The resolving power of a telescope is one of the most important characteristics. There are many misconceptions about the resolving power. The purpose of this paper is to clear this misconceptions. An amateur should understand this term and how it affects his observations.
The resolving power is defined as the minimum angle two stars of equal magnitude should have for us to be able see them as separate stars. The resolving power also affects observation of the Sun, the Moon and the planets. In this case, the resolving power of our telescope determines how small a detail we can see on the sunspots, on the surface of the Moon or on the surface of a planet. This is also true for galaxies, star groups, comets, asteroids, etc. The resolving power of the telescope affects practically every mode of operation. We need to study what we call resolving power.
The resolving power of the telescope is a consequence of the wave nature of light and the phenomenon of diffraction. Consider a refracting telescope where all the areas through which the light travels are perfectly circular. As a consequence of the wave nature of light, the image of a star, which we see as a point, will not be a point. The image of a point star will be a small disk and a series of rings, the diffraction rings. The disk is brighter at the center and its illumination falls towards the edge. The illumination of each ring is higher at the center and falls both sides. The diffraction rings are dimmer as they are farther from the disk. If the optical system is perfect, 84% of the light that comes from the star falls in that small circle. If the tolerance of the surfaces is within 1/8 of a wavelength, 68% of the light from the star falls in the small circle. If the tolerance of the surfaces is within 1/4 of a wavelength, only 40% of the light from the star falls in the small circle. The rest goes to the rings. This is further complicated in reflecting telescopes because of the obstacles in the path of light.
Recall that tolerance is the distance between the theoretical surface of a lens or a mirror and the real surface of the lens or mirror. Let us give an example. Imagine we consider that a mirror is a paraboloid. If it is not exactly a paraboloid, less of the light from the star falls in the circle mentioned above. The same star will be brighter in a better mirror than in a poorly formed mirror of the same diameter. This is the importance of the tolerance. Smaller tolerances can be achieved normally with more careful work and better equipment.
It is interesting to study what this has to do with observing. A point in favor is that the eye cannot observe small differences of luminosity. On the other hand, we know that if the image is a circle, the larger the circle for the same conditions, the less luminosity it will have. Then, we are interested in finding out what makes that circle larger or smaller. The diameter of the circle formed by a star, considering as always the yellow light, if given by the following relation:
d = 0.000146 f / D
where d is the diameter of the disk in centimeters; f is the focal length of the system in centimeters; and D is the diameter of the objective in centimeters. Let us analyze this equation from the practical point of view. We know that the larger the circle the dimmer it is. If we want to have sharp images of the stars, we need small circles. A small circle can be obtained by reducing the focal length of the system, by increasing the diameter of the objective, or both. Notice that the relation between f and D is the focal ratio of the telescope. That is to say, we need that the f/ratio of the system is a smaller number. This is the reason why a short focal length telescope gives sharper stars than a long focal length telescope of the same diameter. Let us make a small table:
| f/ratio | Diam in mm. |
| f/1 | 0.00146 |
| f/2 | 0.00292 |
| f/4 | 0.00584 |
| f/8 | 0.01168 |
| f/10 | 0.0146 |
Now, let us go back to the resolving power. We said that when we observe a star with the telescope, the star is seen as a disk and a series of rings that get dimmer as they get further from the disk. If we have two stars, very close together, the two disks would mix into one and we are not able to see them as two stars. Lord Rayleigh determined that we are able to see the two stars as separate stars if the center of the disk of one falls on the first minimum of the other. R. W. Dawes made many experiments to determine if it was really necessary that the stars are so far apart. He determined that it was possible to see as two separate stars even when they were closer than that rule. He determined an empirical formula, as follows:
a = 11.4 / D
In this formula a represents the minimum angular separation between the stars, given is arc-seconds, to be seen as two stars, with an objective of diameter D. The constant 11.4 is when the diameter is given in centimeters. Use 4.5 if the diameter is in inches. This formula is empirical, as it was mentioned above. It is valid only when we consider two stars of 6th magnitude and of the same color. Let us consider an 8-inch telescope. This separation is a little over half an arc-second. In order for the eye to appreciate this separation we need to use a lot of magnification.
Let us find out how much magnification is needed to see two stars as separate stars. The eye is an optical system that forms the image on the retina. The retina is formed by a large number of cells, each one sending signals to the brain, where the sensation of the image is formed. If the images of two points fall on the same cell, the brain is not able to separate the two points. Theoretically, this separation is when the two points are separated by one are-minute. Experiments were made with persons with good eyesight or with the proper correcting lenses. They could separate points that were between two and four arc-minutes, depending on the person and his or her training. If the two points are very bright, the separation must be larger. This is complicated more by the color. If the two points are of different color, or of different brightness, they need to be further apart to be recognized as separate points.
From the practical point of view, let us consider the value of 4 arc-minute. Working in centimeters, we make the value a in the formula above equal to 4 arc-minute, 240 arc-second, by applying a magnification m to the telescope; then,
240 = 11.4 m / D
orm / D = 21
This is to say that we need a magnification of x21 for each centimeter of diameter of the objective, which is a lot. It is more than x400 for an 8-inch telescope. When the two stars are very different in brightness, the problem increases. The case of Sirius is well known. Sirius has a companion at a distance of 31.6 arc-seconds that should be easy to see. The problem is that Sirius is very bright, magnitude -1.5, and the companion is very dim, magnitude 14. It is quite a challenge to see the companion with small telescopes. Let us put the results we have in a table for different diameters of the objective of a telescope, in inches.
| Diam | Separation | Stars | Moon |
| 4-inch | 1.125 arc-sec | x200 | x50 |
| 6-inch | 0.75 arc-sec | x300 | x75 |
| 8-inch | 0.56 arc-sec | x400 | x100 |
| 10-inch | 0.45 arc-sec | x600 | x150 |
The table above includes a column labeled Moon. Let us explain it. In the case of the details on the surface of the Moon, the Sun, or on the planets, the problem is different. The reason is that all these objects are bright. The luminosity of the details is different. Some are dimmer than others. We wish to distinguish differences in illumination. The best magnification to use is that which produces a disk of exactly one arc-minute. This is such that we see all the disks as points. This is obtained with a magnification of x5 for each centimeter of diameter of the objective; that is, x100 for an 8-inch telescope. Going to higher magnification make the observation more comfortable, but at the cost of loosing details. An easy example is looking at a picture in a computer. If we use a magnifying glass to study the picture, or we enlarge the picture, we will not see more details. The dots will be larger, that is all.
A further note is needed. We have been mentioning the diameter of the objective without making any qualifications. In the case of the resolving power, we need to consider the actual diameter of the objective, without any regard to obstructions or shadows. We can emphasize this by indicating that the theoretical resolving power of a telescope will be the same if we put a mask in front of the objective, covering most of it, except for two sectors at the ends of a diameter. Consider also two spherical mirrors mounted in such a way that they are acting as the ends of a diameter of a large mirror. They will have the same resolving power of the large mirror. That is to say, if we take two 8-inch spherical mirrors and mount them 30-inch apart and we align them perfectly; they would have the same resolving power of a 30-inch telescope. The resolving power also shows the importance of the pupil of the telescope. If the pupil of the telescope is larger than our pupil at the moment, our pupil will act as a diaphragm, reducing the diameter of the objective. In summary, the resolving power of a telescope for details depends on the total diameter of the objective.
Let us analyze the practical aspects of what have been said. Let us use a very dramatic example. It is very easy to hear that using a large telescope for looking at the Sun or the Moon is a waste. The Sun and the Moon are very bright. We need solar filters or Moon filters to reduce the luminosity. It is also easy to read recommendations for observing the Sun saying that the best thing to do is to put a cardboard disk at the entrance of the telescope with only a small opening covered by the filter. According to the explanation above, those persons are only considering the luminosity of the Sun. If the purpose of the observation is a casual look at the Sun, that is correct. If the purpose of the observation is looking for details on the sunspots to either draw or count them, they are wrong. To see those details, the whole diameter of the telescope is needed, the larger the better. A solar filter is needed to reduce the luminosity. A magnification of x5 for each centimeter of diameter of the telescope should be used. Then the details of the sunspots will be clear and beautiful.
Let us consider a numerical example. Imagine using an 8-inch, f/10 telescope to look at the Sun. We will be able to resolve details with 0.56 arc-sec separation. We should use an x100 for best results. If we put a mask with a 2-inch hole, we will have an f/40 telescope. The luminosity of the image will be much lower. We will be able to resolve only detail with 2.25 arc-seconds separation. That is to say, the smaller details we will see are four times larger than the ones we could see with that telescope. This also means that many small sunspots will not be seen.
If interested in observing the Moon and the planets, high magnification is needed, but not too high. A long focal length telescope is useful, but compute what magnification is the best for the observing conditions. On the other hand, to observe binary stars, to determine their angle and their period, use as much magnification as the conditions permit. Observing details in galaxies and nebulas is similar to observing details in the surface of the planets, with the added constrain that the galaxies and nebulas are not too bright. As large a telescope as possible is needed to observe details in dim galaxies. Use as much magnification as possible without washing the galaxy or the nebula. This is normally not too high.
We have been talking about visual observing. We need to talk now about photography. The problem is very similar. In photography, whether with film or with a CCD camera, the eye does not enter. We need to consider the sensitive surface instead of the retina of the eye. Consider first, that the sensitive area is placed on the focus of the telescope. In film photography the sensitive area is formed by grains of a silver compound suspended in gelatin. These grains are very small. The resolving power of film photography is very high. We need only to think that a film photograph can be enlarged very much before the grains are seen. The limitation on film photography is more in the ability of the film to record the details we want to see, rather than in the resolving power.
The case of CCD photography is different and similar. It is similar in the sense that there are individual elements recording the light. These elements, or pixels, have a finite dimension. Most CCD cameras use chips where the pixels are a square between 5 and 10 microns in side. Some cameras have rectangular pixels. The resolution of a 5-micron pixel is very close to the resolution of the eye.
Photography makes clear the problem mentioned above about the resolution of stars of different magnitude. The problem is that each photon that gets to the film reduces a grain of silver. In CCD photography, each photon that gets to a pixel releases an electron. The longer the exposure, the more silver is reduced and the more electrons accumulate in a pixel. This has a limit. In a film, as the exposure increases, the stars get larger because the photons reduce adjacent silver grains. In CCD, as the exposure increases, the pixel gets full and starts spilling to adjacent pixels. In film photography this is called saturation. In CCD photography this is called blooming. The effect is the same. It limits the resolution of the picture. Consequently, to obtain the maximum resolution of the picture we need to give an exposure long enough to register the details, but not so long that produces saturation.
Magnification was mentioned when talking about visual observing. We need to give the right magnification so the details fall on different cells of the retina. The requirement still exists in photography. The problem is that the camera is at the main focus of the telescope. The only way we can change magnification is by inserting a Barlow lens before the camera. Most camera mounts do not permit us to do this. There are two solutions. One is to use an adapter for the camera that ends in a 1 1/4 inch barrel. With this adapter, the camera can be mounted like an eyepiece. It is possible to put a Barlow lens ahead of the camera. The other possibility is to use projection. It is possible to find adapters that permit to connect the normal T mount from the camera to an eyepiece. They are called eyepiece projection adapters. They normally come with several small tubes threaded on both ends. Different combination of tubes produce different distance between the camera and the eyepiece; thus, different magnification. Focusing the camera under these circumstances is rather delicate.
In summary, the influence of the resolving power is when the details are of interest. The larger the diameter of the telescope, the more details. The magnification should be adjusted to the conditions of observation. When planning an observation it is necessary to consider what details we wish to observe. We need to determine how the different factors affect the resolution for those details.