This article is divided into two parts, “Resolution Limits” and “Magnification, Contrast and the Human Eye“. For a better understanding, they should be read in order.
Resolution Limits
First, we look at the relationships that exist between various resolution criteria, magnifications needed to access these limits, and the contrast required to do so in the context of using an astronomical telescope visually. Next, let us look at the various resolution limits.
Where 550nm is the choice of lambda (l, wavelength of light) and 0.206 is the conversion factor from radians to “arc, and D is in mm, then:
Rayleigh limit: 1.22 • 550 • 0.206/D = resolution “arc.
(ex: 1.22 • 550 • .206= 138.23/127mm= 1.088″arc)
Simplified: 138/D in mm = resolution “arc or 5.45/D in inches
Dawes Limit: 1.025 • 550 • 0.206/D = resolution “arc
(ex: 1.025 • 550 • 0.206 = 116.13/127 = 0.91″arc
Simplified: 116/D in mm = resolution “arc (or 4.56/D in inches)
Sparrow’s Limit: 0.94 • 550 • 0.206/D = resolution “arc
(ex: 0.94 •550 • 0.206= 106.50/127mm= 0.839″arc
Simplified: 107/D in mm = resolution “arc (or 4.2/D in inches)
There is nothing controversial about the above within optical science; these figures are essentially uniform across the literature.
For a perfect optic, the Rayleigh limit suggests a 28% drop in contrast differential between two Airy Disks, 5% at the Dawes Limit, and 0% at Sparrow’s limit. An MTF of zero between Airy disks is one of the formal definitions of the Sparrow limit.
“Airy Disk” is a mathematically derived theoretical term only (as strictly applied). “Spurious disk”, on the other hand, is that portion of the theoretically derived Airy disk as seen visually through an actual optic.
The reason it can be important to use two different terms is because the spurious disk can vary somewhat in extent from the theoretically calculated Airy disk– example: optical aberrations present in an actual optic can cause such a variance.
This business only becomes relevant (that is visually observable in a plainly differentiated way) when dealing with the most challenging double star resolution scenarios.
The Abbe Limit
1) Also known as the “Diffraction Limit”, Ernst Abbe developed the Abbe limit in 1873. The formula for microscopes is 0.5l/NA, which translates to 113/D = resolution “arc for telescopes (4.46/D inches), putting it between the Dawes and Sparrow limits. (Rayleigh=0.61, abbe, 0.50, or 82% of rayleigh, 138.23(0.82)=113.3).
It can be safely considered to represent a hard floor for optical resolution that cannot be surpassed by any observer using any telescope.
Rayleigh Limit
This is the “gold standard” of optical resolution, widely used throughout professional optical science. It was derived mathematically with experimental confirmation by John William Strutt (Lord Rayleigh) in 1896.
Rayleigh was a genius-level physicist and mathematician and a Nobel Prize winner. The Rayleigh Limit is based upon the performance of an optic delivering an 0.82 Strehl at the focal plane, the so-called “diffraction limited” standard of an optic meeting the minimum aberration requirements for spherical aberration defined by the Rayleigh Criteria (an optical construction standard).
Dawes Limit
1 – The Dawes limit mainly is considered empirical, not only from the way it was derived in the first place (by observation) but also by virtue of the fact that it de facto assumes a contrast threshold for the human eye (on bright high contrast objects of equal brilliance) to be 5%.
Of course, on an individual basis, the contrast threshold of the eye will vary. So, in a given optic, one person with exceptional contrast sensitivity might be able to detect a Dawes limit separation– while another with less than average contrast sensitivity would not.
2 – Only in use by the amateur astronomy community, Dawes Limit represents a good empirical limit on double star resolution for amateur telescopes of high quality.
3 – Properly applied only to equal magnitude pairs of about 6th magnitude, it is probably most accurate with apertures in the range of 3 to 8 inches.
Sparrow Limit
1 – As the Sparrow limit is defined by a zero contrast differential, and contrast differential is required for any resolution to take place, it follows that the Sparrow limit is not a resolution limit at all but rather a *non*-resolution limit, a point above which resolution first becomes theoretically possible and *at* which it is impossible.
2 – A simple web search will show that the Sparrow limit is most commonly used in the context of microscopy rather than astronomy–at least on a professional level.
It has also come into some vogue in amateur astronomy circles, rather inexplicably I’m afraid, as considering the ever present contribution of the atmosphere to astronomical observation, any limit less than dawes becomes de facto an exercise in simple wishful thinking.
3 – There are errant figures circulating the web defining the Sparrow limit. (The most common misinformation seems to be 70/D in mm.)
Since the Rayleigh Limit is defined as Dq=0.61l/a=1.22l/D or in microscopy r = 0.61l/NA the Sparrow limit is defined as D = 0.47l/NA so we have 1.29:1 (ratio) = 0.61:0.47= 1.22:0.94 = 138:107. (references below) Thus 107/D in MM is the correct formula for Sparrow’s Limit at 550nm l. (for whatever use that is to an amateur astronomer, anyway)
References:
Airy Patterns and Resolution Criteria Olympus Corp. Microscopy Tutorial (an excellent optical science resource generally)
National Optical Astronomy ObservatoryJennifer M. Lotz, Ph.D.
University of Minnesota, Institute of Technology Characterization Facility
Biography Lord RayleighSchool of Mathematics and Statistics University of St Andrews, Scotland Sidgwick, “Amateur Astronomer’s Handbook”
Magnification, Contrast and the Human Eye
Here is why increasing magnification above the theoretical “resolving” magnification for the human eye (13x/inch) is useful: it is because as magnification increases, the spatial frequency of a discreet detail is increased from a zone where the eye has no contrast sensitivity into a zone where the human eye has contrast sensitivity.
This effect is somewhat easier to understand with the help of an illustration:
Here is the Campbell-Robson diagram, which illustrates the varying contrast sensitivity of the human eye, its contrast sensitivity threshold curve, and below that, an MTF of a perfect optic. The various colored dots, the blue line, and the spatial frequencies are plotted to correspond between the two charts.
As frequency decreases (bars/spaces become wider) the eye will detect progressively smaller contrast differentials between the dark & light bars. (contrast differential decreases as you progress downward on the chart).
This is because the eye has a variable contrast response per frequency and degree of contrast. The U-shaped curved line represents this. The eye cannot resolve outside this line as the sensed contrast becomes zero.
The “average” eye has a best photopic acuity of 60″arc for 20/20 vision and an absolute best resolution capability of 30″arc at 100% contrast on a bright target (20/10 vision). This latter point is marked by a blue dot on the Campbell-Robson chart above.
This is the best resolution the eye achieve under ideal circumstances; for instance, an eye exam. It occurs at about 60 spatial frequency. For clarity, the spatial limiting is 50 cycles frequency for the eye in the remainder of this discussion as this equates to 20/15 vision, the most common “sharpest” vision. (20/10 is very rare and less than that does not exist) The chart below provides a correction for various acuity figures for fig. 1:
Note: The last column represents equivalent apparent resolution and does not mean that someone with 20/10 vision can exceed the diffraction limit for that aperture. This is because the ultimate resolution in the focal plane of the telescope is set by the physics of the telescope (the diffraction or Abbe limit) not the eye.
All that additional eye acuity can offer is to lessen the amount of apparent separation needed for resolution. Example: in a 4.5″ telescope a 1 arc” double would be detected as double by a person with 20/20 vision at about 225X (50x/inch) magnification at a 225″arc apparent separation; but a person with 20/10 vision would be able to detect duplicity at half that magnification or 112X (25x/inch) at an apparent separation of only 112″arc.
The well observed increase in double star separation from the 60″ separation that native eye acuity would seem to predict, to a practical value double that amount (~120″arc) is a result of the point source diffraction pattern, not decreased acuity due to decreased illumination as often supposed.
The resolution of a reasonably bright lunar observation (set at approximately -5 magnitudes from a specific telescope’s limiting magnitude) remains at 60″ arc, with full photopic acuity when the diffraction pattern is interpreted using line pairs (see illustration below):
But neither observer would be able to resolve a pair separated by less than 1″ arc as this level of resolution is not present in the telescope’s focal plane. In other words, no matter how sharp your eyes are, you cannot resolve what is not already resolved within the focal plane.
A telescope can resolve to a much higher degree of native resolution than the eye because it possesses much bigger & (usually) better-corrected optics. The focal plane of a telescope will resolve down to a frequency of about 90 cycles compared to the eye’s 50 cycles.
However, the increased resolution this gain in frequency represents is accessible to the eye only through the magnification of the telescope’s focal plane. The focal plane’s highest resolution is represented by a red dot on the MTF chart above and corresponds to the Dawes Limit.
What all this means:
In order for the eye to be able to see the resolution the telescope offers at 90 cycles and at 5% contrast, it must be translated somehow to a lower spatial frequency in order to bring the finest image detail within the bounds of the eye’s contrast sensitivity at the 5% level. In looking at the Campbell-Robson chart, we can see that we want to hit a spatial frequency at or above the eye’s 5% contrast threshold and inside the curve of the eye’s contrast sensitivity. (see below)
We can accomplish this by magnifying the image at the focal plane, uniformly decreasing the spatial frequencies of the resultant image.
For instance, applying 13x/inch magnification to the focal plane will provide a spatial frequency of about 50 to the image. The yellow dots represent this. Now, notice this still is not enough enlargement for the eye to resolve the telescope’s image because the yellow dot still lies outside the eye’s contrast threshold in the Campbell-Robson diagram. The image must be further enlarged to an even lower spatial frequency and brought within the limits of the eye’s ability to detect contrast. Without contrast, there can be no resolution. (see below)
The green dot represents the point where sufficient magnification (about 50x/inch) begins to bring the image above the eye’s contrast threshold, within the curve, making resolution of the finest detail in the focal plane now accessible to the eye.
This occurs at about 12.5 cycles. Since the eye’s contrast threshold curve bottoms out at about 5 cycles, higher magnification is possible, but as you can see on the Campbell-Robson chart, it is limited to about 100x/inch, after which point the spatial frequency of the image at the 5% contrast point begins to leave the eye’s contrast curve on the opposite side, into the area of zero contrast and “empty” magnification again. (see below)
For an example, we’ll walk through this process using an imaginary perfect 4.6″ telescope in perfect seeing having a maximum resolution of 1″arc and examining a perfect double star of equal 6th magnitude and having a 1″ separation.
Applying 13x/inch (about 60 power in this case) to a one second true stellar separation enlarges the apparent angle of separation to 60″arc, the eye’s theoretical resolving power under perfect conditions. However the image is not perfect.
We do have excellent inherent target contrast (none better than a double star) but even so the telescope, thanks to its inherent diffraction, can only deliver this frequency at maximum of about 5% contrast on it’s focal plane.
The pair now lies on the yellow dots and we still can’t resolve it as the contrast level present between the double is not yet within the eye’s contrast sensitivity range. (refer to diagram below)
So we increase magnification to 50x/inch or about 230 power (green dots). This enlarges the apparent separation of the pair to about 230″arc and at this point the star begins to hint at being resolved. In an attempt to gain a better resolved image, we up the magnification (and the apparent separation of the double along with it) to 75x/inch (purple dots) and then to 100x/inch (pink dots).
This brings better resolution (more contrast sensed between the pair) as the image is now enlarged to coincide with the eye’s peak sensitivity at the lowest possible contrast threshold, at about 5 cycles and 5% contrast.
If we try for even more definition by adding much more magnification from the 100x/inch point, the image will degrade rather than improve as by doing so we pushed the image to too low of a frequency, below 5 cycles and beyond our eye’s ability to sense contrast at this 5% level that the telescope gave us to examine.
The 50x/Inch upper magnification Limit
So an answer to “Why is 50x/inch so often quoted as the maximum magnification limit?” is this: 50X per inch is simply a round figure for 53.8x/inch, which is the magnification/inch figure required to bring the focal plane’s maximum resolution frequency (found at 13.453X/inch at the Abbe limit for a 20/20 eye) to an optimal median of 7.5 cycles within the 20/20 eye’s maximum acuity range of 5-10 cycles/degree.
Let’s take a look at how the 53.8x/inch aperture figure was derived (using simple math and averaging).
The ratio between magnification/inch (m) and resolution of the eye (a) is m : a = 13.453: 1 when a in arc minutes. This ratio derives from the ratio in arc seconds which is m : a = 1: 4.46 (from the difffraction, or Abbe limit, 4.46/D). We will simplify the ratio to 13: 1 for the following. (This corresponds to about a 4.62/D resolution metric, which is likely a more realistic figure for the majority of real world optics used under real world skies…)
Where ƒ= visual cycles/degree:
_x_
(2y) = ƒ
The formula to convert the eye’s resolution (acuity) in minutes of arc (a) and where the constant (x) is 60 in minutes, to cooresponding frequency in cycles/degree (ƒ) is:
60_
(2a) = ƒ
Applying the ratio to the constant (60•13 = 780) for magnification/inch (m) to cycles/degree (ƒ) gives:
780_
(2m) = ƒ
Plugging the numbers in for eye resolution of 1’arc we have:
60_
(2•1) = 30c/d
and for 13x/inch magnification
780_
(2•13) = 30c/d
Then, expanding the eye resolution to 4 minutes apparent, as is the case ~50x/inch magnification, we have:
60_
(2•4) = 7.5c/d
and for magnification/inch:
780_
(2•52) = 7.5c/d
Any native eye acuity/telescope resolution/magnification per inch result can be worked out more precisely using the following:
r=4.46/D
m=a/r
m¹ = m/D
where:
r = telescope resolution in seconds of arc
m = telescope magnification
m¹ = magnfiication/D
D = telescope aperture in inches
a = naked eye resolution (photopic acuity) in seconds of arc
Now you know that the 50x/inch upper magnification limit is not some random number pulled out of some long gone astronomer’s hat because you know where it came from and how it was derived!
Effects of Seeing on Maximum Magnification Range
As seeing decreases from perfect (essentially, this is when FWHM seeing equals the telescope’s best resolution, i.e., for a 4inch telescope seeing that is 1″ FWHM is essentially “perfect”), the magnifications which fall into the 5% zone of visual contrast sensitivity uniformly decrease.
For example, (see fig. 4) as seeing worsens from perfect, the range would go from 50-100X/inch of perfect seeing to 25-50X/inch in mediocre seeing and to even lower ranges as seeing deteriorates further.
This is because seeing impacts the focal plane image of the telescope, as seen visually, primarily at the mid to higher spatial frequencies of the focal plane (45-90 cycles), the same frequencies where the target’s finest high contrast detail lies.
With this detail obliterated from the focal plane by atmospheric turbulence, less magnification is required to bring what is left within our contrast sensitivity curve. For example, (see fig. 1) in excellent seeing, there is detail to be had lying within the upper-frequency ranges of the focal plane (as in the MTF chart) all the way up to the red dot of the Dawes limit at 90 cycles.
Since we have to translate this to a much lower frequency in order to see it visually, we need to apply large amounts of magnification to the focal plane image, thus 50X/inch and upwards to about 100X/inch (as shown).
However, when seeing worsens, the red line of the telescope’s contrast function moves to the left– and instead of ending at 90 cycles, falls at a steeper angle to a lower point on the scale, somewhere below 90 cycles, progressing to a lower value as seeing worsens.
When this happens, the amount of frequency reduction required to bring the best available detail in the focal plane to within our best contrast sensitivity becomes proportionately less–as moving from, for example, 75 cycles to 10 cycles in a seeing impacted focal plane image requires about 25% less magnification than moving from 90 cycles to 10 cycles in the perfect case.
This is why the maximum useable magnification decreases as seeing worsens.
Notes on Variances in Biological Systems
When dealing with a science such as basic physics, figures that extend to several decimal places can often be taken as relatively accurate. But when dealing with the biological sciences, especially those dealing with systems (such as the human eye and cognitive system that interprets its signals) that are not often the case.
In this latter case, what science gives us are somewhat “rough” ranges rather than hard points. This is simply a function of the variation between individuals; some people have 20/20 vision, some 20/10, some 20/80, etc, and fall into all sorts of groups regarding contrast perception as well. But the individual commonality is that essentially healthy individuals all fall within certain ranges (in biology, this is typically a variety of Bell curve), and the characteristics of the system overall do apply to all of us.
What this means about magnification and contrast is that although all people do possess a contrast sensitivity function that is expressed as a curve, as visualized in a Campbell-Robson chart, the exact cutoffs at any given frequency will vary, within a certain range, between individuals.
So, in the case where magnification first brings the spatial frequency of the finest resolution into one’s contrast sensing range (fig. 2), this point will vary within a range for most people between about 45 and 55X/inch.
In reality, these numbers represent nothing but an instructive average within the broader range of individual values found within the human population.
Note: the conventional spatial frequency unit of measure differs slightly between the eye (cycles/degree) and that commonly used for an optical instrument (cycles/mm). This difference was disregarded in the above discussion for clarity. In any case, the effect of ignoring this modest difference of unit measure does not change the character of the interrelationships of the operative concepts as discussed.
References:
Webvision , John Moran Eye Center, University of Utah
Campbell-Robson Contrast Sensitivity Chart & referencesVisual Neuroscience Laboratory, Ohzawa Lab, Japan
The Contrast Sensitivity FunctionPeter Wenderoth, Macquarie University, Sydney, AU
Sidgwick, “Amateur Astronomer’s Handbook”
Rutten & Venrooij, “Telescope Optics”
