A little elbow grease and damp sponges cleaned off most of the external gunk. Once in the lab, we took out the mirror and found the aluminum coating to be in bad shape. Certainly time for it to be recoated. But first, we'll have fun with it doing a Foucault test to check its figure. Mirrors with pretty bad optical surfaces can still produce images.
A Foucault test is an optical test at center of curvature, often used by amateur telescope makers while "figuring" a mirror to its final shape. The test uses simple equipment to assess the shape of the mirror to within 1/4 wavelength of light or so. It is not as accurate as surface metrology done with an interferometer, but instead of needing $50,000 worth of equipment, you can use stuff that is probably lying around the house. Or lab.
We upgraded the usual Foucault instrument to include a CCD camera to capture the upstream view of the mirror for all to view. In the "classic" Foucault test the procedure is to eyeball it.
Here you see the LED gooseneck lamp that is illuminating a slightly off-axis frosted 1 mm diaphram opening. The frosting is achieved with a piece of Scotch-brand magic tape. This diffuse light source in turn illuminates the mirror, which is located two focal lengths (1 radius of curvature) downstream. Near the focus is a probe on a mounting post. A knife edge is often used. We used a chisel from the toolbox since it comes with a built-in handle for mounting. Later on, Jacob suggested using a pin instead and swapped out the chisel for a mechanical pencil with its 700 micron diameter lead extended, which is what you see here:
Looking upstream from before the pencil, here's what the cell phone camera captured. The bright light on the telescope mirror is the out-of-focus image of the frosted diaphragm, which is located behind and to the left of the camera. To take this picture, I moved the camera until I could see the image of the frosted diaphragm go into the lens of the camera.
Of course, it's not really an experiment without data, so here's the data off the camera when the probe is inserted at a point very near the focus. A perfect mirror would show a uniform disk with varying brightness as the probe was moved to the focal point. But an imperfect mirror doesn't focus all the light at the focal point, some misses. So with the probe at the focal point, the light that misses keeps on going and enters the camera. What you can see here is that the light coming from the edges of the mirror is not blocked, while the light coming off the center is mostly blocked.
The outer edge has a different focal length, by a small amount (several mm) than the middle. By playing with the probe, Paul found that the edge rays came to a focus farther out, so the edge of the mirror has a longer radius of curvature than the middle. This so-called "turned-down-edge" is a common aberration in amateur-class telescopes. The amount present here is pretty small, and probably doesn't noticeably affect image quality for visual observations with this telescope.
Finally, some boardwork on the Zernike functions used to describe aberrations of optical systems. These functions, defined over the unit circle in RxR, form a linear vector space much like sines and cosines do. An arbitrary function on the unit disk can be written as a linear combination of Zernikes. The usefulness comes from the ability to map physical effects onto Zernikes when doing "forensics" in an effort to understand and potentially correct aberrations. Low order terms correspond to mis-pointing, out-of-focus, off-axis, and mount-induced aberrations.
