Stockton Astronomical Society
Valley Skies - June 2000 Issue
The Telescope Nut
by Jeff Baldwin
Big Dude Update / Sagittal Volume
Eric Reichenbach, Leo Radcliff, Roger Radcliff and I spent Saturday, April 15th, sandblasting my 40" blank into the proper curve of f/3.6. Sisto Gelsomini, owner of San Joaquin Monument Company, allowed us to use his commercial sandblaster to remove glass from the blank in order to shape the glass to the general curve of the sphere from which it will be parabolized.
Curve-generating would take a very long time to do by conventional methods since 436 cubic inches of glass needed to be removed. At 1/12th of a cubic inch per hour -- the usual rate of curve generating by hand -- I would have expected to complete this task after nearly 5000 hours of work. Not an option. Using a high speed grinder, I think I could have pulled it off in a few hundred hours, after using a few hundred dollars worth of disks. The sandblaster was definitely the ticket.
We have a template with the correct curve in it. After blasting much of the material away, we would hold it up to the glass, and wherever the glass touched the template we would mark with a wax pencil. Then we would blast off the glass on those marks, remark it, and continue. After a while we noticed that the template was matching the glass quite well, and we discontinued blasting. It is possible that blasting too deeply would ruin the glass by going under the level at which we wanted to be.
It is important to know how many cubic inches of glass need to be removed from
a mirror blank, especially if it is large, fast, or both large and fast. If we
didn't know how many cubic inches of material had to be removed, then we would
just start grinding until we either got it deep enough, or died of old age. As
you can see, 5000 hours of hand grinding would have caused the second result. How
do you know when you are deep enough? If your mirror is of radius r, which is
half the aperture, and its radius of curvature is R, which is twice the focal
length, then there are two sagittal numbers you need to know. The first one is
how deep you need to go, or the sagitta,
.
If the mirror's sagitta is this deep and the mirror is spherical, then you are
done curve generating. Of course, a very important number if you have a large
mirror is the sagittal volume, or the number of cubic inches of glass that must
be removed. This is important because it tells you the amount of work that needs
to be done, translating into how long it will take you to do it. This amount is
.
Let's look at a couple of examples. An 18" f/4.5 mirror has a radius of
9" and an RoC of 162". The sagitta of this mirror is 92/324
= 81/324 = 0.25". If the mirror is spherical and the depth is 1/4",
then you are done curve generating. How much glass is that? The sagittal volume =
= 32 cubic inches. This is quite a bit of glass to remove. This much glass may
require you to use a high speed method rather than conventional hand grinding
methods.
Let's look at the standard 8" f/6. Sagittal depth is 16/192 =
1/12th inch; the sagittal volume is
= 2 cubic inches, not much work.
This is how we accounted for the glass removal for the 40" mirror. It is
an f/3.6, so its focal length is 144", making its RoC = 288". The
sagitta is 400/576 = 0.694" (almost 3/4"), and the sagittal volume is
= 436 cubic inches. That told us that conventional grinding was not going to
happen, so we chose a high speed method, since this is the same amount of glass
that would be in a 7.6" cube (that's a lot of glass).
To derive the sagitta, I start with the cross section of the paraboloid being
the parabola
, or
since R, the radius of curvature, is twice the focal length f. If
x is the radius of the mirror, then
, or
.
Why does R = 2f? The cross-sectional parabola can be written two
ways; one way is y = ax2 and the other way is
.
This means that
.
The slope of the parabola will be the derivative of the parabola, which will be
m = y' = 2ax. The line that passes through the point (x0,f(x0))
will have a slope normal to that line, the slope now being -1/y'. The line of
tangency at the point x0 will be y - y0 = m (x - x0),
or 
or 
or 
or 
.
This is a line with a slope of
,
and the y-intercept is
.
The radius of curvature for any zone will be the y-intercept, and the radius of
curvature of the center zone of the mirror will be the limit as x0
approaches 0, which will be
.
Since 
,
then this radius of curvature, or R, will be
,
implying that R = 2f.
To derive the sagittal volume, I used the disk method involving solids of
revolution. If the cross sectional parabola is
,
this allows the axis of symmetry for the parabola to be the y-axis. If this is
revolved about the y-axis, then the result is a paraboloid, and the disk method of
revolved solids is used. Let's deal with it as
,
the sum of the volumes of disks that make up the paraboloid, and when there are
an infinite number of disks we'll get the Reimann sum
.
Since
, then x =
,
making x2 = 2Ry. This leads to
Stuff to remember:
is the sagittal depth.
is the sagittal volume.
is the relationship between the radius of curvature and the focal length.One more note on magnitudes from the last article. In talking to Mel Bartels, he mentioned that if he places a 7mm aperture mask on a 20X telescope he can see down to about mag 7.6, which means that there is a magnitude limit placed on us by the magnification of the star. That's the same aperture as the human eye, only with power, and instead of seeing only down to 6 mag, he could see down to 7.6 mag. That would imply that the magnitude formula would change from 8.8 + 5logA to 10.4 + 5logA. This may be why a 24" scope should be able to see down to mag 15.7 but actually was able to reach mag 17.1. Therefore I am changing the formulae to:
Check the ATM website for the updated magnitude table by following the links to "Aperture vs. Magnitudes".
Clear Skies...Jeff Baldwin
For more information on Telescope Making jump to the
ATM page.
Copyright © 2001 by Jeff Baldwin
Last Updated: 3/9/2001
http://astro.sci.uop.edu/~sas/Newsletter/TTN_BigDude.html