January 1999 Newsletter

Stockton Astronomical Society

Valley Skies

Valley Skies - January 1999 Issue

The Telescope Nut
by Jeff Baldwin

Conic Sections and the Foucault Test

Telescope mirrors start out being spherical, and are then either left spherical, or are altered to other conic section shapes. The most common shapes are #1 Paraboloidal, #2 hyperboloidal, #3 spherical, #4 prolate ellipsoidal, #5 oblate ellipsoidal (sometimes misreferred to as an oblate spheroid).

What are these things?

To make conic sections, you start by making a cone by revolving a line about a point, the line sweeping out a cone (what most people think of as 2 cones with their vertices touching). Then we slice the cone with a plane. If the plane is cut through the cone perpendicular to the cone's axis, then the intersection is a circle. If we tilt the plane a little, we get an ellipse. If we tilt the plane so far that it is parallel to the tilt of the generating line, then we get a parabola. If we tilt the plane until it is parallel to the cone's axis, then we get an hyperbola of two sheets, since there are two parts that are disjoint.

Let's go a little further now and rotate these conic sections about their vertices. If we rotate a circle around in and out of its plane, we create a sphere, also known as a spheroid. If we rotate a parabola in and out of its plane about its vertex, we get a paraboloid. It's shaped like a bullet slug. If we rotate the hyperbola in and out of its plane about its vertices, we get an hyperboloid of two sheets, and if we only examine one of those sheets, we have the optical equivalent to an hyperboloid. If we rotate an ellipse around in and out of its plane, we create an ellipsoid. However, there are two types of ellipsoids: prolate ellipsoids and oblate ellipsoids.

When we rotate the ellipsoid in and out of the sheet, we can rotate it about the prolate axis or the oblate axis, creating either a prolate ellipsoid or an oblate ellipsoid.

OK, that was the Cliff Notes version of optical conic sections. Here's the important part of this: for Newtonian telescopes we will use paraboloidal mirrors. We start with a spherical mirror and alter into a paraboloid, which will focus infinitely distant objects at the center of the field of view into their theoretically smallest dot. This is the simplest reflecting telescope for amateurs to make. We'll get to the others in future articles.

The test that we will start discussing is the Foucault test. Let's start with a sphere. If there is a light at the center of curvature of a sphere then the light will progress outward to the sphere, reflect off of the sphere, and return directly to the light source. OK, let's do that again, only this time we're going to let the light pass by a razor blade, reflect off of the mirror, and return and slide by the razor blade. There are three cases that can occur: #1, the razor is closer to the mirror than the center of curvature, #2, the razor is farther from the mirror than the center of curvature, or #3, the razor is exactly at the center of curvature. On each of these cases we move the razor sideways into the light path and observe what the mirror looks like. If the razor is too close to the mirror, then when the razor is moved into place the light from the razor's side of the mirror will be obstructed making the mirror go dark on the same side as the razor. If the razor is too far from the mirror, then when the razor is moved into place the light paths have already crossed over and the opposite side of the mirror will go dark. If the razor is exactly at the center of curvature, then when it is moved into place the entire mirror will darken uniformly.

This takes practice and it would be nice if you saw this at an ATM workshop. Brief explanations of the Foucault test rarely suffice in grasping it completely.

Note that the above illustrations are applicable only to a Foucault test on a spherical surface. If the mirror is a paraboloid, it doesn't look like these pictures.

A Foucault test on a paraboloid will look more like this:

This is a test image of a 12.5" f/4.1 mirror that is almost a paraboloidal surface. There is a mark at the center that is used for references. Notice that the light energy is not uniform but somewhat of a doughnut in appearance. This is a result of the fact that paraboloids have different radii of curvature, rather than the same radius of curvature throughout the mirror. Next month we will discuss measuring these radii of curvature in order to determine if the mirror is really paraboloidal or not.

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