the detecting element, the width of the Cassini division does not. The width is small, even smaller than the limit of resolution of many small scopes. With exception to its linear dimension, Cassini is a small, very high contrast image requiring fine resolution that fits the definition of a point source.

Extended objects are easier seen if high contrast is present. It is a combination of superb contrast and resolution that allows some observers to exceed some of the point source limits.

An added linear dimension to a feature helps make the extended image easier to see.

The Cassini division separates the brightest portion of the B ring from the brightest portion of the A ring. The dark Cassini division is bounded on either side by the brightest features within the ring system. The Cassini division may just be the highest contrast extended planetary feature in the entire Solar system.

A larger aperture will provide a smaller diffraction disk. Greater magnification will reduce the extent of the diffraction fringe. High contrast will improve the ability to see the division. And finally, an added linear dimension to the feature helps make the image easier to see. All these things will have an affect on the ability to see the image of the division.

Any feature that does not have angular dimension greater than the limit of the telescope’'s ability to resolve cannot be observed as anything larger than the Airy disk. Since this is the smallest image spot size that can be achieved by a given telescope, no matter how large that image is magnified, it cannot be construed as having width. You would not be magnifying a resolved feature. You would simply be magnifying the smallest image spot size.

A telescope with an aperture that will produce an Airy disk smaller than the observed feature is required to see the feature as having dimension. Smaller telescopes may observe the feature, but will not have the resolution required to see the feature as dimensional. The smallest telescope that significantly exceeds the necessary criteria is a 9” scope. I suspect it takes at least a 9” scope to achieve true dimensional effect.

No objective lens will reach the limits of its maximum resolution unless sufficient magnification is employed. With increased image scale higher magnification utilizes more of the ability of the available aperture. The telescope will realize full potential resolution with a magnification of approximately 1300Dmeters. This is near the same as 30Dinches.

The telescope will realize potential resolution with a magnification of 1300Dmeters. For a 9” scope 1300D produces a magnification of 297x. Since the limit of resolution slightly exceeds that needed for the resolvability of the feature, the division may be seen as having dimension at lower magnification. As the aperture of the telescope increases, and the limit of resolution increases, the magnification needed may be even less than 1300D to achieve dimensional resolution of the feature. However keep in mind at magnifications below optimum, optimum resolution is not achieved.

Some observers may achieve eye resolution of 3 arc minutes or 180”, in which case they could perceive dimensional width at a magnification of 180” / 0.75” = 240x. It is more likely that most observers have eye resolution in a middle of a range needed and would require 4 arc minutes or 240”, resulting in a necessary magnification of 240” / 0.75” or 320x to see dimension in this feature.

Credits

Beiser, Aurthur, “Modern Technical Physics”, Cummings Publishing, 1973.

Kitchin, C. R., “Telescopes and Techniques” Springer-Verlag, 1995;

Liller, William, “The Cambridge Guide To Astronomical Discovery”, Cambridge University Press, 1992;

Porcellino, Michael R., “Through The Telescope” TAB Books, McGraw Hill, 1989;
Price, Fred W., “The Planet Observer’'s Handbook”, Cambridge University Press, 1994;

Seeds, Michael A., “Foundations of Astronomy”, Wadsworth Publishing, 1997;

Sidgwick, J.B., “Amateur Astronomer’'s Handbook”, Dover Publications, 1971

Suiter, H.R., “Star Testing Astronomical Telescopes”, Willmann-Bell, Inc. 1994-2001

Jeff Medkeff, web article on the Cassini division and resolution: Why & When to use High Powers, 2002; http://www.roboticobservatory.com/jeff/observing/why_high_power.htm

Ed Zarenski, web article for a discussion on Testing for Visual Acuity, May 2003;

http://www.cloudynights.com/ubbthreads/showthreaded.php?Cat=1,2,3,4,5,8,9,10&Board=binoculars&Number=2076&Forum=All_Forums&Words=acuity&Match=Entire%20Phrase&Searchpage=0&Limit=25&Old=allposts&Main=2076&Search=true#Post2076

Thanks to David Knisely for his clear explanations in our correspondence which helped lead me to a better understanding of the difficult subject of resolution.

Drawings by Ed Zarenski

Airy disk image generated by Abberator, a freeware software written by

Cor Berrevoets, Ritthem, The Netherlands, http://aberrator.astronomy.net/index.html

Photography:

Saturn 10-23-03 by Wes Higgins, originally published on Astromart: CCD Imaging and Processing http://www.astromart.com/messages.asp?message_id=128342&page=

Saturn 10-23-03 at 10:35 UT from Tecumseh Oklahoma. 14.5" Starmaster @ F/27, Toucam Pro Webcam, 801 images from one Avi of 1600 stacked in K3CCDTOOLS, Imagesplus, Photoshop 7.

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