Activity 2
The Telescope

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Background Information

In a refracting telescope, the objective lens makes a real image of the object. This image is made within a focal length of the eyepiece, which then acts as a magnifier. Remember that a real image is made by focusing light that hits the lens all over its surface, so these rays reach the image from many different directions. If the image is not projected on a screen, the rays keep going and spread out as they would from an object. In this case, the real image made by the objective is the object of the eyepiece. These two lenses are separated by the sum of their focal lengths. Since the real image is just outside the focal length of the objective, it is just inside the focal length of the eyepiece, which therefore makes a magnified, virtual image.

The magnification is given by the ratio of the focal lengths of the objective and eyepiece.

For the real image made by the objective, the image and object heights and distances are related by this equation (derived in the Background Information section of Activity 1):

Solving for the height of this image gives

H_i is proportional to f_o
The image is almost at the focus of the object, D_i is approximately f_o, the focal length of the objective. Therefore, the height of the image made by the objective is proportional to the focal length of the objective.
H_i is proportional to f_o
The second diagram shows a magnifier. By similar triangles

or, solving for H_i

In a telescope, the eyepiece magnifies the real image of the objective. This real image is located almost at the focus of the eyepiece (see drawing above), so for the virtual image made by the eyepiece
H_i is proportional to 1/f_e
To find the effect of both objective and eyepiece working together, simply multiply the proportionalities.
H_i is proportional to f_o/f_e
Since the magnification M is proportional to the image height, M is given by

Note that this number is a relative magnification to compare one telescope to another. As with any engineered instrument, the design of the refracting telescope contains tradeoffs. To make the magnification M as large as possible the refractor must be long, since M is proportional to the focal length of the objective. A long telescope is awkward to mount and to aim. In addition, to make a bright photographic image, an optical system must have a low f-number (The f-number is the focal length divided by the aperture). Thus, the telescope can be designed for high magnification only at the expense of its speed (as in a “fast” lens). Moreover, any lens system is plagued by distortions, including the following:

Spherical aberration and coma: The edges of the lens have a longer focal length than the portion near the axis.

Chromatic aberration: Blue light is brought to a focus closer to a lens than red light, for the same reason that a prism creates a spectrum—the index of refraction of the glass depends on the wavelength of the light. Building up a composite lens of different kinds of glass reduces chromatic aberration.
Refractors have one important advantage over reflecting telescopes (which collect light with a large mirror). The reflector’s mirror cools gradually during observations at night. Since the cooling starts on the outer surfaces, the resulting temperature differences inside the mirror causes distortion in the shape and therefore in the images the telescope produces. Reflectors, with lenses partly sealed inside a long tube, suffer much less from this kind of distortion.