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# How Smooth is a One Mile-Wide Mirror?

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### #1 Bob Myler

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Posted 17 January 2013 - 05:11 PM

Would someone please check my math?

We start with an 8" astronomical mirror - figured to within 1/8th wave surface accuracy:

A. Wavelength of green light: 550NM (0.00000055 Meters)
B. Convert this to inches (x's 39.37): 0.00002165"
C. 1/8th of the above would be 0.0000027"

So - just how smoooooth is 2.7 millionths of an Inch?

If we were to expand the diameter of our 8" mirror to One Mile - its (6 to 1 ratio) edge would stand almost 300 yards tall - and its finished face would have a surface accuracy of approximately 1/50th of an inch.

Folks, is that a reasonable approximation of reality?

### #2 GlennLeDrew

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Posted 17 January 2013 - 06:09 PM

No matter how large the mirror, the surface accuracy must still be to within the same fraction of a wavelength of the part of the electromagnetic spectrum observed. A 3", 300", or 30,000" mirror for visual spectrum work would all need to be figured to the same 1/4 wavefront error.

The allowable error does not scale up with the aperture. If it did, big mirrors would be easier to figure than small ones (discounting the handling issues.)

### #3 Bob Myler

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Posted 17 January 2013 - 06:26 PM

Understood.

I'm simply asking: Is a fiftieth of an inch surface error (for a mirror with a diameter of 63,360 inches) - the scaled-up equivalent for another - accurate to an eighth of the wavelength of green light, and having an 8 inch diameter?

### #4 George Kiger

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Posted 17 January 2013 - 06:59 PM

Your math is correct. Interesting way of looking at it.

### #5 Bob Myler

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Posted 17 January 2013 - 07:11 PM

Thank you. Using the very large to explain the very small - seems to help some better visualized the very small...

### #6 Jon Isaacs

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Posted 17 January 2013 - 08:14 PM

Understood.

I'm simply asking: Is a fiftieth of an inch surface error (for a mirror with a diameter of 63,360 inches) - the scaled-up equivalent for another - accurate to an eighth of the wavelength of green light, and having an 8 inch diameter?

I once saw a similar comparison. As I recall, it went like this:

If the Hale 200 inch at Palomar were scaled up so that it covered the surface of the United states, the bump would be about 4 inches.

Jon

### #7 dpwoos

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Posted 17 January 2013 - 11:41 PM

I always tell folks that if the diameter of the mirror was 1 mile, then it wouldn't deviate from an ideal paraboloid by more than the thickness of a playing card anywhere on its surface.

Similarly, I tell them that to a giant the Earth is smoother than a basketball.

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