April 2006 Newsletter

Stockton Astronomical Society

Valley Skies

Valley Skies - April 2006 Issue

The Telescope Nut
by Jeff Baldwin

Micrometers, Sagittas and Errors

Wow, another installment of The Telescope Nut! What's next, new episodes of Bonanza? Hmm, that would be cool.

When telescope mirror-makers grind the initial curve into the mirror blank, they need to know when the curve is deep enough. Initially it is spherical, but after the mirror has been polished spherical, it is then deformed into a paraboloid with approximately the same radius of curvature. The sphere will have the same radius of curvature wherever it is measured, but the paraboloid will have shorter radii of curvatures in the central zones as compared to the outer zones. However, they are so close that when using mechanical methods to measure them they are immeasurably close.

You can measure the radius of curvature of the mirror using the paraboloid formula, or you can use the sphere formula. The paraboloid formula is much easier, so we use it. Here's how we derive each of them.

For the sphere, we recognize that the equation of a circle (the sphere intersected with a plane which passes through its center) is (x - h)² + (y - k)² = r², where the point (h, k) is the center and r is the radius. If we want the center to be on the y-axis and the circle be in contact with the origin (0, 0), then we get (x - 0)² + (y - R)² = R², where R is the radius of curvature of the mirror (twice the focal length). That gives us x² + (y - R)² = R². That leads to (y - R)² = -x² + R², which leads to , which leads to . The + part of that is the other side of the circle, so the equation is . This gives us the height of the circle at position x away from the origin. If you're trying to figure out what R is based on the fact that you already know the position x and the depth y, then we have to solve for R, which would be . In optics we call the x value r and the y value σ, so we get . If we have a 10" mirror with a 0.104" sagitta, then the RoC is (5²+.104²)/.208, or about 120.24".

For the paraboloid, we recognize that the equation of a parabola (the paraboloid intersected with a plane which passes through the axis of symmetry) is . Since R is twice the f.l., then it becomes . So, if y is the sagitta and x is the radius on the glass at which we are measuring, then we get . Solving for the RoC, we get . If we use the exact same mirror as we did in the sphere problem, a 10" with a sagitta of 0.104", then the RoC would be 25/.208 which is 120.19". Notice that these two are different by about 0.05", 1/20 of an inch. Using the same spherometer we can calculate two different RoCs, but they are extremely close, much closer than we need to know them. So, we use the easier of the two formulae to calculate it. That would be the paraboloid formula .

OK, we're ready to try one out. Pretend you are making a 16" f/4.5 mirror. If has a focal length of 16 times 4.5, which is 72". That makes the RoC = 144". That makes 2R = 288". Your sagitta needs to be r²/2R, or 8²/288", or 0.222". If your spherometer has legs that are 2" from the mike post, then your micrometer reading needs to be 2²/288 = 4/288 = 0.0139". Keep digging the mirror deeper until you are at this reading.

Sometimes we calculate this by suspending a measuring object in the middle of the mirror, and sometime we use a spherometer.

A spherometer has three legs in the arrangement of an equilateral triangle, with a micrometer head in the middle to measure the drop below the plane of the three legs. A side view is shown in the illustration. The radius is the distance from the micrometer to one of the legs. This sagitta can be measured to a thousandth of an inch on most micrometers, and interpolated to the nearest ten-thousandths.

Some mikes are better. If you use a micrometer with legs that are too close together, then the number you get may be erroneous. Also, if the micrometer is cheap and imperfect, you may also find that your RoC is not what you expected. Let's check out some error analysis on these micrometer readings.

Consider a micrometer that has legs 2" from the center post and a micrometer that has thousandth of an inch intervals. Also consider the 72" focal length mirror. Aperture doesn't matter here since the micrometer is so much smaller than the mirror. Our sagitta ought to be 4/288 = 0.0139". Let's plug 0.0139" into the equation to see what we get. R=4/(2X0.0139) = 143.88". That worked out, there's a little round off stuff, but it worked out pretty good. Hmmm, I wonder what would happen if we misinterpreted the micrometer by a little. It looked like it was 0.0139", but what if it were off by a ten thousandth of an inch one way or the other. That would mean that it could have been 0.0140 or 0.0138. Let's check them both to see what a misread of a ten thousandth of an inch would do to our RoC determination. 4/(2X.140) = 142.86, and 4/(2X.0138) = 144.93". Not bad, an error of a tiny misread doesn't seem to booger it up much, it goes up or down about an inch each way. So if your error was in a tenth of a thousandth of an inch, your RoC would be off by 1". That's a ten thousand to one ratio from micrometer error to RoC error. Now I wonder what would happen if your post to micrometer measurement was off by a little. When you drill a piece of metal to put a post in it, you are going to miss by a small amount, and if you're a machinist, that would be small, but if you're me, that might not be small. Let's pretend that you put the post 1/16" off either too close or too far. That would be 1 15/16" or 2 1/16". Using the same 0.0139" sagitta and the original 143.88" RoC, we now get (1 15/16)²/(2X.0139) = 135.03" RoC for the small radius error, and for the 2 1/16, we get (2 1/16)²/(2X.0139) = 153.02". Wow, that's an enormous error in RoC considering the small error in post positioning. We're talking 9 inches. OK, let's try it with a machinist. Let's have him put the posts in and they are positioned to the nearest thousandth of an inch. That would give us 1.999" and 2.001". This now gives us 143.74 and 144.03. This time our errors are reduced to about a seventh of an inch in RoC.

Most micrometers have ball feet so they won't scratch the glass. This tends to produce an error in the RoC determination because the ball contact on the slope of the mirror's surface produces an error. The height is not measured at the end of the leg but rather a slight distance shorter, and this produces a large error in RoC determination.

Here's the deal, the smaller the spherometer the wilder the errors if a mismeasurement occurs. Furthermore, mismeasurement is not the only place an error can be generated. If the micrometer is warm or cold, it expands or contracts, so does the glass, and all these can change the readings. Warming it in your hand can do it. Moving inside from outside can do it. Also, the legs may not be an equilateral triangle, or the post may not be exactly in the center nor perpendicular to the plane of the legs. All this stuff could also compound together to give you an estimate of the RoC at best. Another glitch is that when most mirror-makers polish glass they reduce the focal length because they tend to increase the depth of the glass as they work. The deeper the sagitta, the shorter the focal length. It's nice to start with a slightly longer RoC than with what you intend on finishing. Also, you may not want to be too particular about the exact focal length you desire. If you are making a mirror that needs to fit a focal length specification very closely, then this becomes a problem. Some ATMers wet the glass with a spray of water to make it shiny and focus the Sun onto a platform and measure from the mirror to the image to ensure their focal length. I have found this to not be very accurate when the mirror is rough. Also, water tends to sag on the glass and it gives the illusion that your focal length is specific when in fact it is a different length when the glass is polished out.

The secret: Make the largest radius and most accurately measured posts on your micrometer that you can get away with. Use your spherometer and sagitta measurements to estimate the RoC but be aware of the magnitudes of the possible errors and accept that they will occur. Determine your sagitta with more than one measuring device or method. And one more, enjoy making the mirror and using the telescope without being too worried about the exactness of the focal length, your eyeball isn't going to know the difference anyways.

One last note about accuracy. You need to establish when the spherometer is measuring a flat surface, sort of like "zeroing it out". If you can find the perfect flat surface, great! That may be harder to find than you think unless you make optical flats. If you are making a mirror, then your tool is the same RoC as your mirror, only convex instead of concave. If you measure the mirror's micrometer reading with your spherometer, then measure your tool's micrometer reading, then "flat" ought to be their average. Add them up and divide by two, and you will have the spherometer's micrometer setting for "flat", and your depth measurements for the mirror will now be more precise.

Clear Skies and Smooth Glass.
Baldy