# Telescope mirrors under tension.

In this article Geoffrey Landis proposes builing a space tower using pressurized structures:
THE TSIOLKOVSKI TOWER RE-EXAMINED
Journal of The British Interplanetary Society, Vol 52, pp. 175-180, 1999. http://www.aeiveos.com/~bradbury/Authors/Engineering/Landis-GA/TTTR.html
He notes that typically materials have higher failure or ultimate strength in tension than in compression so this would allow higher towers to be built by using pressurized towers that translate the vertical compressional forces into tensional forces that tend to expand the pressurized structure outwards. See section 4.3 and Fig. 3. I wanted to use a similar idea for creating large telescope mirrors. However, the formula for deflection of a mirror due to self-weight depends on Young's modulus, not ultimate strength and Young's modulus is about the same in tension and compression. Still it may be possible to create larger mirrors because the deflection formula also is dependent on density which will be much less with a gas filled structure:
Deflection ~ Density*(1-Poisson's ratio^2)/Young's Modulus
However, all three of these quantities will likely change for a two material structure as with a pressurized mirror. (BTW, perhaps someone can answer this for me, in the Landis article he just calculates the tensional outward force that would need to be supported, but surely the outside walls would still need to support some vertical compression force. Is this just a minor component and can be neglected?) Some proposals for inflatable membrane mirrors for use with space telescopes are given in this conference report:
Ultra Light Space Optics Challenge: Presentations. http://origins.jpl.nasa.gov/meetings/ulsoc/presentations.html
Another possibility for a mirror under tension is given by this surprising fact:
Parabolas and Bridges. "If you hang a flexible chain loosely between two supports, the curve formed by the chain looks like a parabola, but isn't. It is a catenary, a more glamorous curve which can be represented algebraically by hyperbolic functions [y = A (cosh kx - 1)]. In this case, the vertical load on the chain is uniform with respect to arc length. A whirling skipping rope is another example of a catenary. "The load on a suspension bridge is (approximately) uniform with respect to the horizontal distance. In this case, the curve is a parabola ..." http://www.du.edu/~jcalvert/math/parabola.htm
Shape of a suspension bridge cable.
http://aemes.mae.ufl.edu/~uhk/SUSPCAB.jpg
Hanging With Galileo. "Take a flexible chain of uniform linear mass density. Suspend it from the two ends. What is the curve formed by the chain? Galileo Galilei said that it was a parabola, and perhaps you made the same guess. This time Galileo was not correct. The curve is called a catenary. However, it is easy to see how he could arrive at this answer through casual observation and incomplete deduction. ... "We can get back to the chain solution later. First consider this extension. What about the curve formed by the cables of a suspension bridge? Is it too a catenary? No, it is a parabola. So, what gives? How can this be a parabola while the other one is not?" http://whistleralley.com/hanging/hanging.htm
Then to form a parabolic surface you could have the suspension cables arranged in concentric circles hanging from the mirror supporting a weight. In this case gravity would be working to *form* the surface shape. A problem is that just with liquid mirrors you might need to keep the mirror horizontal so it would have to be zenith-pointing. However, it may be that by varying the cable lengths you could maintain the parabolic shape.
Bob Clark
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Robert Clark wrote:

Funny, I could have sworn it was the other way around.
Fused silica glass is about 20x stronger in compression than in tension (1100 MPa vs. 50 MPa). See
http://www.sciner.com/Opticsland/FS.htm
and look at "compressive" and "tensile" strength on that page.
Mark
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In the world.

Under such conditions of awareness, but I know that.

The biggest difference is in the reliability of the hardware and software. You can look forward to very stringent manufacturing parameters and programming discipline.

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