February 1999 Newsletter

Stockton Astronomical Society

Valley Skies

Valley Skies - February 1999 Issue

The Telescope Nut
by Jeff Baldwin

Figuring the Paraboloid

When you measure a mirror to see if it is a paraboloid, you are checking to see where the perpendicular reflection rays cross the optical axis. At a particular zone, the light will darken neither from the left nor from the right, but at the same time. If you look at the image on the left you can see that it has been corrected from a sphere to some figured shape. There is a distinct circle around the mirror in which the crest of these shadows exist. Outside of this circle the mirror is dark on the right and bright on the left, which means that the razor is inside the radius of curvature for that area (razor is on right unless otherwise specified). Inside this circle the mirror is dark on the left and bright on the right, implying that the razor is outside the radius of curvature for that area. The razor is right on the radius of curvature for the crest of this image. If the bottom of the mirror is flipped over, as on the image in the middle, we can see where the exact location of the crest is. This is not done in Foucault testing, but it's nice for this example.

What we do in the shop is mask the mirror everywhere except for the zones we are examining, as in the image on the right. Can you tell which is darker or lighter, the right zone or the left zone? In the shop you can tell, and when you have the razor positioned to the spot where the two are the same intensity, you record the micrometer position of the razor. This is done for all the zones on the mirror.

These micrometer position need to satisfy the equation y = r²/2R ± Rr /r where r is the zone radius, R is the radius of curvature for the center zone, and r is the radius of the "Airy disk", the smallest field a scope can resolve. This r = 1.22 times the wavelength of the light you're observing times the focal ratio of the telescope. Light that humans can see averages 554.4 nm, or .0005544 mm, or .0000218 inches. So, r = 1.22 x .0000218 x F/d. Each mirror has a different calculation for that since each mirror has a different focal ratio.

When you plot the micrometer positions on a graph against radius positions of the zone holes in the mask, you get the following "tornado" graph.

In this graph you see four curves. Three of them are together toward the bottom. They are plots of the micrometer readings for the perfect paraboloid , the uppermost acceptable limit and the least acceptable limit. The fourth line at the top is the from readings when the mirror is spherical. Since all the readings are the same for a spherical mirror, the graph plots as a horizontal line. As the mirror is corrected and the radii of curvature shorten towards the center of the mirror, this line starts to tilt downward as in the next graph.

As this correction continues, eventually the mirror maker will target the correction so that all the micrometer readings are within this tornado, as in the next graph.

This mirror is an acceptable mirror and is ready to be aluminized.