On a clear summer night, look up at the Milky Way. Gigantic clouds of innumerable stars form a stairway to heaven. Grabbing binoculars, the clouds of stairs reveal light and dark streaks, and fuzzy condensations . But only with a telescope do these clouds reveal their true nature: countless stars forming the spiral arms of our galaxy. What telescope design best stirs our passion, delighting us at the eyepiece?
Or to rephrase the question, what aperture will show the greatest number of stars in an eyepiece? Larger apertures reach fainter magnitudes but have narrower fields of view. The answer hinges on counts of star at particular magnitudes, or what is called star density.
Here are star counts based on the same ratio of aperture to magnification as the unaided-eye: (figures adapted from Glenn, Sky and Telescope, 1980, adjusted for increased field of view of the latest eyepieces.)
|Aperture||Field (60 deg eyepiece)||Stars||Field (100 deg eyepiece)||Stars|
|eye (7mm)||60 deg||1300||100 deg||3600|
|50mm||8 deg||2500||14 deg||6900|
|4 inch||4 deg||2700||6 deg||7600|
|8 inch||2 deg||2800||3.4 deg||7800|
|16 inch||1.1 deg||2700||1.8 deg||7600|
|32 inch||0.5 deg||2200||0.9 deg||6200|
As you can see, the best result is broadly centered on 8 inches aperture.
It is said that a person can see 2000 stars at a dark site with their unaided-eyes. More than that are visible in a single eyepiece field of view. Telescopes used with widest angle eyepieces that yield an exit pupil of 5-7mm are what we call Richest Field Telescopes - the richest field of stars possible in any single view.
The idea was originally proposed by Mr. Walken in Knowledge (now Discovery) in 1916. He writes in part, "My chief part was in perceiving and publicly pointing out how every aperture could be made 'an' RFT, of that aperture, for a given observer and that there was one of all these which, in connection with the curve of star density against magnitude of stars, was uniquely 'the' RFT for the observer, in respect of maximum countable number of star per apparent square degree." He concludes, using star counts published in Knowledge, 1914, that a 2.5 inch aperture has the greatest star count, though he goes on to say that the view through a 6 inch is "little inferior" and that the views through larger apertures "is decidedly more attractive and "richer."" He calculated a star count of 423 given a 50 degree eyepiece. Incidentally, he concludes that a magnification that yields the maximum possible exit pupil is best: too low of magnification wastes light and too high of magnification narrows the field of view dropping the star count.
It's plausible that the table of star counts or density in 1914 set the stage to conceive of richest field telescopes. If so, then this is an example of information driven revolution. So perhaps we can set the origination date to 1915.
In 1936 an updated star count is used to revise the RFT aperture to 4 inches with the count of stars as high as 711. Further discussion about the values of star density resulted in the conclusion that the RFT aperture is closer to 7 inches. Finally there's a note that a brightened background due to the Moon (today, we sadly have to add light pollution) dramatically increases the optimum aperture perhaps to 10 to 12 inches.
Clyde Tombaugh, discoverer of Pluto, built a 5 inch RFT in 1935. Clyde describes the views as "truly marvelous", mentioning how "dark nebulae in Sagittarius stand[s] out beautifully, as it does on a moderate exposure photograph." While Clyde extols the virtues of observing in Sagittarius, he says that the most beautiful and richest star fields are in the Cygnus region.
H.R. Suiter notes in 1996 that the best RFT aperture has varied from 1.5 inches to 12 inches depending on revised star density values. Because of this to and fro with aperture, he concludes that a rich-field telescope is best defined as one that can be used with an eyepiece yielding the maximum exit pupil that your eye can open to. He recommends personal experimentation since the aberrations of the eye are so substantial that star counts may not increase beyond 5 to 5.5 mm exit pupil. He makes an interesting comment that earlier observers would be awe struck by today's short focal ratio refractors and high quality high angle eyepieces, but that modern observers would envy earlier observers' dark skies even more!
In a 1980 Sky and Telescope article, Glenn Shaw takes up the question of the best RFT aperture, using the latest star density values. He concludes that an aperture of 9 inches is best though he employs a scope of 5 inches aperture. He adds an interesting graph where only stars brighter than 5th magnitude in the eyepiece are counted. This favors apertures closer to 24 inches. Since Glenn's numbers are the latest that I'm aware of, I use these, adjusted for wider angle eyepieces available today.
Faint stars don't impress me. Hmm, OK, then increase the aperture. The star count is about the same, but the stars will be brighter with larger aperture, the best aperture being broadly centered on 24 inches.
How about counting the total starlight, not just the star count? Then the best views will be with the greatest aperture. The counts of the dimmest Milky Way stars do eventually bottom out: an aperture of 24-60 inches works best.
What about increasing the magnification? A complicating factor is the issue of magnification. With the unaided-eye at a dark sky site, the faintest stars visible are about 6.5 magnitude. But if we could use a telescope with the same aperture of the eye at greater than unit power, we would see stars down to 8 magnitude. How is this so? The increased magnification dims the sky background but does not dim the stars. The background glow is spread over a larger area thanks to the increased magnification. Spreading the same amount of light over a larger area dims it. Stars are not dimmed because they are point sources - so far away that no practical magnification can reveal their disks. The diffractive nature of light and turbulence in the atmosphere combined with instrument defects do enlarge a star's pinprick of light, but only extreme magnifications make this detectable. So, the blackened background enable us to see fainter magnitudes.
To achieve adequate sky background darkening, we need to magnify the view at least three times more. This is 1/9 as much field of view, consequently we must reach 9 times greater star count to stay even. Assuming star counts that stay roughly constant across varying aperture, we need to triple our effective aperture. On the other side, 1.5 magnitude deeper penetration (8 mag - 6.5 mag = 1.5 mag) computes to doubling the aperture or 4x greater star count. So, we fall short: doubling the aperture doesn't make up for the required tripling, or, 4x the number of stars doesn't make up for 1/9th the number of stars. Conclusion: use the lowest power possible.
OK, so increasing the magnification doesn't work - how about decreasing the magnification? Human eyes when dark adapted have a pupil size from 5mm to 9mm; 7mm being taken as an average. Here's a chart depicting exit pupil versus age: http://www.egroups.com/files/amastro/Pupil.jpg. All the light from the optics must fit into this 7mm opening, otherwise the light is wasted. The eye's ratio of aperture in inches to magnification is 3.6x. We must keep this ratio regardless of aperture or waste light.
But what if we do waste light in this manner? We do gain extra field by going to lower power. How does the extra field compare to the aperture the eye can use? The effective aperture is found by using the formula: lowest power magnification = 25.4/7 = 3.6x per inch of aperture ratio.
This formula is derived from:
magnification = telescope focal length * 25.4 / eyepiece focal length
magnification = telescope aperture * focal ratio * 25.4 / eyepiece focal length
magnification = telescope aperture * focal ratio * 25.4 / (focal ratio * 7mm exit pupil)
magnification = telescope aperture * 25.4 / 7mm exit pupil
magnification per inch of aperture = 25.4 / 7mm exit pupil = 3.6
If we lower our 3.6x per inch of aperture magnification by 1/3, the telescope's exit pupil increases from 7mm to 9mm. The aperture actually used by the eye decreases by 1/3. Our field of view increases in width by 1/3. It's as if we started with a telescope of 1/3 less aperture and used the 3.6x per inch of aperture magnification for that lowered aperture. So, we get the richest field possible for that lowered aperture. If we have a 9 inch telescope but instead of using it at 32x, we use it at 22x, then we get an effective aperture of 6 inches, and the corresponding wider field of view of a 6 inch rich field telescope. It's like having two richest field scopes in one! We cannot stretch these lowered magnifications and increasing exit pupils too far because our Newtonian diagonal shadow eventually becomes too large and because the physical size of the eyepieces, in order to maintain the wide apparent field of view, becomes too large and heavy and expensive.
Here are the eyepiece focal lengths for different focal ratios and exit pupils. At the faster focal ratios, Newtonians need coma correctors. The TeleVue coma corrector multiples the focal ratio by 15%. Using this corrector alters the values.
Planetary viewing: my 6" f/4 will resolve the Galilean moons at 300x. I did spend close to two months working on the mirror, summoning all my skill to figure the mirror as close to perfect as possible. It is possible but difficult to make a f/4 mirror perform as well as a f/6. All factors such as collimation and focusing accuracy are all that much tighter with a f/4 compared to a f/6.
What about those large diagonals that fast wide angle telescopes call for? First of all, slightly larger diagonal sizes do not significantly impact image quality, rumors to the contrary. Also, at f/4 and particularly at f/3, the illumination fall off, that is, how much does the light dim as you look towards the edge of the field, is very gradual. So, a diagonal size that fully illuminates only the center of the field will do nicely. My 13 inch f/3.0 RFT uses the same diagonal size as Coulter's original 13 inch f/4.5. An 18 inch scope calls for a diagonal to eyepiece distance of 12", and if the mirror happens to be a f/4, then a 3" diagonal will work fine. This is a diagonal to primary ratio of 6:1. At this ratio, image degradation due to the diagonal obstruction is practically insignificant.
You've convinced me about stars, how about nebulae? I understand that the greatest number of stars are seen with the largest exit pupil that my eye opens to. And that the star counts don't vary that much with common apertures. But I'm uncertain how best to view nebulae: lowest power, medium power, high power; more aperture, less aperture? We need satisfy two conditions: 1) that the nebula be recognizable and 2) that it be detectable.
For the nebula to be recognizable it has to fit substantially within the field of view. Consider perhaps the grandest nebula of all, our Milky Way Galaxy. Imagine looking skyward on a summer night from atop a ladder in field with distant horizons. Looking up at our galaxy overhead we imagine ourselves floating in space 30,000 light years from the monster black hole that dwells in its center. We imagine the unimaginably immense galaxy stretching out in front of us. Though we cannot sense it, we are spinning around the black hole at 150 miles per second. We can see the central hub peaking up and the giant spiral arms twisting in front of us. At a particular time in the spring, we can see the entire galaxy rim the horizon. It's impossible to make these wonderful observations using a high powered telescope pointed at a tiny fragment of our galaxy. So the nebula has to fit in the field of view to be distinguished.
How about showcase objects as examples? The Sagittarius Star Cloud is about 2x1 degrees in size. A scope with a 2 degree field of view can be up to 12 inches aperture with a 100 degree eyepiece at F/3, 8 inches aperture with a 70 degree eyepiece at F/6. Larger scopes simply cannot frame the entire star cloud.
The North American Nebula's Gulf of Mexico and immediately surrounding area is about a degree in size. With the popular F/4.5 size and a 82 degree eyepiece, aperture can be as great as 18 inches yet still frame the object.
The Horsehead Nebula is about 8x6 arcminutes in size but needs the nebulosity that it's embedded in to be framed, so let's say something closer to a half degree. An aperture close to 50 inches looks to be ideal. That we can see it in scopes as small as 5-6 inches and that we can see it nicely in 20+ inch scopes is a testament to our eyes. Before we pat ourselves too much on the back let's analyze the smallest aperture that we successfully detected the Horsehead. 5-6 inch aperture at 6mm exit pupil with a 70 degree eyepiece yields an apparent size close to 3 degrees, which matches our three degree apparent size rule from above!
The Milky Way is full of bright and dark nebulae of all sizes. Telescopes of all apertures will give exciting views if they are used at large exit pupils and widest angle eyepieces. If aperture is smaller then the field of view will contain many Messier and NGC objects: the point being to observe the panorama taking in the surrounding nebulosity and stars. If the aperture is larger then the field may contain one or two objects: the point being to observe detail.
Finally, many experienced observers think of a threshold aperture close to 12 inches that resolves most globular clusters even at low power into sprinkles of stars.
I built my 13.2 inch [34cm] F3 telescope because I wanted to explore F3 telescopes with the potential of building a much larger telescope that would not require a ladder to reach the eyepiece. I did not expect the rich and detailed views of low contrast wide angle objects. It is something to resolve two, even three NGC globular clusters in the same field of view into light sprinkles of star dust interwoven with the Milky Way's flowing tendrils of dark and light. Before tackling a much larger mirror, I decided to make two more smaller mirrors, a 6 inch [15cm] F2.8 and a 10.5 inch  F2.7 to explore the sub-F3 world.
The very first object I zeroed in on with the new 6 inch was the Pleiades. I was astonished to see an irregular bubble of faint nebulosity surrounding the Pleiades, visible by virtue of the 4+ degree field of view and the 6 inches of aperture. Later it took me some time to find a deep digital image that showed the bubble as faint extensions of the Milky Way. Wow!
What makes F3 and faster scopes so special? It has to do with the latest in eyepiece and coma corrector design.
Currently the widest apparent angle eyepieces on the market are TeleVue and Explore Scientific eyepieces with 100 degrees apparent field of view. The lowest power eyepiece in the TeleVue Ethos set is the 21mm. In the Explore Scientific 100 deg line it is the 25mm. The focal ratio that gives the widest field of view using this eyepiece is about F3 (assuming an eye that opens to 6mm exit pupil and a coma corrector that increases the focal ratio by 15%). This creates a huge "WOW" factor when viewing.
The image is expanded over a wider field resulting in more detail. There is larger aperture for the field of view meaning brighter stars and nebulae and more resolution.
Apparently not all focal ratios are created equally.
It's unlikely that monster 100 degree eyepieces will ever be commonly available in significantly longer focal lengths. Imagine such an eyepiece: it would be a foot long, several inches wide and weigh 20 pounds not to mention costing thousands of dollars. As the following table and images illustrate, there is a major difference in viewable field area between widest fields of view at various focal ratios.
Compare the Richest Field Telescope field of view and the focal ratio.
The old rule of thumb that telescopes with a range of focal ratios can achieve Richest Field performance as long as a suitably matched eyepiece is used is no longer valid. Eyepieces of extreme apparent field of view are now available, but only in shorter focal lengths.
Here's a table showing how eyepiece apparent field of view and focal length impact the RFT experience for varying focal ratios.
Table generated for aperture = 13 inches, exit pupil = 6mm. The telescope focal ratios optimized for several popular eyepieces.
|Telescope F.R.||Eyepiece||Coma corrector X||Eyepiece focal length mm||Apparent FOV deg||Telescope focal length inches||Eyepiece field stop mm||Actual FOV from field stop deg||Actual FOV from field stop + coma corrector deg||FOV area deg^2||Mag|
Notes on derivation:
Most columns are published values from the manufacturer.The "Coma corrector X" is the magnification factor built into the coma corrector.
The exit pupil is the eyepiece's focal length divided by the focal ratio, further divided by the coma corrector magnification factor.
The "Actual FOV from Field Stop deg" is given by the formula: field stop in inches / focal length in inches * 57.3
There are three keys that work in concert:1. Shorter eyepieces allow faster scopes to maintain 6mm exit pupil.
2. Wider apparent fields of eyepieces allow shorter eyepieces to achieve the same field stop as longer focal length eyepieces.
3. Since the field stops are essentially the same, the faster focal ratio results in a shorter telescope focal length which results in a larger field.
Here are the widest fields possible (each at 6mm exit pupil) for the above focal ratios through 13 inches aperture observing M31 (image from Stellarium):
F/2.5 - F/3.6
2.5 square deg field
100 deg Ethos/ES
F/3.8 - F/5.2
1.5 square deg field
82 deg Nagler
1.1 square deg field
70 deg wide field
Another interesting way to look at it is to calculate the maximum aperture possible for different focal ratios given a field of view. The focal ratios are optimized for widest angle eyepieces.
field of view = 1.8 deg, exit pupil = 6mm
|Telescope F.R.||Eyepiece||Eyepiece Focal Length mm||Apparent FOV deg||Eyepiece Field Stop mm||Coma corrector X||Max Mirror Diameter|
Going down to f/3.6, f/3.0 or f/2.5 means jumping up in aperture from 10 inches to 13 inches. In other words, what we could see previously with 8 inch scopes and wide angle Erfle eyepieces in the 1960's to 1990's and with 10 inch scopes equipped with Naglers in the 1990's and 2000's is now seen with 13 inches aperture. This increase in aperture increases the limiting magnitude by a whole number.
Formula is: mirror diameter = eyepiece field stop * exit pupil * 57.3 / (field of view * eyepiece focal length * 25.4)
From: field of view = eyepiece field stop / telescope focal length; focal length = focal ratio * mirror diameter; eyepiece focal length / exit pupil = focal ratio
A New Relationship: Maximum Aperture or Field of View Based on Varying Focal Ratio While Holding Exit Pupil Constant
The impact of greater aperture for the same field of view is dramatic. Objects like the Horsehead, barely detectable in an 8 inch, are readily detectable in a 13 inch. Otherwise invisible galaxies pop into view. Instead of a faint object or two, many objects are visible at once.
Notice how the field of view is darker at higher magnifications? You can see more detail not only because the image is magnified but also because the background is darker. Do you ever stop the aperture down in order to darken the field at the given magnification? No?
Let's say you are observing Orion's Great Nebula, M42. And let's say that the magnification is set at say 100x. Would you choose a 6 inch scope with a darker background at this 100x or a 12 inch scope with a brighter background at 100x? Choose the 12 inch?
Then doesn't it follow that you should settle on a single eyepiece and a progression of apertures, say 6 inch, 12 inch and 24 inch?
The only question then is what is the optimum exit pupil which sets the background brightness in the field of view. Next time you observe with a Richest Field Telescope, see if you can experiment with stopping it down to a smaller aperture which gives you a smaller exit pupil. What exit pupil is best?
Now decide on a single eyepiece taking into account apparent field of view, eye relief, quality and price. Next settle on a focal ratio for your progression of telescopes. Now go out and buy or make at least three scopes to fill out the progression.
Observing an object then consists of setting up all three scopes, jumping back and forth between them based on optimal magnification and field size. Too onerous? Then consider a radical new telescope design where there are several apertures side by side, perhaps sharing a common focuser to be used with your eyepiece of choice.
I call this "one eyepiece - many telescope" or "eyepiece mode" as opposed to the universal "many eyepieces - one telescope" or "telescope mode". Provocative? My three RFTs are easy to setup, all three being table top telescopes. Jumping between them while aimed at a single object is a compelling experience.
Sky and Telescope magazine, March 1980, by Glenn Shaw, pages 192-4
Amateur Telescope Making Book Two, by Ingalls, pages 623-630.
ATM III, pages 389-417, by Ingalls, publisher Willman-Bell