Practical Astronomy Focus: Concave (Spherical) Mirrors and The Reflecting Telescope

 

                                                   Basic reflecting telescope with 4 1/4" mirror

                 Basic approaches & options for making one's own reflector

                                          Harry Bayley Observatory 14" reflector telescope

The telescope most often used in more advanced amateur astronomy is the reflector - which uses a spherical mirror for its objective.  The principle is simple: parallel rays of light, i.e. from a star, fall upon the curved surface of the spherical (concave) mirror which is coated with silver or aluminum to make it highly reflecting. Each ray of light is then reflected according to the law of reflection. If the mirror has the correct concave shape, all the rays are reflected back through the same point - the focus of the mirror. 

This will be slightly different for the Newtonian reflector (top 2 images) and the Schmidt-Cassegrain type- which is used at the Harry Bayley Observatory (bottom two images above).  A diagram showing the difference in the reflected light paths is shown below:

Note from the diagrams the Newtonian reflector uses a plane mirror to merge incoming rays to a focus, while the Cassegrain telescope uses a plan0-convex mirror. In either case the image of the object appears at the focus.

In more detail, the light acted upon by any spherical mirror will conform to the laws of geometrical optics as shown in the diagram below:

The following points apply:

 i) any light ray which passes through the center of curvature (C) of a spherical mirror en route to the mirror surface, will be reflected back upon itself; 

ii) all light rays which approach the mirror in paths parallel to the optical axis are reflected through a common point on the optical axis known as the principal focus; 

iii) any light ray which passes through the focal point on its way to the mirror will be reflected parallel to the principal axis.

The basic law for image formation is:  1/s  + 1/s’ = 1/f.

Since the radius of curvature R = 2f, this can also be written as:

1/s  + 1/s’ = 2 /R

Which is called “the Mirror equation.”

If an object is very far from the concave mirror, say effectively at infinity (s = ¥) then we have a situation peculiar to that for reflecting astronomical telescopes:

                                                          

For the case depicted here: 

1/f = 1/s  + 1/s’ Þ 1/s =  1/ ¥  =  0  and s’ =  R/2

Numerically, the distance s for an image corresponds to 1 degree in the sky and is given by the equation:  s = 0.01744f  

 Where f is the focal length of the mirror. The scale of an image can then be computed from the above formula.

Example:  What is the image size of the Moon in the Mt. Palomar reflecting telescope? (Focal length 660 inches).

Solution:

Using:    s = 0.01744f

In normal application this applies to an image corresponding to  1 ^o  in the sky.

Then for f= 660 in. we get s = 0.01744 (660 in.) =  11. 5 inches per degree.

But the Moon subtends  1/2 ^o  so that the resulting image scale for the Palomar reflector would be 1/2 what it is for one degree in the sky, or:

½ (11.5/ deg) = 5.75 in/ deg

Suggested Problems:

1) A small object lies 4 cm to the left of the vertex of a concave mirror who radius of curvature is 12 cm. Find the magnification of the image.  Include a basic sketch to show the relation of the image to the object position relative to the mirror.

2) What would be the size of the image of the Moon produced by the 14-inch (356 mm) Harry Bayley Observatory Cassegrain telescope? (Focal length = 154 inches)  What would be the radius of curvature R for the telescope mirror?

3) Repeat the above computations for a 6-inch Newtonian reflecting telescope with a focal length of 48 inches.