Rich Field Telescopes: Design Considerations

The Rich Field Telescope or Richest Field Telescope or RFT has a long history. The idea behind the RFT is "how low of a power can I profitably use with a given objective". The old Amateur Telescope Making series in Scientific American (edited by Albert Ingalls) had many articles on the RFT. It later became the series of three "Amateur Telescope Making" books.

There are many uses for an RFT or an approximation. One use is just what the name suggests: wide-field views that are rich with stars. The Pleiades with a six-inch scope at 22x is an impressive sight. Another use is to maximize the brightness of extended objects: this is especially important when using narrow-band nebula filters, which I do much of the time.

The RFT is a telescope well-suited to dark sites and less-suited to light-polluted city locations, because many RFT target objects are of low surface brightness and a bright sky background will "wash out" such objects. (The Pleiades at 22x looks good most anywhere.) I discuss the sorts of observing that is best-suited for RFTs in a companion article at this site and I discuss the particular merits of RFTs with nebula filters in another article that also lists some of my favorite RFT targets.

An interesting question is "How rich can a rich-field telescope be?" What are the limits on either maximizing the number of stars visible in the field (the main concern of Amateur Telescope Making) or of maximizing the apparent surface brightness of an extended object like the Veil Nebula? That is the primary concern of this article.

Although conceptually a RFT can be of any aperture, in normal usage, an RFT is relatively small. There is a "lowest" power for a 30-inch telescope (about 110x) but such a telescope, with a field-width of less than a degree, is not normally considered a RFT.

I have recently built two Rich Field Telescopes: a six inch folded refractor and a eight-inch or four-inch folded refractor with flip-mirror switching.

For a somewhat different view of the Rich Field Telescope than this one, see Mel Bartels RFT page.

Fitting the Light into your Eye

The goal of an RFT is the lowest power for the widest field, and the lowest power for the brightest view of non-point sources.

What produces a lower limit on the power? The exit pupil. The exit pupil is the bundle of light that leaves the eyepiece, headed for your eye. If it does not get into your eye, through the pupil, light is wasted. Thus your pupil determines the lower limit on magnification. The standard nominal value for pupil diameter in the dark is 7mm. It decreases a bit with age; I measured 6.5mm at age sixty. When your eye-doctor dilates your eyes, your pupils can grow to 9mm or even 10mm diameter -- but this isn't useful, because your eye's lens isn't that big.

All that a too-big exit pupil does is waste light. If you have a four-inch aperture 20-inch focal length objective and you use a two-inch focal length eyepiece, you will get 10 power and a 10mm exit pupil. What bad is happening? If your pupils are only 7mm, then you are only using the inner 7/10 part (linear) of the light. Half of the incoming light (1 - 49/100) is missing your pupil and thus you are wasting half of your four-inch telescope's light-gathering. This may be OK -- maybe you really want 10x and don't have a smaller telescope. But you wouldn't plan for this, because bigger aperture is a major cost.

Along with wanting to fill your pupil, you want to use all of the pupil. Your pupil is a circular disc. You not only want to match the diameter, but the shape. Fortunately, most telescope apertures are round. Unfortunately, many have holes in the middle: secondary obstruction. From the viewpoint of filling your pupil, the "hole in the middle" is a pure waste that cannot be recovered. (Further, the center of your pupil has the best acuity, so you are also losing eye resolution.)

Thus if you want to max-out the RFT, you need an exit-pupil that matches your eye and does not have a hole in it. This could be a refractor or an off-axis reflector. We will later see that we would like a short focal-ratio, on the order of f/6 or so. That makes an off-axis reflector exceedingly expensive.

The above discussion is the critical one: decide what field of view you want or what objective aperture you can afford, and then create a system that gives you a 7mm (or your measured pupil-size) exit pupil: exit pupil diameter = objective diameter / power, or alternatively exit pupil diameter = eyepiece focal length / primary f-ratio

More On Refractor vs Reflector

The reader might think "Come-on, in the case of a reflector we are only talking about a small donut hole in the exit pupil here. How much does it really affect the throughput?"

Indeed the design of the optimal RFT might be considered a bit of "overkill". But the remarks above are real. Let me expand them a bit:

The central obstruction of a reflector, as I noted, is not something that can be recovered. It is a pure loss. How big a loss? Well, reflectors certainly exist with a central obstruction of about 0.25 linear, or 1/16th of the area. But remember we are talking about a short focal ratio and wide field telescope. That changes the story about a central obstruction: both the wide field and the shorter f-ratio require a larger secondary.

We would like an optical system with coverage of the field-stop of a two-inch eyepiece without vignetting. Take as a base-case an eight-inch f/6 Newtonian reflector with an eyepiece with a 46mm field-stop. Assume a telescope tube with a radius one inch larger than the mirror (to allow for tube currents) and assume that your eyepiece is five inches away from the tube (room for focusser and for nebula filters, perhaps in a filter-slide). If you perform a required-diagonal calculation (or ray-trace) for this short Newtonian, you will see that a non-vignetting Newtonian diagonal needs to be 79mm minor-axis, 82mm or so with holder. That is a 40% linear obstruction, 16% light-loss.

Mel Bartels has a useful diagonal-size calculator at his website. He notes that one should accept a 0.3 magnitude light-loss at the edges of the field. 0.3 magnitude is a factor of 1.32, so that means your eye receiving 100 units of light rather than 132 units. His analysis is quite reasonable for a general-purpose reflector, but again, we are talking about a RFT here.

The obstruction is worse for a faster system because the "cone of light" is fatter. The obstruction is worse for a smaller-aperture system because the room for the focusser does not scale with the aperture. Perform the same calculation for a 6-inch f/5 telescope and you will get a 28% light loss. Inverting this, a six-inch refractor will have 39% more throughput than the reflector.

In practice, Newtonian telescopes accept considerable field-edge vignetting in order to make the obstruction smaller, but this of course hurts the RFT-ness, and we are discussing RFTs here.

Also, a reflector will have at least two reflecting surfaces. Reflecting surfaces are less efficient, today, than multi-coated transmissive surfaces. So the reflections will be an additional loss. This can be ameliorated with special coatings like "enhanced silver" -- but such a coating is expensive (and fragile) on a primary mirror.

So a fast refractor will be 20-40% more efficient, overall, than the corresponding reflector with a non-vignetting Newtonian diagonal. And remember, one cannot say "Well, I can make that up by increasing the aperture a bit" because you are limited by the eye-pupil diameter.

One final facet of a reflector with a central obstruction is that the annulus-shaped exit pupil seems to makes it more difficult to keep the exit pupil aligned with your eye's pupil -- the "I am off center" visual feedback seems weakened. This is especially the case if you are using a super-wide-field eyepiece where you cannot see the field-stop.

Buying Eyepieces -- and Focal Ratio

The above is really the only crucial calculation: Use all your pupil. But there are other practical concerns.

One is eyepieces and focussers. Recall the second formulation about exit pupil: exit pupil = eyepiece focal length / objective f-ratio.

Thus if you have a f/15 refractor and you want a 7mm exit pupil, you need to use an eyepiece of 105mm focal length. Such an eyepiece is not practical. For it to have a decent apparent field of view, it would have to be four inches in diameter. We are talking bulk, weight, cost.

If we would like the 68 degree apparent field of a TeleVue Panoptic, the longest focal length available is 41mm, due to the limitations of a two-inch barrel: for a 7mm exit pupil this means we need an f-ratio of six or under.

The Mel Bartels RFT page, mentioned earlier, notes the benefits of a super-wide eyepiece. Indeed if the goal is maximum nomber of stars visible in the apparent field, then an increased field (at the same exit pupil) is a clear win, since the "extra" field is just replacing blackness: this is assuming apparent field that you have to turn to see fully is valuable. The downside of the extra wide field is that for a 100-degree apparent-field eyepiece that fits in a two-inch focusser, the maximum possible focal length is about 27mm and the maximum for sale focal length is 21mm; with a 21mm Ethos eyepiece you need a primary f-ratio of three or under to get a 7mm exit pupil.

It is difficult to create a refractor with a short focal ratio that has good optical quality -- "chromatic aberration" is a particular problem. And "apo" refractors, which are much better than doublets at color-correction, are quite expensive. Fortunately, in our RFT we are going to be using the lowest feasible powers, so we are not stressing the image-forming capabilities of our objective, and a decent doublet will be satisfactory.

Geometry and Vignetting

In the preceeding section we discussed how our RFT goals interact with real-world considerations about eyepieces and eyepiece size.

Related to eyepiece size is the other elements of the path from objective to eyepiece -- in particular the focusser and the star-diagonal of a refractor.

We noted that for a 68 degree apparent field Panoptic, the longest focal length available is 41mm, because a longer focal length would require a field lens that would not fit in a two-inch barrel.

We can extend that geometric discussion to the rest of the system and in particular to the focusser and the star-diagonal. Suppose we have a f/6 system with a 41mm Panoptic eyepiece, giving us a nice 7mm exit pupil. The field-stop of the eyepiece is 46mm in diameter. If we ray-trace from the objective to the eyepiece, we see that the photons fill a frustrum of a cone, with the big end at the objective and the small end at the eyepiece. The cone expands as it goes from eyepiece to objective. If we have an eight-inch (203mm) f/6 objective, then three inches from the eyepiece field-stop, the light-cone will have expanded from 46mm to 56mm diameter. The optical path-length through a two-inch star-diagonal is about three inches. So the clear aperture of a star-diagonal needs to be about 56mm to avoid any vignetting. That is well over two inches: a "two-inch" diagonal will vignette the edges of the field with this eyepiece and objective. The situation will be similar if you use a focusser with a two-inch drawtube. There is likely three inches of focusser-tube beyond the eyepiece. It will vignette like the star-diagonal. One of my RFTs has a 2.7-inch diameter focusser, the other has a three-inch focusser.

It is easy to draw a simple sketch of the converging beam, from objective to eyepiece, and see how big a diagonal you would need. Or, turning it around, how much a two-inch diagonal limits the fully-illuminated field at the eyepiece and hence the "richness" of the field of view.

Since that 41mm Panoptic will vignette with a two-inch diagonal, a 35mm focal length Panoptic eyepiece with a smaller 39mm field stop is a better choice if you use a two-inch diagonal. With that eyepiece, your objective focal ratio needs to be f/6 for a 6mm exit pupil, f/5 for a 7mm exit pupil.

Narrowing It Down

We have narrowed our space of "ultimate" RFT systems to refractors of f/6 or shorter: slower than that and we require an unreasonable eyepiece.

Since the shorter the refractor the more expensive and the harder it is to control aberrations, that effectively means a refractor between f/4 and f/6.

Our aperture should be greater than 50-70mm (below that we can easily use binoculars). We are limited at about 200-250mm at the high end due to unavailability and cost.

Thus our ultimate RFT will be a refractor of between 80 and 200mm aperture, f/4 to f/6, with the appropriate eyepiece to create a 7mm or 6.5mm or 6mm exit pupil.

Actual Examples

Given the above, you might expect that my RFT refractors would fit the constraints listed above. Indeed they do. They are a six inch f/5 and a flip-mirror switching RFT that is both a eight inch f/6 and a four inch f/6.

Two Pupils May Be Better Than One

One way you can beat the "size of your pupil" limitation is to realize that you have two of them.

Hence the appeal of large binoculars and binocular-telescopes.

How much you gain by using two eyes varies from person to person. Experiment before you go this expensive route. It is easy to experiment: just use astronomical binoculars and compare your observations using both eyes to your observations with one eye shut or one objective capped.

You can find a somewhat longer discussion of this at my page on binoculars for astronomy.

This is different from using a beam-splitter binocular eyepiece holder on a telescope. The latter cuts the overall brightness at each eye in half (really worse, considering losses) and so does not offer the improvement of binoculars.

 

Contacts

I would be happy to correspond about telescope design with interested individuals.

I can be reached via email at astroayers@gmail.com