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The Telescope Nut
by Jeff Baldwin

Parabolizing

Correcting the sphere to a paraboloid is a requirement for most Newtonian telescope mirrors. At this point I won't go into extreme detail on the calculations, but rather just give an overview. I imagine that in future articles I will accumulate details that as a whole will cover this completely.

A Newtonian telescope with a paraboloidal mirror must be of some exactness or the image will be degraded. It is an industry standard that the mirror will be within 1/4 wavelength of visible light, peak-to-valley, from a perfect mathematical paraboloid. A wavelength is about 1/50,000 of an inch, so 1/4 of that is 1/200,000 of an inch. 'Peak-to-valley' means that, as the shape of the mirror deviates from a true paraboloid, the highest error to the lowest error will have a difference of 1/200,000 of an inch. If the error is larger than that, the light will be bundled in a larger spot than the "airy disk", the theoretically smallest spot that the mirror can make a star image. (I'll get to all that over the next few years or months. Let's just get some basics in for now.)

The mirror starts as a sphere and we are going to change it into a paraboloid, which is very close to a sphere over the regions in which we are working. The larger the mirror is in diameter, the larger the difference is between the sphere and the paraboloid. Also, the faster the mirror is -- which means the shorter its focal length as compared to its diameter, -- the greater the difference between the sphere and the paraboloid. For example, a 6" f/12, which is a very small, slow telescope, requires no parabolizing from a sphere because the spherical mirror is already within 1/4 wave of a paraboloid (1/25 wave), whereas a 16" f/4 sphere has 2.8 waves peak-to-alley error. Without any parabolizing work, the 6" f/12 would outperform the 16" f/4 in clarity and resolution. The 6" f/12 would need no correction and could be sent to the aluminizers as a sphere. The 16" f/4 would require 11.2 waves of correction to bring it down from its current 2.8 waves peak-to-valley error to a perfect paraboloid. It could not be sent to the aluminizers as is or the mirror would be performing much lower than it should in clarity and resolution.

Each aperture and focal ratio has a sphere-to-paraboloid deviation that is different. Also, the peak-to-valley error of a sphere multiplied by 4 gives you the amount of correction from the edge to the center that needs to be done. (Notice the 16" f/4 has a peak-to-valley error of 2.8 waves, and 2.8 times 4 is 11.2, the number of waves of correction needed to fix that error). Here is a table of peak-to-valley errors of different mirrors along with the number of waves of correction needed to make the spheres into paraboloids. These tables are based on light with a wavelength of 554.4 nanometers.

Difference between Spheres and Paraboloids in waves. 554.4 nm
Aperture f/3 f/3.5 f/4 f/4.5 f/5 f/5.6 f/6 f/8 f/10 f/12 f/15
4.25 1.76 1.108 0.742 0.521 0.38 0.27 0.22 0.092 0.05 0.03 0.014
6 2.485 1.565 1.048 0.736 0.536 0.382 0.31 0.131 0.07 0.04 0.019
8 3.314 2.087 1.398 0.981 0.715 0.509 0.414 0.174 0.09 0.05 0.026
10 4.142 2.608 1.747 1.227 0.894 0.636 0.517 0.218 0.11 0.06 0.033
12.5 5.178 3.261 2.184 1.534 1.118 0.796 0.647 0.273 0.14 0.08 0.041
14.25 5.903 3.717 2.49 1.749 1.275 0.907 0.737 0.311 0.16 0.09 0.047
16 6.628 4.174 2.796 1.963 1.431 1.019 0.828 0.349 0.18 0.1 0.053
17.5 7.249 4.565 3.058 2.148 1.565 1.114 0.906 0.382 0.2 0.11 0.057
18 7.456 4.695 3.145 2.209 1.61 1.146 0.932 0.393 0.2 0.12 0.059
20 8.285 5.217 3.495 2.454 1.789 1.273 1.035 0.436 0.22 0.13 0.066
22 9.114 5.739 3.844 2.7 1.968 1.401 1.139 0.48 0.25 0.14 0.072
24 9.942 6.261 4.194 2.945 2.147 1.528 1.242 0.524 0.27 0.16 0.079
25 10.356 6.522 4.369 3.068 2.237 1.592 1.294 0.546 0.28 0.16 0.082
28 11.599 7.304 4.893 3.436 2.505 1.783 1.449 0.611 0.31 0.18 0.092
30 12.428 7.826 5.243 3.682 2.684 1.91 1.553 0.655 0.34 0.19 0.099
32 13.256 8.348 5.592 3.927 2.863 2.038 1.657 0.699 0.36 0.21 0.106
36 14.913 9.391 6.291 4.418 3.221 2.292 1.864 0.786 0.4 0.23 0.119
40 16.57 10.435 6.99 4.909 3.579 2.547 2.071 0.873 0.45 0.26 0.132

Correction to a Paraboloid from a Sphere in waves. 554.4 nm
Aperture f/3 f/3.5 f/4 f/4.5 f/5 f/5.6 f/6 f/8 f/10 f/12 f/15
4.25 7.04 4.432 2.968 2.084 1.52 1.08 0.88 0.368 0.19 0.11 0.056
6 9.94 6.26 4.192 2.944 2.144 1.528 1.24 0.524 0.27 0.15 0.076
8 13.256 8.348 5.592 3.924 2.86 2.036 1.656 0.696 0.36 0.2 0.104
10 16.568 10.432 6.988 4.908 3.576 2.544 2.068 0.872 0.44 0.26 0.132
12.5 20.712 13.044 8.736 6.136 4.472 3.184 2.588 1.092 0.56 0.32 0.164
14.25 23.612 14.868 9.96 6.996 5.1 3.628 2.948 1.244 0.64 0.37 0.188
16 26.512 16.696 11.184 7.852 5.724 4.076 3.312 1.396 0.71 0.41 0.212
17.5 28.996 18.26 12.232 8.592 6.26 4.456 3.624 1.528 0.78 0.45 0.228
18 29.824 18.78 12.58 8.836 6.44 4.584 3.728 1.572 0.8 0.46 0.236
20 33.14 20.868 13.98 9.816 7.156 5.092 4.14 1.744 0.89 0.52 0.264
22 36.456 22.956 15.376 10.8 7.872 5.604 4.556 1.92 0.98 0.57 0.288
24 39.768 25.044 16.776 11.78 8.588 6.112 4.968 2.096 1.07 0.62 0.316
25 41.424 26.088 17.476 12.27 8.948 6.368 5.176 2.184 1.12 0.64 0.328
28 46.396 29.216 19.572 13.74 10.02 7.132 5.796 2.444 1.25 0.72 0.368
30 49.712 31.304 20.972 14.73 10.74 7.64 6.212 2.62 1.34 0.78 0.396
32 53.024 33.392 22.368 15.71 11.45 8.152 6.628 2.796 1.43 0.83 0.424
36 59.652 37.564 25.164 17.67 12.88 9.168 7.456 3.144 1.61 0.93 0.476
40 66.28 41.74 27.96 19.64 14.32 10.188 8.284 3.492 1.79 1.03 0.528

The highlighted boxes are at or near 1/4 peak-to-valley error without correction. Even though a mirror may be at or below peak-to-valley error as is, remember that there are two mirrors in a Newtonian telescope. The two mirrors together will have a sum error that will degrade the image if they are combining to a high degree. To figure out what their sum total error might appear as, you can add the error of the primary mirror to the product of error in the secondary mirror and the distance ratio it has comparing the focuser's distance to the secondary and the primary's distance to the secondary. What the heck? OK, if the primary mirror has an error of 1/20 wave, and the secondary mirror has an error of 1/8 wave, and the distance from the focuser to the secondary mirror is 8 inches and the distance from the secondary mirror to the primary is 48 inches, then the expected error might be 1/20 + 1/8 X (8/48) = 1/20 + 1/48 = 12/240 + 5/240 = 17/240 = about 0.3/4 wave, or 1/3 of 1/4 wave, which is fine. This is only a judgment calculation, because there are some errors of small degree that can have tremendous results, and large errors that can have minor results, so it is just a rule of thumb. One way out is to make the primary mirror as good as humanly possible. If it has no measurable errors, then in combination with the secondary mirror it will perform well.

The first step of correcting the mirror has been done -- deciding how much correction is needed. If you are making the standard 8" f/6 beginner's mirror, then you can see that it is almost 0.41 waves out of correction and needs about 1.7 waves of correction to bring it into perfection. When you decide on what size and focal ratio of mirror to start your mirror making adventures, you may want to review these tables. You may want to start with one that requires no correction, or one that requires a little correction so you can have that experience, or you may want to say "the heck with it" and go for whatever mirror you want to look through. Texereau suggests an 8" f/6 because it requires the beginner to do a small amount of correcting, while I usually recommend going for whatever telescope you will be happy to have when you are finished regardless of the degree of difficulty.

When we meet again, we'll blab about how to correct and how to measure how far you've corrected, the idea being that you will correct the mirror until it is right on the nut, and then quit.

Clear Glass...Jeff Baldwin
For more information on Telescope Making jump to the ATM page.

Copyright © 2000 by Jeff Baldwin
Lasted Updated: 12/12/2000
http://astro.sci.uop.edu/~sas/Newsletter/TTN_Parabolizing1.html