Sommers-Bausch Observatory - University of Colorado

Telescopes & Observing

This essay is excerpted from the Sommers-Bausch Observatory's "APAS 1010 Laboratories - Introduction to Astronomy" lab manual, 1996, by Keith Gleason.


Telescopes come in two basic types - the refractor, which uses a lens as its primary or objective optical element, and the reflector, which uses a mirror. In either case, light originating from an object (usually at infinity) is brought to a focus within the telescope to form an image of the object. The size of a telescope refers to the diameter its primary lens or mirror, rather than its length.

By placing photographic film at the focal plane, the objective lens or mirror forms a camera system. If instead we position an eyepiece lens at an appropriate distance behind the focal plane, we form an optical telescope.

The optical arrangement of a refracting telescope is shown below. The image is formed by the refraction of light through the lens. The refractor has an advantage over reflectors in that there is no central obscuration to produce diffraction patterns, and therefore yields crisper images. However, refraction introduces chromatic abberation, which is corrected by using two-element (doublet or achromat) or three-element (apocromat) lenses. The lens complexity makes the refractor very expensive; hence, refractors are much smaller in diameter than comparably-priced reflectors. All of the Sommers-Bausch Observatory (SBO) finder telescopes are of the refractor type.

A reflecting telescope focusses and re-directs light back towards the incident direction, and therefore requires additional optics to get the image "out of the way". This central obscuration reduces the amount of light reaching the primary, and adds diffraction patterns which degrade the resolution. However, the telescope does not suffer from chromatic abberation, since only mirrors are used. Reflectors can be built much larger since the mirror is supported from behind (while a lens is mounted only at its edges). Furthermore, only one objective surface must be ground and polished, making the reflector much less expensive. As a result, virtually all large telescopes are of the reflector type.

The Newtonian (invented by Sir Isaac Newton) is the simplest form of reflector. It uses a diagonal mirror (a plane mirror tilted at a 45 angle) to re-direct the light out the side of the telescope to the eyepiece. The placement of the eyepiece at the "wrong" end of the telescope limits it practical size, and its asymmetric design precludes the use of heavy instrumentation. The newtonian is used principly by amateur astronomers since it is the "cleanest" as well as least expensive reflector.

In the Cassegrain reflector, a convex secondary mirror intercepts the light from the primary and reflects it back again, reducing the convergence angle in the process. The light passes through a central hole in the primary and comes to a focus at the back of the telescope. The location of the Cassegrain focus makes it easy to mount instrumentation, and the folded optical design permits larger diameter telescopes to be houses within smaller domes; hence this design is most widely-used at professional observatories. All three of the permanently-mounted SBO telescopes (16", 18", 24") are Cassegrain reflectors.

Telescopes that use a combination of lenses and mirrors to form images are called catadioptric. One form is the Schmidt camera, which uses a weak lens-like corrector plate at the entrance to the telescope to correct for off-axis image abberations; this specialized telescope can't be used for viewing but does produce panoramic sky photographs. SBO has an 8" Schmidt for astrophotography.

One of the most popular, albeit expensive, telescope designs is the Schmidt-Cassegrain. As the name suggests, it looks like the Cassegrain telescope but with the addition of a Schmidt corrector plate. SBO has six of these smaller, portable telescopes: an 8" Meade, two 5" Celestrons, and three 3.5" Questars.

The fundamental optical properties of any telescope are described by three parameters: the focal length, the aperture, and the focal ratio (f/ratio). The focal length (f) is the distance from the objective where the image of an infinitely-distant object is formed. The aperture (A) of a telescope is simply the diameter of its light-collecting lens (or mirror). The f/ratio is defined to be the ratio of the focal length of the lens or mirror to its aperture:

Obviously, if you know any two of the three fundamental parameters, you can calculate the third.


As shown in the diagram below, an object that subtends an angular size (theta) in the sky will form an image of linear size h given approximately by

The image scale is the ratio of linear image size to its actual angular size:

Thus, the scale of the image of any object depends only upon the focal length of the telescope, not on any other property: two telescopes with identical focal lengths will produce identically-sized images of the same object, regardless of any other physical differences. Furthermore, the image scale is directly proportional to f: doubling the focal length will produce an image twice the linear size (and four times the areal size).

The plate scale (used in photography) is usually expressed as the inverse of the above - the angular size of the object (usually in arc-seconds) corresponding to a linear size (usually in millimeters) in the focal plane.

For a compound telescope (with more than one active optical element), we refer to the effective focal length (EFL) of the entire optical system, and treat it as a simple telescope with single lens or mirror of focal length f = EFL.


The aperture A of a telescope is the diameter of the principle light-gathering optical element. The light-gathering ability, or light grasp, of the telescope is proportional to the area of the objective element, or A2. For point sources such as stars, all of the collected light will (to a first approximation) converge to form a point-like image; hence, stars appear brighter, and fainter stars can be detected, with telescopes of larger aperture. Telescope aperture is the principle criteria for determining the limiting visible stellar magnitude.

The resolving power (ability to resolve adjacent features in an image) of an optical telescope is also chiefly a function of aperture; the larger the aperture, the smaller the diffraction pattern formed by each point source, and therefore the better the resolution. The theoretical diffraction-limited resolution of a telescope is given by

where (lambda) is the wavelength of the light used, assumed to be 5500 .


Although the amount of light gathered by a telescope is proportional to A2, the collected light is spread out over an area at the focal plane that is proportional to h2, and therefore propertional to f2; hence, the image brightness of an extended (non-pointlike) object will scale linearly with the ratio (A/f)2, or inversely with the square of the f/ratio = f/A of the telescope:

The f/ratio of a telescope is therefore the only factor that determines the image brightness of an extended object. For example, if one telescope has twice the aperture and twice the focal length of another, both telescopes are geometrically similar and would use exactly the same photographic exposure to produce identically-bright images of, say, the Moon. Of course, the first telescope would produce an image twice the linear size (and four times the area) as the latter, but both would have the same photographic "speed". Note that the smaller the f/ratio, the brighter the image and the faster the speed: a 35mm camera using an f/ratio of f/2 will photograph the Milky Way in less than 5 minutes, while the same film on an f/15 telescope will require an hour or more to capture the diffuse glow! On the other hand, individual stars will record much better using the telescope.

The optical speed of a compound telescope is simply the ratio of its effective focal length to its aperture. Both effective or actual f/ratios are a measure of the final angle of convergence angle of the cone of light before it forms the image; the smaller the f/ratio, the greater the convergence, and the more critical the focus; this is why "slow" optics, with a slowly-converging beam, exhibit a larger "depth-of-field".


Equation (4) gives the theoretical limit to the angular size that can be resolved by a telescope. This limit is almost never achieved in practice, since atmospheric turbulence is always present to blur our image of the object. The quality of the seeing is measured by the angular separation needed between two stars for their images to just be resolved. Average seeing in the Boulder area is about 2 arc-seconds, making it a marginal task to "split" the 2 arc-sec pairs of the "double-double" star Epsilon Lyrae. Good seeing in Boulder is when the atmosphere is stable enough to resolve 2" separations. Poor seeing conditions can be as bad as 5" or even worse. By comparison, half-arc-second seeing occurs routinely at several optimally-located major observatories, but it is rare for any ground-based telescope to experience 0.1" conditions.

One can estimate the seeing of a night simply by glancing up. If the stars glow solidly, the seeing is probably "good"; if they twinkle, the seeing is "average"; if bright stars dance and planets flash with color, the seeing is "poor". Another measure is to look towards Denver - if there are lots of particulates in the air, and Denver is on smog alert, you will see a bright horizon glow; in this situation, you can usually count on good seeing! These conditions are created by an inversion layer of stagnant air, which permits rock-solid astronomical viewing.

The best conditions for good seeing are the worst for sky clarity. Good clarity implies dark skies due to a lack of light-scattering dust particles, and an absence of water-vapor haze. Clarity usually improves in Boulder after the passage of a thunderstorm, which clears out the dust. Of course, the conditions that improve clarity usually destroy seeing.

Since ideal nights of good clarity and seeing are rare, an observer must be prepared to take advantage of the best properties of the night, if any. Lunar and planetary observing requires good seeing conditions, since planetary disks are small, only 2-40 arc-seconds in diameter. Poor clarity is not a major problem for bright solar system objects, although low-contrast features may be washed out. On the other hand, good clarity (and the absence of strong moonlight) is essential to observe galaxies and diffuse nebulae, since the light from these objects is faint and dark skies are needed for contrast. Seeing is not critical, since these objects are fuzzy patches to begin with. When both seeing and clarity are poor, it's best to focus your attention on bright star clusters and widely-spaced double stars.


An eyepiece or ocular is simply a magnifier that allows you to inspect the image formed by the objective from a very short distance away. The eyepiece is placed so that its focal plane coincides with the focal plane of the objective, so that the rays from the image emerge parallel from the eyepiece. As a result, the telescope never forms a final image - the lens of your eye does that.

By selecting appropriate rays for both the objective lens (obj) and the eyepiece lens (eye), we can see that rays incident at the telescope at an angle theta(obj) will emerge from the eyepiece at a larger angle theta(eye); the observer perceives that the object subtends a much larger angle than is actually the case. A little geometry gives the angular magnification M(theta) produced by the arrangement:

That is, magnification is the ratio of the objective focal length divided by the eyepiece focal length. For example, the 16-inch f/12 telescope has a focal length of 192 inches, or 4877 mm. If we use an eyepiece with a focal length of 45 mm (engraved on its barrel), the "power" of the telescope will be about 108 X; by switching to an 18 mm eyepiece, we will have 271 X. The answer to the question "what is the power of this telescope?" is "whatever you want - within the range of the available eyepiece assortment".

Eyepieces come in a variety of designs, each representing a different trade-off between performance and cost, ranging from the inexpensive short-focal-length two-element Ramsden (R) to the 6-element triple-doublet Erfle (Er). Other types include: the Kellner (K), an improved Ramsden; orthoscopic (Or), good at moderate focal lengths at reasonable cost; and the expensive low-power wide-field designs - the Plossl (including Clave), the Konig, and the Televue.

Besides focal length (and image quality), the other important characteristic of an eyepiece is its field-of-view - the size of the solid angle viewable through the ocular, or when used on a particular telescope, the actual angular size of the observable sky field. The field available with a given telescope-eyepiece combination can be measured directly by positioning a bright star just at the northern edge of the field; after noting the declination, you move the telescope northward until the star is at the southern boundary of the field, note the declination again, and calculate the difference in angle. The field can also be calculated from the image scale of the telescope (equation (3)): measure the diameter of the field stop (the ring installed within the open end of the eyepiece) , equate that to the linear image size h, and calculate the corresponding angle (theta).

The Barlow is a telecompressor (concave or negative) lens that can be installed in front of the eyepiece. It reduces the angle of convergence of the objective light cone and hence increases the f/ratio and the EFL of the telescope, thus increasing the magnification (by a factor of 2 to 3) from a given eyepiece. It helps achieve high magnification without sacrificing eye relief (see below).


The choice of eyepiece is one of the few factors that an observer may control to enhance the visibility of astronomical objects.

The Observatory's eyepieces range from focal lengths of 70 mm down to 4 mm. The longer focal lengths are in the 2"-diameter format, which fit directly into the large telescope tubes; the shorter (higher magnification) eyepieces have 1-1/4" diameter barrels, but can be used in the 2" tube using an eyepiece adapter plug.

Although equation (6) implies that it's possible to have any magnification one desires, there are practical limits. The entrance pupil of the human eye (the diameter of the iris opening) is about 7 mm for a fully dark-adapted eye (5 mm for older individuals). The exit pupil of an telescope (diameter of the bundle of light rays exiting the eyepiece) can be shown to be

If the exit pupil is larger than the entrance pupil of the eye, the eye can't intercept it all and some of the light is wasted. This occurs if the magnification is too low. If the exit pupil just matches the eye pupil, all of the light is utilized and we have the brightest-possible, or "richest-field" arrangement. This is why 7x50 (7 power, 50 mm aperture) and 11x80 (11 power, 80 mm aperture) binoculars are excellent for night observing - their exit pupils optimally match the dark-adapted human eye. At higher magnifications, all of the light enters the eye but is spread over a larger solid angle which dims the field.

The average human eye can just barely resolve two objects separated by about 1 arc-minute, although a separation of about 4' (240") is much more comfortably perceived (about 50 such 4 arc-minute "pixels" comprise the face of "the man in the Moon"). Hence, one criterion for a telescope's maximum useful power M(max) is "that magnification that matches a 250 arc-second visual separation to the diffraction limit of the telescope"; from equation (4) this equates to

By this "rule of thumb", the 16-inch telescope has a maximum useful mag-nification of 800 X on a night of perfect seeing, which would be achieved with a 6 mm eyepiece. Any greater magnification will be "empty" - that is, the image will become larger, but the enlargement will contain only "blur", not additional detail. (Note: poor-quality 2.4"-refractors are provided with 4 mm eyepieces plus a cheap 2.5X barlow lens so that the manufacturer can advertise "600 power" - 5 times beyond any useful application!)

The above magnification limit is somewhat optimistic for telescopes of moderate-to-large aperture, since detail is almost invariably limited by atmospheric seeing rather than by telescope resolution. For example, if the seeing is about 1", a 250" perceived separation implies a useful maximum magnification of about 250 X - or an eyepiece in the 18 - 24 mm range for the 16-inch telescope. For 2" seeing, a reasonable choice is about 125 power, or eyepieces in the 32 - 45 mm range.

There are several additional considerations related to high magnification. On the negative side, short-focal-length eyepieces require critical focussing and eye placement, making them difficult for inexperienced observers to use. In addition, they exhibit short eye relief - the distance behind the lens where the eye is positioned to see the entire field-of-view. In particular, eyepieces shorter than about 12 mm focal length are problematic for eye-glass wearers, since glasses prevent the eye from being placed close enough to avoid "tunnel vision".

On the positive side, high magnification reveals fainter stars. Remember that stars are unresolved point sources whose light is concentrated (to a first approximation) at a single point in the image regardless of magnification. The glow of the background sky is a diffuse source, which will be spread out by higher magnification, reducing its brightness. Although high magnification doesn't increase the brightness of faint stars, it improves their contrast against the sky, making them more visible! By this same token, high magnification helps diminish the intense glare from bright solar system objects, making the Moon less painful to look at, and the bands of Jupiter easier to see.

Finally, there is the consideration of the type and angular extent of the object being viewed. High magnification certainly helps resolve fine planetary detail and separates close double stars, provided that seeing conditions permit. Open clusters usually extend over a large patch of the sky, and hence need low power (a wide field-of-view) to encompass them - in fact, the small finder scopes may present a more pleasing view than the main telescope. High power won't help the contrast between sky background and diffuse nebulae (since both are extended sources), and in fact may keep the observer from seeing the glow since all aspects of the image are dimmer. Low power is essential for extremely large nebulae, since these objects can fill the field-of-view - creating a situation where "you can't see the forest for the trees". Globular clusters can be appreciated at a variety of magnifications: low power creates an impression of a "cotton-ball" floating in space, while high magnification shows a magnificent sparkling array of bright points reminiscent of distant city lights. In the end, your personal experience and judgement should be your guide.


Observing through an eyepiece is an unnatural experience; good observing techniques come with time and experience. Here are some hints to help you get the most out of your telescope time: