Telescopes and image formation

The astronomical refractor

An astronomical refractor is two converging lenses on a shared axis. The objective, the large lens at the front, catches light from a distant object. That light arrives as near-parallel rays, so the objective brings it to a real image in its focal plane, one focal length behind the glass. The second, smaller lens, the eye lens, is then used as a magnifying glass to inspect that little image.

In normal adjustment the eye lens is positioned so that its focal plane coincides with the objective's. The real image then sits exactly one focal length in front of the eye lens, so the light leaves it parallel again and the final image forms at infinity. Your eye stays relaxed, focused on the far distance, for hours at the eyepiece. Add the two focal lengths and you have the tube length, f_o + f_e$f_{o} + f_{e}$f_o + f_e.

Because the object and the final image both live at infinity, the telescope cannot make anything bigger in the ordinary sense. What it magnifies is angle. The angular magnification is defined as the angle the final image subtends at your eye, divided by the angle the object subtends at the unaided eye. In normal adjustment that ratio collapses to the focal lengths:

so a long-focus objective and a short-focus eye lens give the biggest angular gain. Look at the geometry in the figure and you can see where that comes from. Both angles share the same image height h in the focal plane. The small angle coming in is roughly h/f_o$h/f_{o}$h/f_o, the steeper angle going out is h/f_e$h/f_{e}$h/f_e, and dividing one by the other leaves f_o/f_e$f_{o}/f_{e}$f_o/f_e.

The Cassegrain reflector

Serious telescopes use mirrors, and the classic layout is the Cassegrain. A large concave primary mirror collects the light. Its surface is ground to a parabola, because a parabolic mirror brings every ray parallel to its axis to one sharp focus, however far from the axis the ray strikes. Before the light reaches that focus, a small convex secondary mirror intercepts it and reflects it back down the tube, through a central hole in the primary, to a focus just behind the main mirror where the eyepiece or camera sits.

The fold is the point. Reflecting the beam back on itself packs a long effective focal length into a short, stiff, steerable tube, and the eyepiece ends up in the most convenient place possible, behind the telescope where an observer or an instrument can sit. The angular magnification works exactly as before, with the effective focal length of the mirror pair playing the part of f_o$f_{o}$f_o.

Reflector or refractor

Lenses carry two built-in flaws, and both are examinable by name. Chromatic aberration comes first. Glass refracts blue light more strongly than red, so a single lens gives each colour its own focal length and no one sharp focus exists. Every bright star wears a faint coloured fringe.

Then spherical aberration. A lens or mirror ground to a spherical surface focuses rays through its edge slightly short of rays through its centre, smearing the focus along the axis. Mirrors escape both problems more cheaply than lenses do. Reflection is independent of wavelength, so a mirror has no chromatic aberration at all, and grinding the primary to a parabola removes the spherical error for light arriving parallel to the axis.

This is where the big-magnification boast collapses. Magnification only stretches the light you have already caught, and stretching a dim, blurred image gives a bigger dim, blurred image. What you actually pay for is how much light the telescope gathers and how fine the detail it can resolve. Both belong to the objective's diameter, the subject of the next lesson.