A Figure of Merit for Telescope Mirror Materials.

This is a talk given by Larry Shaper at Stellafane, Springfield VT in Aug 2013. Stellafane does not put the printed version of talks nor the videos of the talks on it's website. I liked Larry's talk and offered to host it on my site AustinTek, untill a more suitable site is found for Stellafane talks.

Joseph Mack, Oct 2013

The last time that I bought a disc of Borosilicate glass, it was about 7 times the price of the glass you get at the corner glass store, called plate, float, soda-lime, window glass or, the green tinted stuff. For seven times the price, what do you get? The first thing that comes to mind is a low coefficient of thermal expansion. So the question becomes: why is the coefficient of thermal expansion important?

When it comes to telescope mirrors, there are two different situations to consider:

Here is the table that will be used throughout the talk

As the night cools, the mirror gets smaller in every dimension. If the mirror cools evenly, which we should be able to assume because the night cools slowly, and the mirror is relatively uniform in thickness. In the usual case, the center is thinner, but there are no very thin projections along with relatively massive sections. If that assumption holds then all linear dimensions of the mirror will shrink by same proportion, which means that a paraboloid remains a paraboloid, and the only noticeable change to the observer is a shorting of the focal length.

Although I want to compare Borosilicate glass with the common Green glass, I have also listed Beryllium and Aluminum. Beryllium is there because it is the material from which the Webb Space Telescope mirrors have been made. Aluminum is there because it has the worst (greatest) coefficient of thermal expansion of any material that could be conceivably be used to make a telescope mirror.

The 1th row shows the coefficients of thermal expansion.

The 2th row shows the change in focal length assuming a 72" focal length, exposed to a 10 degree C drop in temperature (18 degrees F). For long-exposure astrophotography, especially in critical work, there would be an advantage to using a low thermal expansion mirror, like Borosilicate glass (or better yet, one of the exotic zero expansion materials not considered here), but for visual observers like me, adjusting the focus by 0.017 in. over the course of several hours is a very slight inconvenience. I am always fiddling with the focus and would not notice the effect at all. But, the drop in temperature effects the entire telescope, not just the mirror. If you have an Aluminum truss tube telescope, even if you have a zero coefficient of thermal expansion material, the truss tubes will shrink and the image will be out of focus. Actually, in that case, you would be better off with an Aluminum mirror. That way, everything would shrink together and the image would stay in perfect focus. Therefore, my conclusion is that, for observing, the low thermal expansion coefficient of Borosilicate glass is not an advantage and may even be a disadvantage.

There is another phenomenon affecting the choice of material for a telescope mirror, and that is the convection layer of warm air that forms above the mirror surface, as it cools during the night. We want the mirror to adjust as quickly as possible to the temperature of the ambient air. The usual way to reduce the cooling time is to make the mirror thinner. If the mirror is thinner, it contains less heat that needs to be released, and that will make its temperature adjust to ambient air more quickly. This makes for difficulties in figuring the mirror; I will discuss that later. The PLOP program, free software on the internet, tells you what kind of support system you need for the mirror. After playing with the program for a while, you will realize that you can make the mirror as thin as you want, if you are willing to construct an increasingly complicated cell. So the first step in determining the thickness of a proposed mirror is to decide how complicated a cell you are willing to construct. (My compromise is 9 points on a single circle of support.) Once that decision is made, the thickness, for a particular mirror diameter, is determined by the strength to weight ratio of the material.

The 3th row shows Young's Modulus of Rigidity for the materials.

The 4th row shows the density (specific gravity, SG) of the materials

The 5th row shows the effect of ν (Poisson's Ratio). Poisson's Ratio is the amount that a wire will shrink in diameter when it is tensioned along its length (cf a rubber band shrinks in cross section when you stretch it). The relevant number here is (1-ν²).

The 6th row gives the appropriate formula for the strength to weight ratio; it turns out that it is actually strength to weight squared that is proportional to the thickness of the mirror. Follwing is the ratio for the various materials. You can see that, in addition to having a lower thermal expansion coefficient, Borosilicate glass also has a better strength to weight ratio than Green glass, so it can be made thinner and therefore cool faster. However, Beryllium is in a class by itself, and that may be one of the reasons why the weight conscious folks as NASA chose it for the Webb Space Telescope.

There is another characteristic of materials which affects the amount of heat in a mirror; it is the specific heat. Specific heat is a measure of the amount of heat that you need to add to a material in order to raise (or lower) its temperature one degree C. A low specific heat is good to have.

The 7th row shows the specific heat of the materials, and you can see that, once again, Borosilicate glass comes out the best, if only by a little bit.

There is one more characteristic to consider, namely thermal conductivity. High thermal conductivity will insure that the assumption made earlier, that the mirror will cool evenly, holds true. In addition, high thermal conductivity will permit the heat inside the mirror to move easily to the surfaces where it can be conducted, convected, and radiated away.

The 8th row shows the thermal conductivity, κ, of the materials. Once again, Borosilicate has an advantage over Green glass. However, Beryllium and Aluminum are 200 and 100 times better than the glasses.

The 9th row gives the formula for a Time-to-Cool constant. All of the factors considered are linearly related to the time to cool. If the strength to weight ratio is twice as large, the thickness will be one half, and the time to cool will be one half. If the specific heat is twice as large, the heat contained will be twice as large, and the time to cool will be twice as long, et cetera. In the formula, the things that we want to be small are in the numerator, and the things that that we want to be large are in the denominator. Following the formula is the data for the time constant.

The 10th row shows the relative time to cool with Borosilicate at 100.

The 11th row shows a reasonable example with the time to cool for Borosilicate at 20 minutes. In that case, Green glass will take 13 minutes longer to cool. It is for the observer to determine if avoiding 13 minutes of waiting time is worth the difference in price for Borosilicate glass.

The results for Beryllium and Aluminum indicate that the problem of convection currents would be completely eliminated if either of these materials were used for the mirror. But let's take a look at the figuring process.

The basic requirement for figuring is opposite from the case of observing:

For the figuring case, we want the mirror to be massive (thick with a high specific gravity) and have a high specific heat. Also, although we found that a low coefficient of thermal expansion could be either good or bad for observing, depending on other aspects of the telescope, for figuring we want it to be as low as possible. If the coefficient is zero, then no mater how much heat is generated by the polishing and figuring process, the mirror will not change its shape or size.

It is good to have a high thermal conductivity for figuring as well as for observing. It keeps the temperature even throughout the mirror, which means that the thermal expansion will be even for the whole mirror and there will be no lumps and bumps in the surface (creating what is sometimes called "dog biscuit"). Also, high thermal conductivity allows the heat of figuring to travel away from the surface to the interior, keeping the surface temperature rising as little as possible, thereby keeping the thermal expansion to a minimum.

Row 12 shows the formula for a figuring Degree of Difficulty. In the same way as was done for the observing time constant, we want a low degree of difficulty, so we put the things that we want to be small in the numerator and the things that we want to be large in the denominator. Following is the results using the characteristics given in the literature.

Row 13 gives the relative Degree of Difficulty with Borosilicate Glass at 100. It shows that Green glass is between 3 and 4 times more difficult to figure than Borosilicate glass. It is hard for me to put a number on it, but it has been significantly more difficult for me to figure Green glass than Borosilicate glass.

Once again, Beryllium and Aluminum look very attractive, and I was wondering why I have never seen an ATM mirror made from these materials. With a little research I found that Beryllium is very expensive (somewhere between $600 and $1000 per pound in the rough), extremely difficult to form and machine, as well as being a powerful toxin to all forms of life. One can die from breathing Beryllium dust or become seriously ill from touching it. On the other hand, Aluminum has none of these characteristics. Many articles have been written enumerating the difficulties of making metal mirrors. I suspect that that is why I have not seen Aluminum mirrors. I have started work on a 7" Aluminum mirror, and I am having plenty of problems.

Figure 1. 7" aluminium mirror