THERMAL SELF-COMPENSATING TELESCOPE TUBE DESIGN

I. THEORY

The design of the truss-tube scope is inherently suited to compensate for thermal expansion/contraction. The focal length of the scope can be held constant as the scope tube materials expand or contract if the elements of the optical tube assembly (OTA) are made of materials with a proper difference in the coefficient of thermal expansions. This is due to the geometry of the OTA truss.

The geometry of the truss, in its simplest form, is represented in Figure 1.
[A] represents the optical axis of the scope and is to be held constant through temperature variations.
[L] represents the length of the scope tube truss; and,
[B] represents the "half-base" of the truss triangle, typically ½ the mirror box diagonal dimension or the radius of the spider ring.

As the scope cools, the truss and its base both shorten as shown. If the truss and its base are made of the same material, the [A] dimension will shorten in proportion to the contraction of the other dimensions. However, if the coefficient of thermal expansions are significantly different and properly selected for the truss and its base, and the geometry of the truss is appropriate, as the scope cools the [A] dimension can be held constant. For this to occur, as the truss [L] shortens by a distance (b), the ½ base [B] must shorten by the distance (t).

II. DESIGN CONCEPT

For various truss/base ratios the corresponding CTE ratios (K/C) required for constant truss length is shown in Figure 2.
The K/C ratios for some common materials are given in Table 1 below. Common materials having relative coefficients of thermal expansion ranging up to 14 to 1 are available. Combinations of wood & PVC or steel and PVC or perhaps wood and steel can be used with proper design. For longer focal ratio scopes, a truss designed with two or more "stages" can be configured. This could be say, a base of PVC, an intermediate ring of plywood and a spider ring of PVC, or a plywood base and spider ring with an intermediate ring of PVC.

Figure 2

III. DESIGN ELEMENTS

Based on the forgoing, it can be seen that a simple truss tube can be constructed from common materials with the capacity to greatly reduce thermal expansion or contraction, maintaining a near constant focus of the scope as temperature changes.

A fully engineered design will take into account the expansion/contraction of radial dimensions as well as the axial dimensions. It will also account for other material properties and geometry of the design, including the CTE of the primary mirror.

For two common truss designs, a standard V truss and a design where the trusses cross eachother an X truss each with 4 connecting points, the "truss" and "base" geometries are given by the following equations as can be derived from the geometries shown in Figures 3 and 4.

Figure 3

Figure 4

 L2 = R12+ R22 + H2
  and,  

 H2 = L2 - R12- R22

 L2 = R12 + R22 + H2 - 2 * R1 * R2 
                        2(1/2)

 H2 = L2 - R12 - R22 + 2 * R1 * R2  
                        2(1/2)

Common design elements for a newtonian truss tube configuration are show in Figure 5 below. L₃ and L₄ on the optical path (red line), are radial "linear" elements. L₁ and L₂ are axial "linear" elements along the optical path. H₁ is a "geometric" element and is the primary element that sets the tube length.

L₁ is the depth of the mirror box from the mirror support to the truss connections,
L₂ is the depth of the spider ring from the truss connections to the diagonal,
L₃ is the distance from the diagonal to the focuser support, and
L₄ is the distance from the focuser support to the focal plane.
R₁ is ½ the mirror box diagonal,
R₂ is the radius of the spider ring at the truss connections,
T is the truss length, and
H₁ is the truss height derived from R₁, R₂, and T.

Figure 5

It must be noted that the actual focal length of the primary mirror changes as determined by its CTE, as if the focal length were a "physical" element of the scope with a CTE equal to the mirror material. While this is commonly seen as a design problem, as we shall see it can be a design asset (See Note 1). If the mirror has a very low CTE, the sum of the elements prior to the temperature change must be very close to the sum of the elements after the temperature change. For a mirror with a large CTE, the focal length will have a relatively large change and the sum of the elements along the optical path must have correspondingly large change.

IV. CALCULATIONS FOR COMMON DESIGN CONFIGURATIONS

Using a spreadsheet to facilitate the calculations, one can easily "try" a number of design geometries and material combinations. This can provide the basis for designing a practical telescope that will provide a nominally constant focus for the scope over a large temperature range. Refer to Figure 5 to locate the various OTA elements.

Table 2 is a calculation for a typical 10" f/4 newtonian, with plywood mirror box and spider ring, aluminum trusses and a Pyrex mirror for a 10 °F temperature drop.

TABLE 2		Delta temperature (°F) =	0	-10
Element	CTE10-6 °F	Description	Length	Length'
10" mirror	1.81	Primary mirror focal length >	40	39.99928
H1	- - - - - -	Length of truss tube	21.00	20.99708
R1	3.4	½ mirror box diagonal	8.50	8.49971
R2	3.4	½ spider ring diameter	8.50	8.49971
T1	13	V-truss length (calculated)	21.98	21.98175
L1	3.4	Primary cradle depth	9.00	8.99969
L2	3.4	Depth of spider cage to diagonal	3.00	2.99990
L3	3.4	Diagonal to focuser length	5.50	5.49981
L4	13	Focuser height	1.50	1.49981
		Focal path length @ delta T = 0		40.000000
		Focal path length @ delta T = -10		39.996287
		Mirror focal length @ delta T = -10		39.999276
		Focal length - Focal Path @ delta T = -10		-0.002989

Table 3 is a calculation for a 10" f/4 newtonian, with plywood mirror box and spider ring, iron trusses and a Pyrex mirror at a 10 °F temperature drop. Note that the overall thermal impact is just less than ½ that when using aluminum trusses.

TABLE 3		Delta temperature (°F) =	0	-10
Element	CTE 10-6 °F	Description	Length	Length'
10" mirror	1.81	Primary mirror focal length >	40	39.99928
H1	- - - - -	Length of truss tube	21.00	20.99862
R1	3.4	½ mirror box diagonal	8.50	8.49971
R2	3.4	½ spider ring diameter	8.50	8.49971
T1	6.3	V-truss length (calculated)	21.98	21.98322
L1	3.4	Primary cradle depth	9.00	8.99969
L2	3.4	Depth of spider cage to diagonal	3.00	2.99990
L3	3.4	Diagonal to focuser length	5.50	5.49981
L4	13	Focuser height	1.50	1.49981
		Focal path length @ delta T = 0		40.000000
		Focal path length @ delta T = -10		39.997829
		Mirror focal length @ delta T = -10		39.999276
		Focal length - Focal Path @ delta T = -10		-0.001447

Table 4 is a calculation for a 10" f/4 newtonian, with plywood mirror box and spider ring, iron trusses and a soda lime glass mirror. At a 10 °F temperature drop, the overall thermal impact for this combination is less than 1/15^th that of using aluminum trusses and a Pyrex mirror and less than 1/7^th that using steel trusses with a Pyrex mirror.

TABLE 4		Delta temperature (°F) =	0	-10
Element	CTE 10-6 °F	Description	Length	Length'
10" mirror	4.95	soda lime glass mirror focal length >	40	39.99802
H1	- - - - - - -	Length of truss tube	21.00	20.99862
R1	3.4	½ mirror box diagonal	8.50	8.49971
R2	3.4	½ spider ring diameter	8.50	8.49971
T1	6.3	V-truss length (calculated)	21.98	21.98322
L1	3.4	Primary cradle depth	9.00	8.99969
L2	3.4	Depth of spider cage to diagonal	3.00	2.99990
L3	3.4	Diagonal to focuser length	5.50	5.49981
L4	13	Focuser height	1.50	1.49981
		Focal path length @ delta T = 0		40.000000
		Focal path length @ delta T = -10		39.997829
		Mirror focal length @ delta T = -10		39.998020
		Focal length - Focal Path @ delta T = -10		-0.000191

Table 5 is a calculation for a 10" f/4 newtonian, with plywood mirror box and spider ring, wooden trusses and a Pyrex mirror, a 10 °F temperature drop produces a thermal impact 3 times that of using iron trusses and a soda lime glass mirror as in Table 4 above and about 1/5 that of the standard plywood, Al, Pyrex design in Table 2.

TABLE 5		Delta temperature (°F) =	0	-10
Element	CTE 10-6 °F	Description	Length	Length'
10" mirror	1.81	Pyrex mirror focal length >	40	39.99928
H1	- -- - -	Length of truss tube	21.00	20.99949
R1	3.4	½ mirror box diagonal	8.50	8.49971
R2	3.4	½ spider ring diameter	8.50	8.49971
T1	2.5	V-truss length (calculated)	21.98	21.98406
L1	3.4	Primary cradle depth	9.00	8.99969
L2	3.4	Depth of spider cage to diagonal	3.00	2.99990
L3	3.4	Diagonal to focuser length	5.50	5.49981
L4	13	Focuser height	1.50	1.49981
	Focal path length @ delta T = 0			40.00000
	Focal path length @ delta T = -10			39.99870
	Mirror focal length @ delta T = -10			39.99928
	Focal length - Focal Path @ delta T = -25			-0.000433

From the forgoing it is seen that a significant improvement in the thermal performance of a standard truss tube scope can be gained simple by using wood or iron for the truss material rather than aluminum. A further gain is made by using a soda lime glass mirror with iron trusses rather than using a Pyrex mirror with "superior" thermal properties.

V. FOCUS RANGE TOLERANCE

To assess the actual benefit of these gains in thermal performance upon maintaining a focused image, one needs to investigate the relationship of the focal path changes to the actual focus depth of an optical system.

Rodger W. Gordon and Chris Lord, Brayebrook Observatory, in a paper titled "Depth of Focus" (Reference 1), provide a detailed analysis to determine the range over which an optical system remains in focus. In short they reference the late Robert E. Cox's presentation at the 1963 Astronomical League's 17th Annual Convention, held in Orono, Maine. Mr. Cox gives the depth of focus in inches as:
Focal Range = 0.000088 (Focal Ratio)² , (eq. 1).

Gordon and Lord extend Cox's work, and confirm that "to a very high degree of precision depth of focus is proportional to the square of the f/ratio." (Ibid; Depth of Focus, Appendix). They however, refine the analysis to determine that the depth of focus is given more precisely as:
Focal Range @ ^n 8 (Focal Ratio)² , (eq. 2).

The term ^n, in eq. 2 Gordon and Lord define as the "defocusing aberration and may be defined in terms of the acceptable tolerance of the wavefront" (Ibid). Taking the wavelength to be 0.000022 inch, and ^n as ½ , eq. 1 and eq. 2 are the same.

Gordon and Lord note that "in the case of very high quality optics, where the wavefront error is itself not more than 1/10, the tolerable depth of focus is correspondingly reduced" (Ibid). Taking the lead provided by Gordon and Lord, but recognizing that "seeing" is nearly always the limiting optical element, a ^n of 1/5 can be selected as a nominal limit for an high quality optical system. This is a wavefront tolerance of 1/10^th and is 2.5 times more stringent than the focal depth allowed by Cox, i.e. 1/4^th wavelength.

In terms of our scope design, for an f/4 scope with high quality optics and excellent seeing, the allowable focal depth (1/10^th wavelength) is given as:
Focal Range = ^n 8 f/² = 1/5 x 0.000022 x 8 x 4² = 0.00056 inch.

In the examples above, for the materials combinations calculated in Table 4 (iron and soda lime glass ) and Table 5 (Pyrex and wood), the scope will remain within the focal range over a drop in temperature of 10 ^oF. The "common" aluminum/plywood/ Pyrex truss scope [Table 2] will remain within the focal tolerance for only a 2 ^oF temperature drop. Substituting iron tubing for aluminum [Table 3] will extend the capacity to stay within the focal range for a temperature drop of 4 ^oF.

Figure 6 provides graphic comparisons of the thermal defocus at a temperature drop of 10 of for the four materials combinations given above, each calculated for optical f/ ratios of f/3 through f/10.

Here it is evident that by using common materials rearranged to take advantage of their thermal properties, a scope design can maintain an accurate focus over a significant temperature drop. It is also evident that a design that uses materials that work against their thermal properties will have a very poor thermal response.

Figure 6

The forgoing calculations were made for construction of the truss tube scope using common materials. Except for the hardwood/plywood combination, none of the arrangements provide for the athermal action of the truss. Rather they show that significant thermal property gains are made simply by using materials that minimize the thermal effects, that is by using materials with CTEs close to the CTE of the mirror

VI. CALCULATIONS FOR OPTIMUM DESIGN CONFIGURATIONS

Now we will look at the effect of using material combinations and that will compliment the truss geometry with comparisons of various design configurations.

Base lines are provided using all Aluminum, all Iron and all Wood OTA elements. These three base lines isolate the thermal effect to that of the CTE of the materials, and eliminate any effect of the OTA geometry.

A further sub-base line is provided for the various material combinations used to demonstrate the effect of the truss geometery. A "straight" truss design is calculated where the truss is taken to run parallel to the optical axis. This gives a base line with which to assess the effect of the geometries of the V and X truss designs using the same material combinations.

The calculations made are for a 10 inch, f/6 geometry using a Pyrex mirror over a temperature range of 20 ^o F. (Note: the low CTE of the Pyrex mirror better demonstrates the thermal performance of the structure by limiting the corresponding shortening of the mirror's focal length. The results of the comparisons are shown in Figure 7 below. [Note: The first material listed is the frame (mirror box/spider ring), the second material is the truss and the truss configuration.]

Figure 7

There are several findings that are evident from this data. The most significant is that the use of any material with a smaller CTE improves the thermal stability of the structure. While this is an obvious conclusion, the premise that materials of significantly different CTEs can be used to make an "athermal" OTA argues against this conclusion.

This results, not from the materials used in the truss geometry, but from the quite large effect of the "linear" elements of the scope design, refer to Figure 5. These linear elements directly impact the overall thermal performance of the structure and are not offset by the truss geometry. These linear element effects dominate the performance for designs that accommodate any but the shortest f/ ratios. Thus, even for CTE ratios of base to truss as much as 12:1, the corresponding large CTE of the linear elements detract from the gains made by the truss/base geometry.

The objective of the exercise, which was to show that a truss geometry has the capacity to reduce the thermal effects on the scope, is demonstrated in the comparisons of the "straight" truss to the V and X truss calculations. A comparison of a steel "linear truss" to the V-truss and the X-truss designs shows a significant improvement in the thermal performance. Furthermore for a steel frame with a wood truss the calculations show the X-truss provides more than a 10% thermal range advantage over a "straight" truss at the focal range limit. It is also shown that the X-truss design is more effective than the V-truss design in stabilizing the thermal effect of the scope.

Calculations are made for a soda lime glass mirror using the same material combinations and geometries calculated in Figure 7 above. The results for the soda lime glass mirror are shown in Figure 8.

Figure 8

From the forgoing it is evident that there is a great variation in the thermal performance of a scope depending on the materials and geometries of the design. Using this to our advantage it is possible to configure a scope that will have a very flat thermal response. However, as shown the linear elements have a great impact on the performance of any specific design. Therefor it is not possible to provide any more than a few examples to demonstrate this. Each scope design must be configured to its own geometry and peculiar materials.

With a little imagination and clever use of materials, one can find a combination of materials and geometry that will produce a thermally stable telescope having a soda lime glass mirror. As an example, using a "composite" truss with ½ of its length made of hardwood and the remaining ½ of iron, a truss with an effective CTE of 4.5 is obtained. Using this truss in the V-Truss configuration with an iron frame and a soda lime glass mirror, this design will maintain focus over a very wide thermal range, (see the plot for "Fe/Hrdwd optimized" Figure 8 above).

Since the CTE of Pyrex is less than the commonly available design materials, it is more difficult when using Pyrex to provide a scope design with an overall athermal performance than when using a soda lime glass mirror. However, it is possible to design an athermal telescope that uses a Pyrex mirror. Since the linear elements are all made of materials with CTE greater than Pyrex ( 1.8 ppm/ ^o F), the advantage of the truss design must be enhanced by decreasing the length to width ratio of the truss assembly and utilizing materials with low CTEs for the linear elements where possible.

Chuck Shaw (see http://www.ghg.net/cshaw/truss.htm ) uses a design that is ideal for this purpose. His double truss design uses the more effective X-truss and decreases the length to width ratio of the truss assembly(ies) by nearly ½ of the conventional truss design. [Note: Chuck uses a wooden base for the aluminum truss assemble. This is great geometry but just the opposite material combination needed to minimize thermal focus drift. Sorry Chuck.]

Furthermore, the design parameters must be carefully calculated. One needs to make a variety of calculations using various geometric parameters and material combinations to find a suitable design for the particular scope being constructed. A spreadsheet provides a useful tool for this purpose. The spreadsheet used by the author is available for download at :
XL Spreadsheet
Quattro v6 Spreadsheet

As a note of caution in designing a thermally stable telescope, it is necessary to build a mechanically stable OTA as well. Again the distances involved in maintaining critical focus are very small. The scope structure can flex enough to obscure any advantage gained by the forgoing, unless careful attention is paid to the structural design and construction techniques to ensure a very stable OTA.

So much for theory, now it is time to build the thing. With the need to accurately know the CTE of materials used and the requirement to construct a very stable structure, it will be a challenge to make this work. Stay tuned.

Joe Garlitz
jgarlitz@oregonvos.net
http://www.oregonvos.net/~jgarlitz/
Elgin, Oregon USA Jewel of the Blue Mountains

VIII REFERENCES/NOTES

Reference 1:
Depth of Focus - by Rodger W. Gordon & Chris Lord
http://www.brayebrookobservatory.org/BrayObsWebSite/HOMEPAGE/forum/Depth-of-Focus_html
http://www.brayebrookobservatory.org/BrayObsWebSite/HOMEPAGE/forum/depthoffocus.pdf

Note 1:
"If you athermalize the truss to a true zero CTE...then you are
forced to spend money or effort to find zero CTE glass for the mirror.
(That's why I'm thinking of using steel and plate glass...CTE is close
enough that over about a 5C temp shift...I can probably stay within
an acceptable focus tolerance. That's not a true, athermal truss...but it
may be good enough for many applications.)" Tom Krajci

Note 2 Definitions:
CTE = Coefficient of Thermal Expansion: defined in terms of length of expansion per degree of temperature per length of material; i.e. inch/°F/inch, typically expressed as millionth of inch/inch/°F in the english system of measure.

PPM = Parts Per Million, thus CTE expressed as ppm/°F.

OTA = Optical Tube Assembly, defined as the mechanical supporting structure of the telescope optical elements.

Soda Lime Glass = Plate glass with a high CTE used in some telescopes for optical mirrors.

Pyrex = A low CTE glass product commonly used for telescope optical mirrors.

Note 3 CTE of glass products:

Glass -	CTE ppm/°F
Plate/Crown	4.95
flint	4.40
7740 Pyrex	1.81
Fused quarts	0.28
Corning 1737	2.05
Corning fused silica	0.28