My Classical Cassegrain Telescope Designs
My Classical
Cassegrain Telescope
By: Jeffrey
D. Beish
The first and
last Classical Cassegrain telescope I made was a 12.5inch f/4 – f/16 system
that had a 3.5inch 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 16inch 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 precisionmachined 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 3/16thinch cork was
glued inside the entire tube then painted flat black. This also provides
a rough finish to reduce light scatter.
Figure 1. 12.5inch
f/30 Classical Cassegrain LEFT: heavy fiberglass tube and RIGHT: with corklined,
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 1.5inch 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 = 0.00248FR^{4}
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 FR^{4}
The 12.5inch primary
was 2.125inch 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 3inch focuser travel the back focus was set so that the focal plane
would be near the midway 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 12.5inch f/4 – f/30 system: a 14.6% obstruction
from the
secondary mirror delivering
0.57inch 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 1.25inch to 2inches 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:
G_{1}
= (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
(r^{2 }OD  r^{2 }ID), where p
is 3.14159, and r is the radius of the tube I.D., so:
p(8.0625^{2}
– 8^{2 }) = 3.15386 square
inches
With a density
of 0.097 lb/in^{3
}the
weight per inch that is 0.097 x = 3.15386 or 0.3059 pounds per running
inch. A 53inch 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(8.17875^{2
}–
8^{2 }) = 9.08533 square
inches
With a density
of 0.07 lb/in^{3 }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. 97101, 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. 2627 , 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..
