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).

Given that,"C" represents the coefficient of thermal expansion (CTE) of the truss, and "K" represents the CTE of the base; then,
(t) = [B] * K * (T), and
(b) = [L] * C * (T).

Noting that the small triangle {a-b-t} is a "similar" triangle to {A-B-L}; then,
(t)/(b) = [L]/[B] = {[B] * K * (T)} / {[L] * C * (T)}. Thus,
K = C * {L2} / {B2}, or K/C = L2 / B2, for athermal performance of the truss.

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

 TABLE 1 - COMMON MATERIALS USED IN TRUSS TUBE CONSTRUCTION Material CTE 10-6 ºF Ratios of K/C {CTE(B)/CTE(L)} S wood H. wood Plywood Steel Copper Aluminum PVC Softwood (along grain) 2.1 1.0 1.3 1.6 3.0 4.5 6.2 13.8 Hardwood (along grain) 2.7 - 1.0 1.3 2.3 3.5 4.8 10.7 Plywood (parallel to face) 2.4-3.4 - - 1.0 1.9 2.8 3.8 8.5 Steel tube 6.3 - - - 1.0 1.5 2.1 4.6 Copper tube 9.4 - - - - 1.0 1.4 3.1 Aluminum tube 13.0 - - - - - 1.0 2.2 PVC pipe 29.0 - - - - - - 1.0 Truss Material CTE 10 -6 ºF Ratios of Truss length to Half-Base width (T/B) for various material combinations which will provide an athermal truss S. wood H. wood Plywood Steel Copper Aluminum PVC Softwood (along grain) 2.1 1.1 1.3 1.7 2.1 2.5 3.7 Hardwood (along grain) 2.7 - 1.1 1.5 1.9 2.2 3.3 Plywood (parallel to face) 2.4-3.4 - - 1.4 1.7 2.0 2.9 Steel tube 6.3 - - - 1.2 1.4 2.1 Copper tube 9.4 - - - - 1.2 1.8 Aluminum tube 13.0 - - - - - 1.5

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

 For 4 point "X" truss design: (Figure 3) ``` L2 = R12+ R22 + H2 and, H2 = L2 - R12- R22``` For 4 point "V" truss design: (Figure 4) ``` 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

Given that all elements along the optical path equal the mirror focal length when they are all at temperature t1 (i.e. the scope is in focus), then the OTA is taken as athermal if the sum of the elements along the optical path after a change in temperature to t2, is equal to the focal length of the primary mirror at temperature t2.

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 1/15th that of using aluminum trusses and a Pyrex mirror and less than 1/7th 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 1/10th and is 2.5 times more stringent than the focal depth allowed by Cox, i.e. 1/4th wavelength.

In terms of our scope design, for an f/4 scope with high quality optics and excellent seeing, the allowable focal depth (1/10th 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

These calculations need to be explained a bit. All of the configurations using hardwood trusses "over compensate" for thermal expansion. That is as the temperature decreases the focal length of the soda lime glass mirror shortens faster than the OTA shortens. Thus, these designs are too effective in compensating for thermal effects. On the other hand, the designs using iron trusses under compensate for the thermal effects. These results are easily understood because the CTE of soda lime glass is less than iron and greater than hardwood.

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.

Figure 9

By dividing the OTA into two trusses, the geometric effect is enhanced. The truss material must have a much smaller CTE than the base material to provide an athermal OTA. Unfortunately using materials with large CTE for the base works against the athermal effect, (as noted above) in the linear elements of L1, L2, L3 & L4, (referring to Figure 9). This can be reduced by using low CTE materials for the elements L1 (r1), L3 and L4 (r4) and using a high CTE material for r2 and r3 (L2). By keeping the high CTE element L2 (r2 r3) short its negative impact is reduced. Or, if the trusses are attached to L2 as shown in the inset, the high CTE of L2 actually works as a "negative" CTE and enhances the athermal performance of the design, allowing L2 to be used to adjust the overall athermal performance.

For an f/6 10" telescope with a double X-truss design, a Pyrex mirror, trusses of hardwood (CTE 2.6 ppm/o F), mirror box and spider ring of plywood (CTE 3.4 ppm/o F), and aluminum (CTE 13 ppm/o F) for the altitude support framing L2 (r2 & r3), a thermally stable scope is obtained. The result is shown in the plot for "X-truss optimized" in Figure 8 above.

Table 6 below shows the scope elements used for this design.
 TABLE 6 Relative T (°F) = 0 -25 Element CTE 10-6 Description Length Length' M 1.81 Pyrex mirror 60 59.9973 L1 3.4 primary cradle 1.00 0.9999 L2 13 Dec ring 3.00 2.9990 L3 3.4 diag cradle 3.00 2.9997 L4 3.4 diag/focuser 5.50 5.4995 L5 13 focus relief 1.00 0.9997 Sum (Li) 13.5000 13.4979 ------ ------ 4 point X truss ------- ------ H2 +++++ Design length 25.25 25.2494 R4 3.4 diag ½ axis 8.50 8.4993 R3 13 dec ½ axis 10.00 9.9968 T2 2.6 X-truss length 28.46 28.4554 ------ ------ 4 point X truss ------- ------ H1 +++++ Design length 21.25 21.2499 R2 13 dec ½ axis 10.00 9.9968 R1 3.4 prim ½ axis 8.50 8.4993 T1 2.6 X-truss length 24.98 24.9746 Focal path length @ delta T = 0 60.000000 Focal path length @ delta T = -25 59.997248 Mirror focal length @ delta T = -25 59.997285 Focal length - Focal Path @ delta T = -25 -0.000037

By increasing the elements r2 and r3 by 3 inches and substituting iron for aluminum for these elements, this same configuration will provide a thermally stable OTA. Table 7 below shows the design elements for this combination of materials.
 TABLE 7 Relative T (°F) = 0 -25 Element CTE 10-6 Description Length Length' M 1.81 Pyrex mirror 60 59.9973 L1 3.4 primary cradle 1.00 0.9999 L2 6.3 Dec ring 3.00 2.9995 L3 3.4 diag cradle 3.00 2.9997 L4 3.4 diag/focuser 5.50 5.4995 L5 13 focus relief 1.00 0.9997 Sum (Li) 13.5000 13.4984 ------ ------ 4 point X truss ------- ------ H2 +++++ Design length 25.25 25.2490 R4 3.4 diag ½ axis 8.50 8.4993 R3 6.3 dec ½ axis 13.00 12.9980 T2 2.6 X-truss length 29.64 29.6428 ------ ------ 4 point X truss ------- ------ H1 +++++ Design length 21.25 21.2494 R2 6.3 dec ½ axis 13.00 12.9980 R1 3.4 prim ½ axis 8.50 8.4993 T1 2.6 X-truss length 26.32 26.3196 Focal path length @ delta T = 0 60.000000 Focal path length @ delta T = -25 59.996852 Mirror focal length @ delta T = -25 59.997285 Focal length - Focal Path @ delta T = -25 -0.000433

VII CONCLUSION

Using either a Pyrex mirror or a soda lime glass mirror and common materials used in construction of amateur astronomical telescopes, ie. wood, plywood, aluminum and iron (steel), one can design a truss Optical Tube Assembly (OTA) that will greatly reduce or eliminate the effects of drift in focus due to changes in temperature. The flexibility in use of materials and design parameters is more than adequate to allow true athermal performance of a telescope.

A problem in designing such a scope is accurately determining the CTE of the chosen materials. The very small changes in length necessary to stay within the focus range of a telescope requires a good knowledge of the CTE of the materials. A change of a few tenths in the CTE (ppm oF) of a material will alter the capacity of the design to hold the OTA within the range of focus. If one searches the web for CTE information, it is soon obvious that materials have a very broad range of reported CTEs.

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 :

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