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Light Grasp of a Telescope

The primary function of a telescope is to gather light and funnel it into the observer's eye.  The larger the telescope, the greater the amount of light captured.  Light gathering ability is a function of the area of the objective lens or primary mirror.  Thus the aperture, or diameter, determines the light grasp of a telescope.  Since for a circular aperture, Area = πr2, as the aperture is increased, the light grasp increases by the square of the aperture.  This means an 8" aperture collects 4 times as much light as a 4" aperture.  The 8" aperture has an area of 50 in2, while the 4" aperture has only 12.5 in2.

How much more light a telescope gathers compared to the unaided eye is determined by the ratio between the light-gathering area of the telescope and the light-gathering area of the eye.  The aperture of the eye is determined by the size of the pupil (see the section below on The Human Eye for more details).  In general, the average pupil will open up to about 7mm in diameter.  Note that this means if the beam of light coming out of the telescope eyepiece is larger than the maximum size of the eye, the eye becomes the limiting factor and effectively reduces the aperture of the telescope.  This is described more later.  The light gathering area of the 7mm pupil is then 0.06 in2.  For the 8" telescope, this gives a ratio of 50/0.06 = 833, meaning an 8" telescope gathers 833 times more light than the unaided eye.

Above:  The difference in relative size between an 8" (200mm) mirror and the 7mm opening of the human eye

This implies an object seen with the unaided eye will appear more than 800 times brighter through an 8" telescope.  However, the situation is more complicated than that.  While the previous statement is true for point sources (stars) it is not true for extended objects (galaxies, nebulae, planets).  This is because the light from an extended object is being spread out by the fact that the telescope is magnifying the image.  So magnification factors into the equation; light is lost in proportion to the square of the magnification.  There is a minimum magnification allowed by the limiting size of the pupil as described above.  This works out such that the image through a telescope can never be brighter than the image as seen with the unaided eye.  This seems counterintuitive.  However, with optimum magnification (described below) the image not be significantly dimmer and will be considerably larger and more detailed.

An additional advantage of aperture that comes into play when magnification is considered is image brightness at a given magnification.  Through a given telescope, doubling the magnification reduces the brightness of an extended object fourfold.  Doubling the aperture of a telescope makes the image four times brighter at the same magnification, or allows twice the magnification to be used while retaining the same image brightness.

The direct ratio between telescope brightness and unaided eye brightness still holds for point sources.  For this reason, stars will appear brighter than they do with the eye, independent of magnification.  The magnitude scale used to describe the brightness of stars is a logarithmic scale.  Each magnitude is a difference of 2.5 in brightness.  A 1st magnitude star is 2.5 times brighter than a 2nd magnitude star.  A 2nd magnitude star is 2.5 times brighter than a 3rd magnitude star.  And the difference between a 1st magnitude star and 3rd magnitude star is 2.5 x 2.5 = 6.25 times.  A telescope which can make stars appear 833 times brighter than the unaided eye will allow stars 7.3 magnitudes fainter to be seen.  If you can see 6th magnitude stars with the unaided eye, you should be able to see 13th magnitude stars through an 8" telescope.  In actuality, fainter stars can be seen.  This is because of the decrease in brightness of the sky background seen through the telescope.  Sky brightness is also a function of magnification, the sky growing darker as the power is increased.  At a magnification of 100x, the sky background appears 2.7 magnitudes darker than without the telescope, which translates to an extra 2 magnitudes of reach, allowing 15th magnitude stars to be seen.  A darker sky which allows fainter stars to be seen with the unaided eye will of course allow even fainter stars to be seen through the telescope.

 

Resolution of a Telescope

A telescope, even with theoretically perfect optics, cannot produce a point image of a point source, such as a star.  This is because of diffraction, which is caused by the wave nature of light.  As light passes through the telescope aperture, the waves interfere with each other, diffusing the point source.  Imagine a wave of water coming in off the ocean and meeting with a break in a wall as shown below.  The aperture causes secondary waves to be created, which the interfere with each other.  Where crests and troughs of waves coincide, destructive interference cancels out the waves. 

Above:  Wave interference at an aperture

The effect of this in a telescope is that the light from a star does not drop off smoothly at the edge of the star image, but in a rhythmic pattern of interference.  At certain points around the star image, destructive interference causes rings of zero intensity.

Above:  Diffraction pattern of a star image through a telescope and profile of the image brightness

Just like the walls in the ocean wave example, an aperture, such as the edge of a mirror, causes light waves to interfere.  The diagram below shows the two wave patterns that are set up by the edge of a mirror.  Where the waves cross, there is constructive interference and the light is amplified, producing the bright rings of the diffraction pattern.  Halfway between these spots, waves cancel each other out, causing destructive interference and forming the gaps in the diffraction pattern.

Above:  How the edge of a mirror creates a diffraction pattern

The resulting effect, assuming no other aberrations, is for about 85% of the light from a point source to be located within the bright central spot of the diffraction pattern.  This central spot is called the Airy disk.  The outer rings are progressively fainter and are difficult to see under normal conditions.  The first bright ring outside the Airy disk contains less than 2% of the light from the source, and the rest are dimmer still.  The effective resolution of a telescope can then be considered the size of the central disk in the diffraction pattern.

The size of the Airy disk can be determined mathematically.  Specifically the angular diameter of the disk is A = 1.22λ/D radians, where λ is the wavelength of the light.  As will be described later, the human eye is most sensitive to a wavelength of 550 nanometers (nm), in the yellow-green part of the visual spectrum.  Converting to the more useful angular measure of arcseconds, this gives a simple equation:  A = 5.45/D, for a telescope diameter in inches.  It can be seen that the larger the telescope aperture, the smaller will be the Airy disk and the greater the resolution.  As an example, the resolution of an 8" telescope is 0.68 arcseconds.  A 12" telescope has a resolution of 0.45 arcseconds.  For reference, Jupiter is about 45 arcseconds in diameter.

The effect of central obstruction, such as that caused by the secondary mirror in a Newtonian or Cassegrain telescope, is to transfer more light from the Airy disk to the outer rings.  An extreme example would be a 50% central obstruction.  This would cause the first ring to become 4 times brighter while the central disk would drop in brightness by a factor of 2.  It also has the interesting effect of reducing the diameter of the Airy disk to about 80% of its unobstructed size.  However, instead of the brightness difference between the disk and first ring being a factor of 50, it is reduced to only a factor of 10.  Overall the image quality is worse, despite the smaller Airy disk size.

 

Magnification

To obtain the most detail out of your telescope, it is critical to choose the right magnification for the object being observed.  Some objects will appear best at low power, some at high power, and many at a moderate magnification.  This section will describe why different magnifications work better than others, and give information on how to choose the appropriate power for your viewing.

 

Minimum Magnification

There is a lowest useable magnification on a telescope.  This is determined by the exit pupil of the optical system and the size of the observer's pupil.  The exit pupil is the diameter of the beam of light produced by the telescope/eyepiece combination.  The lower the magnification, the larger the exit pupil.  Exit pupil is calculated easily by dividing the telescope's aperture (in millimeters) by the magnification.  Thus, an 8" (200mm) telescope operating at 50x has an exit pupil of 4mm.  At 100x on the same telescope, the exit pupil shrinks to 2mm.  The minimum magnification is limited by the size of the observer's eye.  If the observer's pupil is smaller than the exit pupil of the telescope, the beam of light is cut off and the effective aperture of the telescope is reduced.

Above:  Decreasing magnification (or increasing aperture) increases exit pupil size.  If the exit pupil is too large (right), the observer's pupil will restrict the effective aperture of the telescope.

On average, the pupil of the eye will open to a maximum of 7mm.  This is age-dependent, so observers in their teens and twenties may have pupils as large as 7-8mm, while middle aged observers will have pupils in the 6-7mm range.  By age 80 the pupil opens only to 3.5-5mm.  In general, a 7mm exit pupil is assumed.  To determine minimum magnification, calculate the magnification that yields a 7mm exit pupil.  This is simply the telescope's aperture in millimeters divided by 7.  The table below gives minimum useable magnifications for different apertures.

Aperture

Minimum Magnification

4"

15x

5"

18x

6"

22x

8"

29x

10"

36x

12"

44x

This calculation assumes that the telescope is capable of a magnification this low.  There are a couple restrictions that might prevent this.  A telescope with a very long focal length may require a longer-focal-length eyepiece than is possible.  For example, an 8" SCT with a focal length of 2032mm requires a 70mm eyepiece to reach 29x.  Eyepieces longer than 55mm are very uncommon.  Additionally, even if such an eyepiece were available, telescopes with large central obstructions, such as SCTs, suffer from an effect where very low magnifications allow the central obstruction to be seen in the image.  Refractors, having no central obstruction, do not suffer from this effect and make excellent low-power, wide-field instruments.

 

Maximum Magnification

The most common telescope myth is that more power is better.  And while this is usually untrue, there are some instances where a very high magnification is useful.  The important thing to bear in mind is that the highest useable magnification is most often restricted by atmospheric turbulence.  You will normally run into the limit imposed by seeing conditions before reaching the telescope's optical limit.  On nights of excellent seeing, however, pushing the limits of power can be useful for certain types of observation.

The obvious use of high power is viewing the planets.  Magnifications more than 30x the telescope's aperture in inches are usually impractical for planetary observation.  Most observers find this is sufficient to see all available detail.  More magnification makes the planet appear larger but not necessarily more detailed.  On an 8" telescope, 30x per inch of aperture produces 240x.  This would make Jupiter appear 3° across, or six times the size that the full moon appears to the unaided eye.  The exception to this is smaller telescopes, which can be used at higher power per inch, often 40x to 50x.  The table below shows recommended magnifications for viewing the planets with telescopes of different apertures.

Aperture

Planetary Magnification

4"

180x

5"

200x

6"

220x

8"

240x

10"

300x

12"

360x

Higher magnifications can be used for viewing double stars and, as the next section shows, some deep-sky objects.  Very close double stars may allow magnifications up to 50x or 70x per inch of aperture, as long as the separation is above the resolution threshold of the telescope.  An 8" telescope, for example, has a resolution of 0.68", and empirical data show that double stars can often be detected when their separation is about half the telescope's theoretical resolution due to the nature of diffraction patterns.  This means an 8" scope should (in theory) be able to detect double stars as small as about 0.4".  This depends greatly on the quality of the optics, collimation, seeing conditions, the observer, and the magnitude difference between the stars.  Again, smaller telescopes can tolerate more power.  The table below gives the maximum theoretical magnification for various telescopes.

Aperture

Maximum Magnification

4"

280x

5"

320x

6"

350x

8"

400x

10"

500x

12"

600x

 

Optimum Magnification

There are several ways to determine the ideal magnification for viewing an object.  One simple method is based on the resolution of the telescope.  The human eye can typically resolve objects with an apparent size of 1 arcminute.  The magnification required to enlarge details at the telescope's resolution to 1 arcminute is 13x the aperture in inches.  For an 8" telescope this is 104x.  Empirical evidence suggests that an exit pupil of 2mm produces very detailed views of most deep-sky objects.  A 2mm exit pupil requires a magnification of 12.5x per inch, confirming the above theory.

A more complex theory for the best deep-sky magnification is based on the human eye's response to contrast.  The contrast between the sky background and a faint celestial object is critical for observation.  An object whose surface brightness is such that the contrast between it and the sky is below the eye's detection threshold will be invisible.  Observers have noted that low-contrast objects are often easier to detect at higher magnifications.  It is often assumed that this is due to higher power increasing contrast, but this is not true.  The relative contrast between the object and the sky background is unchanged by magnification (each is affected equally).  However, the eye is more sensitive to low-contrast objects when they appear larger.

Above:  As the apparent size of an object increases, the contrast necessary to see it decreases.  Adapted from Clark.

This feature of the eye's response to contrast implies that more magnification should make an object easier to detect.  This turns out to be true, although there is an upper limit.  At some point, the surface brightness of the object starts to decrease faster than the eye's detection threshold.  There is an ideal magnification, determined by what Roger N. Clark calls the "optimum magnified visual angle."  In his excellent book Visual Astronomy of the Deep Sky, Clark describes the optimum magnified visual angle (OMVA) as "the maximum magnification that will help detection."  The value of the OMVA is a function of the sky conditions, the surface brightness of the object, the size of the object, and the telescope aperture.  Clark dedicates a lengthy appendix in his book to describing the iterative calculation procedure.  Suffice to say for our purposes there are some generalizations that can be made.

In general, a smaller telescope will require a higher magnification to detect an object.  This is an interesting result and should be kept in mind by observers using smaller instruments.  For many objects, the optimum magnification for a 6" telescope is 40% greater than for an 8" telescope.  If using a smaller telescope and an object is not seen, trying a higher magnification may reveal the object.  This is counterintuitive to the idea that lower power yields a brighter image.  While this is true, the contrast effects of the human eye actually allow faint objects to be seen better at higher power even though this causes a loss of surface brightness.

For a given telescope, the following rules apply.  For small objects with low surface brightness (such as galaxies), use a moderate magnification.  For small objects with high surface brightness (such as planetary nebulae), use a high magnification.  For large objects regardless of surface brightness (such as diffuse nebulae), use low magnification, often in the range of minimum magnification (as described above).  Specific recommendations for various telescopes are given in the table below.

Aperture

Small/Low SB

Small/High SB

Large

6"

150-250x

400-600x

25-40x

8"

100-200x

300-500x

30-50x

12"

80-150x

200-400x

45-60x

 

The Human Eye

An understanding of how the human eye functions can give you an edge in getting the best visual observations.  The human eye is a very sophisticated and quite impressive organ which can adapt to an amazing array of conditions.  We are capable of seeing in both sunlight and moonlight despite the full moon being half a million times fainter than the sun.  This huge dynamic range allows us to see detail in the moon and planets as well as subtle swirls and spiral arms in dim nebulae and galaxies.  The workings of the eye can be used to definite advantage, so some of the structures and visual adaptations of the eye, and how they factor into observing, are related below.

Above:  The inner workings of your most important observing tool

 

Rods and Cones

The eye can be thought of as analogous to a telescope.  The lens focuses light onto the retina, just as a telescope lens would focus light onto a focal plane.  Putting a camera at the telescope's focal plane captures the light just as the eye's retina does.  And like a digital camera or CCD, the retina is divided into "pixels" made up of light-sensitive cells called rods and cones.  The differences between these two types of cells determine how the eye sees things, especially celestial objects.

Rods are sensitive only to black and white, while cones come in three types, each sensitive to a different color: red, green, and blue.  Cones are concentrated in the center of the retina, while rods form the periphery of the retina.

Above:  Color-sensitive cones occur in the center of the retina, with black-and-white rods surrounding them

An important distinction between rods and cones is that cones are sensitive to high light levels but not to dim light.  This has a significant effect on what you can see through a telescope.  Since most astronomical objects are dim, the light from them does not stimulate the color-sensitive cones.  This is why most deep-sky objects appear black and white.  While these objects have color, as any deep-sky photograph will show, the color is too faint to be detected by the human eye.  Brighter objects such as the planets and bright stars can easily stimulate the cones and appear in color.

Some deep-sky objects are bright enough to stimulate the cone cells.  However, the eye's sensitivity to color changes in low light levels.  The peak wavelength to which the eye is sensitive shifts toward the green part of the spectrum.  Also, red and blue sensitivity drop off dramatically.  The diagrams below show the sensitivity of the human eye in bright light and low light.  Bright light vision is termed photopic, and low-light vision is called scotopic.

Above:  Photopic--daylight--response of the human eye

Above:  Scotopic--night--response of the human eye.  Note the loss of sensitivity to blue and red wavelengths.

This shift in color sensitivity means that even if an object is bright enough to stimulate color receptors, the dominant color will be green.  In fact, the deep-sky objects that tend to show the most prominent color are planetary nebulae.  This is because planetaries emit 57% of their light from excited oxygen atoms, which give off light in the green part of the spectrum to which the eye is most sensitive.  Most observers see planetary nebulae as blue-green in color.

 

Averted Vision

Another effect of the difference between rods and cones comes from both their relative sensitivity to light and their placement within the retina.  When dark adapted, rods are nearly 40 times more sensitive to light than cones.  This allows them to see four magnitudes fainter than the cones.  However, since the cones are concentrated in the middle 10% of the retina and the rods are almost exclusively outside this zone, the eye is least sensitive in the center of its vision and more sensitive to light falling outside the middle of the eye.

Averted vision is a technique where the observer intentionally looks to the side of an object rather than directly at it.  This places the light from the object onto the rods rather than the cones, increasing the ability to see it.  Most of the rods are concentrated just outside of the center of the retina, the peak occurring about 20° off axis.  This implies the best view of an object will occur when it is slightly, but not excessively, offset from the center of vision.

There is a blind spot in the eye, where the optic nerve attaches.  This blind spot sits about 15° to 20° away from the center of the eye, in the direction of the ear.  Observers should thus avert their vision in the direction of the nose to avoid placing the light from the object onto the blind spot.

 

Dark Adaptation

Everyone is familiar with the effect of the pupil opening wider upon entering a dark room.  This change takes only a few seconds, as you can see using a mirror, a dark room, and too much free time.  The pupil opens from about 2mm to 7mm.  This increases the light gathering ability of the eye only about 12 times.  This is certainly not sufficient to account for the huge range of brightness the eye can accommodate.

In addition to the physical change of widening the iris to allow more light in, there are chemical changes that take place to account for a several-thousand-fold increase in sensitivity.  A chemical called rhodopsin is generated as the eye adapts to the dark.  The greater the amount of rhodopsin, the greater the sensitivity of the rods and cones.  Most of the chemical change occurs within half an hour.  This amount of time should be allowed by the observer for dark adaptation before trying to observe very dim objects.  Dark adaptation continues for as much as two hours.  The ability to dark adapt is affected by exposure to bright light beforehand.  Spending long periods of time outdoors in bright sunlight can hinder the ability to fully dark adapt for as much as several days.

 

Pupil Size

As mentioned above, the pupil can expand through a range of diameters, from less than 2mm to as much as 8mm.  As we age, the pupil's maximum size decreases.  The average adult observer will most likely have a maximum pupil size of 7mm.  Older observers may only have 4-6mm maximum pupil diameters.

Pupil size affects a number of aspects of observing.  Foremost it determines the size of the beam of light the eye can accept.  If the beam coming from the telescope (the exit pupil) is larger than the pupil of the eye, some of the light is blocked and the effective aperture of the telescope is reduced.  (See the exit pupil diagram above.)  For example, if an 8" telescope is used at 25x, the exit pupil will be 8mm.  If the observer only has a pupil size of 6mm, the effective aperture of the telescope is reduced to 6".

The image quality of the human eye (ignoring for now any defects such as astigmatism, etc.) is determined by the diameter of the pupil.  A larger pupil diameter increases the aberrations present in the eye.  While there is little to be done to control the pupil size of the eye while observing, what can be controlled is the exit pupil of the telescope.  Even if the observer's eye is opened to 7mm, if the exit beam from the eyepiece is smaller, only that much of the eye is used, and the effect is the same as if the pupil were actually opened to that size.  For example, a 2mm exit pupil uses only 2mm of the eye, even if the actual pupil size is much larger.  Significant aberrations such as astigmatism are very dependent on exit pupil size.  Since these types of aberrations tend to minimize as the eye's pupil shrinks, using higher magnifications on a telescope has the effect of using a smaller portion of the eye and thus reducing aberrations.  For this reason, most observers with astigmatism find they must wear their glasses (or use corrective optics on the eyepiece) when viewing at low powers and correspondingly large exit pupils.  However, when viewing at high power, glasses may not be required due to the reduction in apparent aberration thanks to the smaller exit pupil.

 

Resolution

In a digital or CCD camera, the spatial resolution is a function of the pixel size on the CCD chip.  Similarly, the resolution of the human eye is dependent on the size of the cells in the retina.  At the center of the eye--the fovea--the smallest and most concentrated cone cells are located.  The average size of these cells is 1.5 microns.  Compare this to the typical 5-10 micron pixel size of most CCD chips.  This works out to a theoretical resolution of about 20 arcseconds.  However, under dark adapted conditions the pupil is opened to its maximum size and is subject to more aberrations.  This limits the resolution of the eye to about 60 arcseconds, or 1 arcminute.  This implies that an object, such as a lunar crater, which is 1 arcsecond across, when magnified 60 times, should be just visible to the eye.  This assumes, of course, that the telescope is capable of resolving the object to begin with.  For example, an object 0.2 arcseconds in size, magnified 300 times, would appear 60 arcseconds across.  However, 0.2 arcseconds is below the resolution threshold of any telescope smaller than 27 inches in diameter.

Certain objects, such as close double stars or thin linear features, may be visible below the threshold of both the telescope optics and the observer's eye, due to the effects of diffraction, the stimulation of larger numbers of cones, reduction of pupil size when viewing bright objects, and contrast effects.  From personal experience, using a high quality telescope, the author has seen a 0.6-arcsecond double star clearly defined in a 6" telescope, which has a theoretical resolution of only 0.9 arcseconds.  As contrast lowers, the resolution of the eye decreases, so planetary detail may not be as finely resolved as stellar or lunar details.  The same is true of deep-sky objects.

Observing with a Telescope



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