# A Simple Telescope

A simple working telescope requires nothing more than a pair of lenses mounted in a tube. The lens in front, known as the objective, focuses an image; the lens in back, known as the eyepiece, magnifies the image. Although it may seem like a crude device, a simple telescope nicely illustrates the basic working principles of more powerful astronomical instruments.

### LENSES

Light normally moves in straight lines, but there are situations in which this is not true. You are already familiar with some: for example, the distortions you see looking through the surface of the ocean occur because light bends as it passes from the water into the air. Long before we understood why light bends as it passes from one transparent material to another, people had used this effect to create lenses: optical devices which can gather light together or spread it apart.

In order to understand how a lens works, you need to know a little about how light behaves in passing from one material to another. Imagine a tank of water on the table in front of you; the surface of the water should be perfectly flat and horizontal. If you shine a ray of light straight down from above, it will pass through the surface of the water without bending. But if you shine the light in at an angle, it will bend as it passes through the surface. Fig. 1 illustrates two important facts about this effect. First, in passing from air to water, the light always bends into the water. Second, the smaller the angle between the light ray and the surface, the more it bends in passing through. The same rules would also apply if the tank of water was replaced with a block of glass.

 Fig. 1. Light rays passing from air into water. If the ray strikes the surface at a 90° angle it doesn't bend (left). But if the angle is less than 90° the light bends (middle); reducing the angle between the light and the surface increases the bend (right).

What if you shine the light up through the water into the air? The answer is very simple; light follows the same path no matter which way it's going! To illustrate this in Fig. 1, all you need to do is draw upward-pointing arrows at the other end of each light ray.

To create a lens which can focus many parallel rays of light to a single point, the idea is to curve the surface of the glass so that all these rays, after passing through, come together at the same place. It's a bit tricky to do this right, but we don't need to worry about the details. The simplest kind of lens is a `plano-convex' lens; one side is flat, while the other bulges out at the middle. Fig. 2 shows how such a lens focuses light. The optical axis of the lens is the thick line which passes right through the middle of the lens; a ray of light traveling along the optical axis is not bent at all. Rays which pass through the top of the lens are bent downward, while rays which pass through the bottom of the lens are bent upward. Thus all these light rays are bent toward the optical axis. If the lens is well-made, all rays meet at the same focal point. The distance between the lens and the focal point, measured along the optical axis, is called the focal length.

 Fig. 2. A simple lens in operation. Parallel light rays come from the right, pass through the lens, and meet at the focal point on the left. The thick line through the middle of the lens is the optical axis; the distance F is the focal length.

### IMAGE FORMATION

A lens which could only focus light rays striking the glass head-on (as in Fig. 2) would be fairly useless for astronomy. Fortunately, most lenses can also accept rays which come in at a slight angle to the optical axis, and bring them to a focus as well. This focal point is not the same as the focal point for rays which are parallel to the optical axis; depending on the angle of the incoming rays, their focus lies on one side or the other of the optical axis, as shown in Fig. 3. But if the lens is well-made, all these focal points will lie on a plane which is parallel to the face of the lens; this is called the focal plane.

 Fig. 3. A simple lens forming an image. The red rays arrive with an downward slant, and come to a focus below the optical axis, while the blue rays arrive with a upward slant, and come to a focus above the optical axis. The vertical dotted line at left represents the focal plane.

There's one slightly subtle consequence of this image-formation process: the image is upside-down! Fig. 4 shows why: rays from the lower part of the subject (on the right) come together at the upper part of the image (left), and vice versa. This is also true of a camera; of course, you turn the prints right way up when you get them back from the store, so you're probably not aware that the image is upside down inside your camera.

 Fig. 4. The image formed by a simple lens is upside-down with respect to the subject. Here the subject (right) is an arrow with a red tip pointing upward; its image (left, at the focal plane) points down.

Your simple telescope kit includes a large objective lens which you will use to study image formation. Take the large lens and mount it at one end of the larger cardboard tube; slide the smaller tube into the other end of the larger one, and use a rubber band to hold a sheet of tracing paper over the open end of the smaller tube. Now point the tubes at the subject we've set up in the lab, and slide the smaller tube in and out until you focus a sharp image on the tracing paper.

1. Observe the orientation of the image on the tracing paper. Which side is up, and which down? Which side is to the left, and which to the right?
2. With the image focused as sharply as possible, measure the distance between the lens and the tracing paper. This distance is basically the focal length F of the large lens.
Record your observations and measurements in your lab notebook. Include a sketch the image.

(Most of the time, professional astronomers use telescopes to take pictures of astronomical objects. The instrument you've just built is a crude model of a professional photographic telescope; if the tracing paper was replaced by photographic film, you could use this equipment to take a picture in much the same way the pros do.)

### EYEPIECES AND MAGNIFICATION

To make a telescope you can actually look through, you'll need to add another lens. This eyepiece lens magnifies the image formed by the large objective lens and directs the light to your eye. Basically, the eyepiece works a lot like a magnifying glass; it enables your eye to focus much more closely than it normally can. The eyepiece on a typical telescope allows you to inspect the image formed by the objective lens from a distance of an inch or less. Fig. 5 shows how the objective lens and eyepiece work together in a simple telescope.

 Fig. 5. A simple telescope. Parallel light rays enter from the right, pass through the objective lens, come to a focus at the focal plane, and exit through the eyepiece. The focal length of the objective is F, and the focal length of the eyepiece is f.

Before installing the eyepiece lens in your telescope kit, you should first measure its focal length, f. Remove the lens from its foam mounting and hold it by the edges. Point the flat side of the lens toward a distant light-source and hold a sheet of paper behind the lens and parallel to its face. Move the paper closer or further from the lens until you see a sharp image of the light-source, and measure the distance from the curved face of the lens to the paper. (This really requires three hands; get your lab partner to help!) Record the focal length f in your lab notebook.

You're now ready to put the telescope kit together. Replace the eyepiece lens in its foam mounting. Remove the tracing paper, and insert the foam mounting in the smaller tube. Point the telescope at a distant target and slide the tube in and out until you get a good focus. Which way is the image oriented?

Compared to the image in your binoculars or a real telescope, the image you'll see with this simple telescope is probably a bit fuzzy; you may also notice bands of color around bright objects. These effects are due to the limitations of the simple lenses used in this telescope kit. You can make the image sharper by installing the paper washer in front of the objective lens, but this will also make the image dimmer since the telescope will gather less light.

#### Magnification

The magnification of a telescope is easy to calculate once you know the focal lengths F and f of the objective lens and eyepiece, respectively. The formula for the magnification M is

M = F ÷ f .

Here you can use any units for F and f, as long as you use the same units for both. For example, if you measure F in millimeters, you should also measure f in millimeters. Using the values for F and f you measured above, calculate the expected magnification of your telescope.

To measure the magnification of your telescope directly, we will set up a target - basically a picture of a ruler with marks a unit distance apart. From the other end of the room, focus your telescope on the target. Now look through the telescope while keeping both eyes open; you should see a double image, where one image is magnified and the other is not. Compare the two images; how many of the unmagnified units fit within one magnified unit? The answer is a direct measurement of your telescope's magnification; record it in your notebook and compare it to the magnification you calculated using the formula above.

### REVIEW QUESTIONS

• Suppose you are looking at an image formed on the tracing paper in the experiment on IMAGE FORMATION above. If you turn slightly to the right while holding the tube, which way do you expect the image to move on the tracing paper? Think about this question and write down an answer before you try this in the lab!

• Our big telescopes have objectives with a focal length F = 1200 mm. We have four eyepieces for each scope, with focal lengths f = 32 mm, 25 mm, 14 mm, and 10 mm. What magnifications do these eyepieces provide?

### REPORT: A SIMPLE TELESCOPE

Do the experiments described in the sections on IMAGE FORMATION and EYEPIECES AND MAGNIFICATION, and write a report on your work. This report should include, in order,

1. a general introduction to the experiments,
2. a description of the equipment you used,
3. a summary of your experimental results, and
4. the conclusions you have reached.

In more detail, here are several things you should be sure to do in your lab report:

• Describe the appearance of the image formed on the tracing paper, and draw a diagram showing how it is oriented (i.e., which way is up, which way to the right, and so on).

• Describe the appearance of objects seen through the finished telescope. Are things right-side up or upside-down? Is the image equally sharp everywhere? Does it appear distorted?

• Include your measurements for the focal lengths F and f of the objective and eyepiece lenses.

• Calculate the magnification M you predict using your measurements of F and f.

• Compare the calculated magnification with the magnification you measured directly by observing the meter stick target.

Joshua E. Barnes (barnes@ifa.hawaii.edu)