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ATM: My Classical Cassegrain


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My Classical Cassegrain Telescope Designs
My Classical Cassegrain Telescope
By: Jeffrey D. Beish

The first and last Classical Cassegrain telescope I made was a 12.5-inch f/4 – f/16 system that had a 3.5-inch secondary. Before I learned to do the calculations my optician figured the parameters out and he made some errors that rendered the secondary mirror too large for me. He chose a back focus that was too far back and missed an important variable in the equations for sizing the secondary mirror. The secondary mirror will be a little larger if the back focus distance is too long and the final linear image diameter will be very large as well. Later the system was redesigned as an f/4 – f/30 system with an effective focal length of 375 inches. This design provided a high performance and produced high contrast images that I desired for observing planets.

The first tube was a heavy fiberglass tube but proved to hold heat and tube currents rendered the telescope useless for hours. I replaced it with a 16-inch I.D. by 53" long thin wall (0.060) aluminum tube that was rolled and welded by a local business. To support the secondary end of the tube a local aircraft salvage yard supplied the precision-machined titanium ring. This ring also helps conform the tube and make centering of the spider and mirror cell very easy. Purchased at a local hardware store a sheet of 3/16th-inch cork was glued inside the entire tube then painted flat black. This also provides a rough finish to reduce light scatter.
 
 

Figure 1. 12.5-inch f/30 Classical Cassegrain LEFT: heavy fiberglass tube and RIGHT: with cork-lined, black aluminum tube.


With the telescope pointed straight up the focuser stands 3.5 feet (42 inches) from the ground and required a regular chair to observe with. The Park’s German equatorial mount, purchased back in the late 1970’s, had 1.5-inch chromium steel shafts with two bearing each axis to provide stability. It weighed approximately 150 pounds [Beish, 1999 and Beish, 2000]. The center of gravity is about 13 inches from the primary end. The 18" long saddle completely a wooden system reinforced by aluminum bands. The rack & pinion focuser, primary cell, secondary holder, spider, and tube counterweight set were purchased from Kenneth F. Novak & Co. A local optician made the optics.

Basic Design Criteria

I chose the Classical Cassegrain design with the parabaloid primary and convex hyperboloid secondary. Reading material on hand, such as Kenneth F. Novak’s Cassegrain Notes and Richard Berry’s first ever issue of Telescope Making Magazine (ATM) I learned the important mathematics of Cassegrain design. The first article in Fall Issue, 1978 of ATM was "Cassegrain Optical Systems," by: Dr. Richard A. Buchroeder and was all that was needed to get started (Novak, Berry and Buchroeder 1978).

So, with the requirements and mechanical design our local optician set out to make the mirrors and the parts were ordered or made. From the available documents the following mathematical treatment was found and calculation began:

Cassegrain Equations: p = (F + B) / (X + 1), p' = pX , B = p' – b, c = Dp / F + Bi/ FX

Where: p = primary focus intercept point, F = primary focal length, b = back focus, X = secondary magnification,
p' = secondary to Cassegrain focus, B = mirror separation, c = secondary diameter, D = primary diameter and
i = final image size, [Buchroeder, 1978]. If the secondary mirror is already made the final image can be determined
by: i = X(c F – Dp) / B
 
NOTE: The separation between primary and secondary mirrors in a Classical Cassegrain reflector is critical. However, a small difference can be tolerated [Cox and Sinnott, 1976] and is computed from:
+/- B = 0.00248FR4
where FR is the focal ratio of the primary and the separation can be of 45% closer (-B) together or
55% further apart (+B) is tolerated; if in centimeters then (B) = 0.063 FR4
The 12.5-inch primary was 2.125-inch thick and the distance from the primary face to the back plate surface of Novak’s Cassegrain mirror cell was found to be 4.625 inches. His Cassegrain style focusers were 4.25 inches high when fully racked in, therefore, the back focus (B) would be a minimum of 4.625" + 4.25" or 8.875". With a 3-inch focuser travel the back focus was set so that the focal plane would be near the mid-way point from fully racked in to out, so the back focus was set at 10 inches. This would accommodate the different eyepiece focal planes. The secondary magnification for an f/30 Cassegrain is 7.5x, so:

                                p = (F + B) / (X + 1) = (50 + 10) / (7.5 + 1) = 7.0588"

                               p' = pX = 7.0588 x 7.5 = 52.9412"

                                B = p' – b = 52.9412 – 10 = 42.9412" (+0.3942 or -0.2857),
                                      where 0.55 x 0.00248 (4)4 = 0.3942"  and 0.45 x 0.00248 (4)4 = 0.2857"

The first iteration for the final linear image diameter was set at 0.5" and then the secondary mirror diameter was calculated to be:

c = Dp / F + Bi/ FX = (12.5 x 7.0588) / 50 + (42.9412 x 0.5) / (50 x 7.5) = 1.7647 + 0.0573 = 1.822"

A manufactured secondary mirror closest to the calculated diameter is 1.83 inches, so the image diameter would work out to be:

i = X(c F – Dp) / B = 7.5 (1.83 x 50 – 12.5 x 7.0588) / 42.9412 = 0.57"

The effective focal length (efl) will be: f/30 x D or 30 x 12.5 = 375" that will produce an angular image of:

tan-1 i/efl = tan-1 (0.57 / 375) = 0.0871° or 313.6 seconds of arc.
 

Figure 2. Cut away diagram of final results for the 12.5-inch f/4 – f/30 system: a 14.6% obstruction from the
secondary mirror delivering 0.57-inch linear image field (313.6 arcsec) with a contrast factor (CF) of 3.93 : 1,
whereas a  CF of 5.25 : 1 is an unobstructed system.
Baffles and Glare Stops

Baffle tubes can be made from ordinary household plumbing fixtures, such as brass and PVC sink traps, that come in several sizes from 1.25-inch to 2-inches in diameter and lengths of 6 to 18 inches. While these make perfect Cassegrain baffles I found 1" aluminum tubing at a large hardware store and that worked out better for the baffle system. Ray tracing various tube diameters and lengths the positions for each glare stop can be determined so to prevent light reflecting from baffle wall down the tube to the focal plane. To calculate the length of the primary baffle tube:

Cassegrain Baffle and Glare Stop Equations: L = (WB + Wb – c b –Bi) / (c - i) and G = (Bi + Zi + c b - c Z ) / (B + B)

Where:  L = length of baffle from face of primary mirror, W = baffle I.D., B = mirror separation, b = back focus,
i = final image size, c = secondary diameter and then G = diameter of glare stop, Z = behind primary face
[Novak, 1978] [Buchroeder, 1978].
A Thus, with a 1" I.D. aluminum tube we can find the length of the primary baffle tube by:

                              L = (WB + Wb – c b –Bi) / (c - i)
                                 = (1 x 42.9412 + 1 x 10 – 1.83 x 10 - 42.9412 x 0.57) / (1.83 – 0.57)
                                 = 8.0672"

There are various materials to make glare stops. The author found that brass is an excellent choice. Aluminum is another excellent choice. Once a baffle tube has been selected it is a matter of machining the stop to fit inside the tube with the correct inside diameter opening or aperture. When they are properly positioned within the tube a light coating of flat black paint will help secure the stops in place within the tube.  Another glare stop can be positioned near the entry aperture of the focuser and up away from the eyepiece barrels. Some eyepieces have stops and some don’t so it is wise to add this final glare stop near the eyepiece entrance and Cassegrain focus.

                              G = (Bi + Zi + c b - c Z ) / (B + B)
                                 = (42.9412 x 0.57 + 5 x 0.57 + 1.83 x 10 – 1.84 x 5) / (42.9412 + 10)
                                 = 0.6890"

In the glare stop equation listed below the sign to the ‘Z’ term can be manipulated to calculate the diameter of each glare stop within the baffle tube. Term ‘Z’ is the distance from behind the primary face to the position of the stop and usually presented in literature to calculate the rear glare stop. Changing the sign to minus (-) would put the stop in front of the primary face or somewhere along the baffle tube towards the secondary.

And the glare stop half way up the baffle tube:

G1 = (42.9412 x 0.57 + (-4) x 0.57 + 1.83 x 10 – 1.84 x (-4)) / (42.9412 + 10) = 0.9032"

While not everyone has such machines available at home many ATMs or other hobbyists may help you with these simple projects or you may wish to take the job to a professional machinist. The holes should be cut or milled as smoothly as possible to avoid diffraction streaks or reflections.
 
 

Figure 3. Cut away diagram of typical Cassegrain baffle and glare stop system. Term ‘Z’ is the distance from the primary mirror face to the glare stop with in the baffle tube. See equations below for calculating the Cassegrain optical system and baffles.


STRENGTH AND STIFFNESS OF TELESCOPE TUBES

While aluminum is 39% heavier than fiberglass it is 2.86 times stronger. Therefore aluminum requires less material (Albrecht, 1989). The weight per inch of a 0.0625" thick, 16" I.D. aluminum tube is; p (r2 OD - r2 ID), where p is 3.14159, and r is the radius of the tube I.D., so:

p(8.06252 – 82 ) = 3.15386 square inches

With a density of 0.097 lb/in3 the weight per inch that is 0.097 x = 3.15386 or 0.3059 pounds per running inch. A 53-inch aluminum tube would then weigh 53" x 0.3059 or 16.2 pounds.

For a fiberglass tube to be as strong as aluminum it must be 2.86 times thicker or 0.179". Therefore, the weight per running inch of a 0.179" thick, 18" I.D. fiberglass tube is:

p(8.178752 – 82 ) = 9.08533 square inches

With a density of 0.07 lb/in3 the weight per inch is 0.07 x 9.08533 or 0.636 pounds per inch and the tube would then weigh 33.7 pounds. We can see that the fiberglass tube weighs twice as much as the aluminum tube does plus aluminum is nearly twice as stiff as fiberglass.

Fiberglass, plastic, Formica, and other insulating materials store heat and are slow to radiate this energy to the outside air. Metals, such as steel and aluminum, radiate and loose heat faster. Also, fiberglass tends to pass more Infrared Radiation (Sunlight) though to the glass and metal components and that tends prolong cooling.

REFERENCES

Albrecht, Richard E. (1989), "The Design of Telescope Structures - I," Sky and Telescope, Vol. 77, No. 1, pp. 97-101, January.

Beish, J.D. (1999), "Design a German Equatorial Mount for the Planetary Telescope," Amateur Astronomy, pp. 52., No. 21, Spring.

Beish, J.D. (2000), "Tubes for Reflecting Telescopes," Amateur Astronomy, pp. 26-27 , No. 25, Spring

Cox, Robert and R.W. Sinnott (1976), "On Focusing a Cassegrain," (REF: Roger N. Clark, Applied Optics, 1,266, May 1976), Sky and Telescope, Vol. 54, No. 4, pp. 293, October.

Buchroeder, Richard A., Dr. (1978), "Cassegrain Optical systems," Telescope Making, Vol. #1, Fall.

Novak, Kenneth (1978), Cassegrain Notes, Kenneth Novak & Co..



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