Astronomy & Electronics Centre

When correctly put into practice, resolving power (or resolution) formulas can be used to test optical and other types of instruments. Generally speaking, there are different kinds of resolving power: optical, spectral, electronic, photographic, and energy resolving power. In computing, for example, the number of dots per unit length (dpi) in which an image can be reproduced on a screen or printer is an indication of its resolution. In photography, the image resolution on the final print depends on the type and quality of the optics, film, developer and printing paper. Optical resolving power, or resolution, is the ability of a telescopic or other system to distinguish (resolve) fine details, or a measure of that ability. Here by a telescopic system we mean not only telescopes, but also spotting scopes and telephoto lenses, as their resolution can also be tested in a similar way. However this article is about the optical resolving power, resolution and sharpness of a telescope system and the factors that can affect them. There are two kinds of optical resolving power amateur astronomers can use to test their telescopes. These are known as angular and linear resolving power. The former, which is measured in arc-seconds, applies to point sources of light such as close double stars; while the latter, which is normally measured in line-pairs per millimetre (lp/mm), applies to extended celestial objects such as planetary and lunar surface details. Unfortunately, the nature and purpose of both the angular and linear resolving power are often misunderstood. For example, some people believe the misconception that, everything else being equal, larger aperture telescopes always yield higher resolving power than smaller aperture ones. As we will see in this article, this is not necessarily the case.

Although the terms resolving power and resolution have a similar meaning, they are not exactly the same thing. In this article the term resolving power will be used only when we mean angular resolving power in arc-seconds. Whenever we refer to linear resolving power, we will use the terms resolution and sharpness. Sharpness, however, isn’t a synonym of resolving power, but its true meaning is closer to linear resolution than angular resolving power. Sharpness or acuity can be defined as thinness of edge or fineness of point. Below we will examine and discuss the most important factors which are likely to affect the visual resolving power, as well as the photographic resolution and sharpness of a telescope optical system. Both visual resolving power and photographic resolution tests, as well as their respective limitations and usefulness, will also be part of this detailed examination.

The angular resolving power of a telescope is the smallest angle between two point objects that produce distinct images. Theoretically, the angular resolving power depends on both the wavelength at which observations are made and on the diameter, or aperture, of the telescope’s objective lens (refractor), or mirror (reflector). The minimum angle can be given by the Rayleigh limit:

where λ and d are the wavelength and aperture (in metres). For an optical system the Rayleigh limit as approximately 0.14/d arc-seconds.

Experiments conducted by the nineteenth-century English astronomer W. Dawes showed that, by dividing 4.56 by the telescope’s aperture in inches, we can find out how close a pair of 6^th-magnitude yellow stars can be to each other and still be distinguishable as two points of light. This is called Dawes limit. Traditionally, the Dawes limit is used by telescope manufacturers to specify the angular resolving power of their instruments. As already mentioned above, this kind of resolving power depends on the aperture, and for a given aperture, is independent of the focal ratio. For an aperture D, and wavelength λ, in nanometres, the angular resolving power is:

For a given aperture of, for example, 150mm, the resolving power in arc-seconds is:

The above formula has taken into account λ, which for green light is 555 nanometres (nm). However, for all practical purposes we can obtain the angular resolving power of a telescope system with the following simpler Dawes limit formula:

Whether D is given in millimetres, or in inches, the theoretical angular resolving power of the above-mentioned 150mm telescope, will be 0.76 arc-seconds. (115.8/150 = 0.76; 4.56/6 = 0.76)

It could be thought that when the angular separation of two stars is very small, using a large enough aperture or sufficiently high magnification, would always resolve the light into two distinct images. This is a fallacy, as the diffraction effect turns the image of each star not into a point of light, but into a disc known as the Airy disc. If the two discs overlap substantially, increasing the aperture or magnification would only produce a larger blur of light. If this happens, the telescope won’t have sufficient resolving power to separate the images. However, the stars will just be resolved when their Airy discs touch. This gives the Dawes limit. Incidentally, it is said that two stars are at a telescope’s Rayleigh limit when the centre of one star’s Airy disc falls on the first dark ring of the diffraction pattern of the companion star.

Below is a list of some of the most common telescope apertures in millimetres, and their respective resolving power in arc-seconds:

60 : 1.93 -- 101.6 : 1.14 -- 152.4 : 0.76 -- 203.2 : 0.57 -- 254 : 0.46 -- 304.8 : 0.38.

These aperture-resolving power figures (obtained by using formula No. 1 - Dawes limit formula) clearly show that, theoretically, doubling the aperture of a given telescope will also double its resolving power. However, when wishing to put theory into practice, by trying to split double stars, we need to be aware of the theoretical and practical restrictions which are inherent to this undertaking.

For example, Dawes limit formula is valid only for white double stars consisting of two sixth magnitude components, observed through a 150mm telescope. The Dawes criterion, for example, doesn’t apply to red double stars. Also, the said criterion applies only when the diffraction pattern has the ideal distribution; and since optical aberrations affect this distribution, the resolving power of the observing telescope decreases, as it does when air currents blur the combined Airy pattern of the double star. Furthermore, the different brightness of the two stars will affect their combined Airy pattern, which will make distinguishing the star images more difficult. Apart from all this, the results of the Dawes limit resolution test can be negatively affected by things such as atmospheric turbulence, warm air current inside the telescope, poor quality and/or misaligned optics, and last but not least, the observer’s lack of visual acuity. (Needless to say, inferior quality oculars would also negatively affect the results of the said visual test.) We should also be aware that rarely will telescopes larger than about 254mm resolve to their Dawes limit. In other words, the image from, for example, a 457mm aperture telescope will offer little more detail than the image obtained from a 254mm one.

Assuming excellent ‘seeing’ conditions and optics, as well as the other requirements specified above, a telescope of 60mm in diameter should easily show the green companion of the red supergiant star ‘Antares’. This is because, according to the Dawes limit formula (115.8 / D) the resolving power of the said telescope is 1.93 seconds of arc, and the angular separation between the two stars is 2.9”. However, the said telescope won’t be able to split ‘Antares’ because, a) its aperture is much smaller than the prescribed 150mm; b) the stars aren’t white; and c) their respective magnitudes differ greatly from each other. On the other hand, everything else being equal, a top quality 100mm refractor, for example, will easily perform this task, which is more than most reflectors of the same and larger aperture can do. The Dawes limit criterion ignores the fact that telescopes with apertures larger than about 229mm are unlikely to achieve the theoretical resolving power ascribed to them by the Dawes limit formula, because of less than perfect ‘seeing’. These and other facts are often ignored by telescope manufacturers and buyers alike. (Manufacturers of cheap, mass produced, large-aperture telescopes makes much of the fact that their ‘light buckets’ have a very high nominal resolving power; and sometimes they list this specification as ‘Dawes Limit’.) Also, the said criterion ignores the important role played by contrast on the resolution of planetary, lunar and solar details.

As already mentioned in the introductory part of this article, it is a widespread misconception amongst many amateur astronomers, that the larger a telescope’s aperture, the higher must be its resolving power/resolution and sharpness. Visually, ‘light buckets’ will certainly show a brighter image than their smaller-aperture counterparts. For example, even the best 102mm refractor on the market can show stars only up to apparent magnitude 11. By comparison, even a cheap mass-produced 406mm Newtonian is capable of showing stars up to magnitude 15. However, the much higher light gathering capacity of the said Newtonian has in itself little to do with its resolution and sharpness. These highly desirable factors are effects - the causes being: superior optical design, top quality material, sound constructional techniques and perfect collimation. As a large aperture, high resolution refractor is financially out of the reach of the great majority of amateur astronomers, their best option would be to purchase a cheap large-aperture Newtonian for the observation of deep-sky objects, and a high quality, small to medium-aperture apochromatic refractor for planetary, lunar and solar observation, as well as high resolution astrophotography of both deep-sky and Solar System objects.

Regarding the application of the Dawes’ formula, despite its theoretical restrictions and other negative factors, it may be used with some success. That is, although resolution readings based on the said formula will often lack accuracy, in some cases useful comparison results may be obtained. Be that as it may, we should be fully aware of the extent and nature of the above-mentioned restrictions and the problems they are likely to cause. Furthermore, we should understand that the Dawes limit formula is only applicable to light point sources such as close double stars, not to extended objects as, for example, lunar and planetary details, nebulae, etc. Amateur astronomers, normally test their telescopes on close double stars. However, apart from everything else, splitting doubles doesn’t always prove that a telescope’s optical system has the ability to resolve details on the surface of the moon, planets, and other extended celestial objects. Incidentally, the Dawes limit formula says nothing about the important role played by contrast on the resolution of these objects.

In my opinion, a more satisfactory and reliable way to test the resolution and sharpness of a telescope is photographically, not visually. It isn’t by attempting to test the angular resolving power of its optics, in arc-seconds, but by finding out the linear resolution of the said instrument in line pairs per millimetre (lp/mm) that can we obtain better results. As mentioned at the beginning of this article, when describing linear resolving power, I prefer to use the terms resolution or sharpness. The angular resolving power depends (at least theoretically) on the aperture of the telescope and is independent of its focal ratio. However, the linear resolution of a telescope system (or for that matter a spotting scope or photographic lens) is independent of its aperture, but depends on its focal ratio. Incidentally, talking about aperture and focal ratio and their effects on the resolution of a telescopic system, we should also brief discuss how these two factors affect brightness. Visually, the larger the aperture of a telescope, the brighter is the image of a celestial or terrestrial object under observation. To be more accurate, telescopes with equal aperture, used at an equal magnification setting, have the same visual image brightness; this is true, regardless of their focal ratios. But, when photographing celestial or terrestrial extended objects, faster focal ratios produce brighter images on film and proportionally shorter exposures. This happens independently of the aperture size of the telescope being used.

Returning to the subject discussed in the second part of this essay, we see that for a given focal ratio f/D and a wavelength λ, in millimetres, the linear resolution is:

of course, the lp/mm readings will vary for different focal ratios and/or different light waves. The same test performed in blue light (λ = 450nm), for example, will give a reading of 366 lp/mm. Of course, these lp/mm results are purely theoretical, and based on the assumption that perfect telescope optics are tested under perfect atmospheric conditions. In reality, optics and atmospheric conditions are far from perfect; therefore, lp/mm readings are likely to be much lower than the above ones. Any optical system can be referred to as close to perfection if - in the absence of diffraction, and without obstructions such as secondary mirror or spider - is able to produce a point image of a point source. This doesn’t mean, of course, that reflective telescopes cannot be tested, but only that the results may not be as reliable as the ones obtained from refractors.

Incidentally, there is a correlation between the linear resolving power (LR) and the angular resolving power (AR) through the focal length, f, of the optical system:

As mentioned above, in astronomy, linear resolution applies to extended celestial objects such as the moon and planetary surface details. Detailed telescope observations of lunar craters, for example - when carried out on a night of good ‘seeing’ conditions, by sharp-eyed observers - will enable them to get a rough idea of the optical resolution/sharpness of their instruments. However, to actually find out how many lp/mm a telescope optical system is capable of resolving, we need to view or, preferably to photograph, a Resolution Test Chart, also called a Resolving Power Chart.

Everything else being equal, pictures of extended objects (the moon, planets, etc.) taken through large aperture telescopes are not sharper than those taken through smaller aperture ones. However, larger apertures usually mean longer focal lengths and larger images on film. For example, a picture of the full moon (angular diameter 31’), taken through a 300mm f/10 (3000mm focal length telescope) will form a 27mm diameter image on a 35mm film frame. On the other hand, the image obtained on film, when using a 60mm f/5 telescope, will only be a tiny 2.7mm in diameter. Naturally, the former image of the full moon will show more details than the latter much reduced image. This doesn’t necessarily mean that (because of its larger aperture) the said 300mm aperture telescope has the ability to produce sharper results than the 60mm one, but simply that the lunar image formed on film by the latter is far too small and compressed to provide a valid indication of its resolution/sharpness. When photographing a resolving power test target, however, the magnification and image size on film can be kept constant by moving the telescope closer to or farther from the said test target. (I think we would find it a bit difficult to follow the same procedure when photographing the moon or any other celestial object!)

As the saying goes, a picture is worth a thousand words; and we certainly shouldn’t rely only on visual tests, which, because of the human factor and other variables, are too subjective. This is why we often hear contradictory opinions about the optical performance of the same brand and model of telescope, spotting scope, etc. Nor should we take too much notice of the usually exaggerated mirror or objective wavelength accuracy claims, as well as restrictions-subjected angular resolving power data. I, for one, prefer the photographic testing method, which I have been using for years with both telescopes and photographic lenses. After all, telescopes are quite similar to refractive and catadioptric telephoto lenses. I am well aware that telescope optics are likely to produce their best results when focused at infinity. However, even when focused on a much closer object - such as a resolution test chart - useful photographic test results can be achieved. The moon and the said chart, for example, are both extended objects. However, while pictures of the moon will give a rough idea of the linear resolution/sharpness of the telescope through which they have been taken, pictures of a resolution test chart can give reasonably accurate readings in line pairs per millimetre.

These days, digital cameras are also widely used in astrophotography, as they are relatively easy to use and lend themselves to electronic manipulation at the computer. An example of this is the stacking of many images together, in order to produce sharper and more pleasing results. This manipulation may produce impressive astrophotographs, but these can hardly be used to evaluate the resolution, sharpness and degree of contrast of a telescopic system. SLR film cameras (35mm or medium format) are still the best choice for taking test pictures of a resolution chart.

A few resolution test charts have been available for some time - one of the best-known being the Edmund Resolving Power Chart, supplied by Edmund Optics - USA. This chart contains reproductions of the USAF 1951 Test Pattern, which is one of the standards of the optical industry. Its proper use makes it possible to assess the performance of an optical system, be it a photographic lens, spotting scope or telescope. The various positions, orientation, and colours of the 25 individual small charts will reveal the performance of the telescope under test. When used photographically, the linear resolution of the said telescope will be recorded on film. Also, colour pictures of this chart will reveal possible chromatic aberrations. The said chart can also be used to detect astigmatism. (Incidentally, the same company also supplies contrast, depth of field and distortion test targets.)

The Edmund Resolving Power Chart can also be used visually; however, for the purpose of this article, it will be only used photographically. The chart consists of a stepped series of three bar patterns called elements; these are arranged together in groups. The coarsest element on each of the said 25 individual charts has the centre to centre spacing of the printed lines at a 4mm separation, meaning that these represent 0.25 line pairs per millimetre. As one proceeds through the elements and groups the lines become progressively closer in a step ratio which is the sixth root of 2. The table printed on the charts itself lists these values for all elements. Fig. 1 shows the complete 914mm x 610mm resolution chart, while Fig. 2 illustrates one of the 25 B&W and colour individual charts.

FIG. 1

Fig. 2

Formula (6) is the standard telescope/telephoto lens linear resolution formula, which also take into consideration the wavelength at which observations of celestial objects are made. However, when wishing to take resolution test photographs of the Edmund Resolving Power Chart described above, we will need to use the following formula:

where fo is the focal length of the telescope under test, and d is the distance from the chart to the mirror or the objective lens of the telescope. Both dimensions, fo and d are to be expressed in the same units. The above formula can also be described as the relationship between the line-pairs-per millimetre (LPM chart), as printed on the Edmund chart, and the resolution on the photographic negative (LPM photo).

As mentioned above, the ideal camera to be used for photographing a resolution chart is an SLR film camera, possibly with lock-up mirror, and interchangeable focusing screens and viewfinders. The camera should be loaded with high-resolution, fine grain B&W film, such as the Ilford Delta 100 or the Kodak Tri-X 100 professional negative film. Unfortunately, very high resolution and ultra-fine films, such as the Kodak Technical Pan 2415, are no longer available; therefore, one of the two films mentioned above will have to be used instead. The process of focusing on the chart should be carried out with extreme care. In fact, it is wise to take at least three pictures of the resolution chart at the same exposure setting, but refocusing each time. Also, in order to obtain a correct exposure, various shutter speed setting should be used.

Needless to say, a very accurate positioning of the test chart, telescope and camera is of paramount importance. The focal plane of the camera in use should be perfectly parallel with the said chart, whose illumination has to be glare-free; two 500w halogen lamps will light the chart nicely. Also, the telescope-camera set-up needs to be mounted on a very sturdy, vibration-free support; and the shutter of the camera should be activated by an air cable release or electronically; and, for exposures longer than about 1/60th of a second, the camera mirror should be locked up prior to the exposure. The distance between the test chart and the primary mirror (in a reflector) or the objective lens (in a refractor) has to be 26 times as long as the focal length (Fl.) of the telescope under test. For example, for an Fl. = 300mm, the distance required is 7.926 metres; while a 3000mm Fl. telescope has to be 79.260 metres away from the test chart. As at distances longer than about 30 metres the telescope set-up or the resolution chart is likely to have to be placed outdoor, atmospheric conditions must be near-perfect, before any test can be carried out successfully. Wind and/or heat radiations will invalidate any optical test.

For reasonably accurate photographic measurements of the optical resolution of a telescope, the combination film-developer must have a much greater resolving power than the said telescope optics. The Technical Pan 2415 film, mentioned above, for example, had a resolving power of 320 and 400 lp/mm, when developed in Kodak HC-110 (Dilution D) and Technidol LC, respectively. Suitable developers for the Hilford and the Kodak B&W professional films, also mentioned above, are D-11 and X-Toll, respectively.

At this point I would like to make it quite clear that using a resolution chart to test the sharpness of a telescopic system, requires great accuracy, care and patience. Amongst other things, the camera-telescope set-up must be properly placed, lighting must be precise, and the film must be developed in a specific manner. If all necessary requirements are met, the amateur, not only can find out a photographic lens, telescope or spotting scope’s resolution, but he/she can check it for various types of distortion and aberration. The exposed film should be properly developed in a fine-grain developer. The resulting negatives should be carefully examined with the help of a high quality magnifying glass, at a magnification power of about 10X to 20X - a good light box will come very handy for this task. Those who find that critical examination under a magnifier is a little difficult, may print the test negatives. This task requires an enlarger fitted with a top-quality lens. Here it should be noted that the enlarging process would add another variable - the enlarger’s lens - and this could give unreliable results. However, if a comparison resolution printed test between different telescopes is all that is wanted (and accurate lp/mm readings are not required), this is the way to go.

Having said all that, let’s now consider the following example. We take a photograph of the Edmund Resolving Power Chart through a 1000mm focal length refractor focused on the chart from the prescribed distance of 26 metres (26x1000mm) from the telescope’s objective lens. Upon the examination of the developed negative, the smallest group-element that was resolved is 1 2, which according to the Resolution Values table on the chart, is 2.24 lp/mm. Therefore, according to formula (7), the linear resolution of our telescope is:

The said telescope is quite sharp, as it can resolve 56 lines per millimetres from a distance of 26 metres.

50mm F/8 Apochromat	150mm F/8 Apochromat
150mm F/8 Newtonian	150mm F/12 Maksutov
250mm F/10 Schmidt-Cassegrain	Mass-Produced 250mm F/5 Newtonian
Please note that these images are reduced in quality due to being converted to internet-friendly size.

The few test pictures shown above, taken, developed, enlarged and printed by the author should be sufficient to show that: 1) Larger aperture telescope optical systems are not necessarily sharper than smaller aperture ones. 2) Photographically, a 200mm f/8 telescope, for example, will produce an image on film of the same brightness as the one obtained with even much larger or smaller aperture instruments of the same focal ratio - a separate or camera-built-in light meter will soon prove this fact. 3) Linear resolution and sharpness are determined by a telescopic system’s focal ratio, not its aperture; and 4) Everything else being equal, an f/10 telescope, for example, should produce a sharper image on film than the one obtained from an f/4 instrument, regardless of the diameter of their respective mirror or objective lens.

In conclusion, whoever has the time, the right equipment, the knowledge and the patience to conduct the tests described above will be able to verify the correctness of these statements.

ANGULAR AND LINEAR RESOLVING POWER