26.5 Telescopes - College Physics for AP® Courses

Learning Objectives

By the end of this section, you will be able to:

Outline the invention of the telescope.
Describe the working of a telescope.

The information presented in this section supports the following AP® learning objectives and science practices:

6.E.4.1 The student is able to plan data collection strategies and perform data analysis and evaluation of evidence about the formation of images due to reflection of light from curved spherical mirrors. (S.P. 3.2, 4.1, 5.1, 5.2, 5.3)
6.E.5.1 The student is able to use quantitative and qualitative representations and models to analyze situations and solve problems about image formation occurring due to the refraction of light through thin lenses. (S.P. 1.4, 2.2)

Telescopes are meant for viewing distant objects, producing an image that is larger than the image that can be seen with the unaided eye. Telescopes gather far more light than the eye, allowing dim objects to be observed with greater magnification and better resolution. Although Galileo is often credited with inventing the telescope, he actually did not. What he did was more important. He constructed several early telescopes, was the first to study the heavens with them, and made monumental discoveries using them. Among these are the moons of Jupiter, the craters and mountains on the Moon, the details of sunspots, and the fact that the Milky Way is composed of vast numbers of individual stars.

Figure 26.23(a) shows a telescope made of two lenses, the convex objective and the concave eyepiece, the same construction used by Galileo. Such an arrangement produces an upright image and is used in spyglasses and opera glasses.

Part a of the figure depicts the internal functioning of a telescope; from left to right it has an upright image of a tree, a convex lens objective, a concave lens eyepiece, and a picture of eye where rays enter. Parallel rays strike the objective convex lens, converge; strike the concave eyepiece, and enter the eye. Dotted lines from the striking rays of the eyepiece are drawn backside and join at the beginning of the final image. Part b of the figure, from left to right, has an inverted enlarged image of a tree, a convex objective, a smaller inverted image of a tree, a convex eyepiece and a picture of an eye viewing the image. Rays from a very distant object pass through the objective lens, focus at a focal point f sub o, forming a smaller upside-down image of a tree of height h sub i, converge and pass through the eyepiece to reach the eye. Dotted lines drawn backwards focus at the tip of the final enlarged inverted image of a tree of height h prime sub i, from the rays striking the eyepiece are also shown. An angle theta, subtended by the rays striking the objective lens and an angle, subtended by the telescopic enlarged inverted image are also depicted.

Figure 26.23 (a) Galileo made telescopes with a convex objective and a concave eyepiece. These produce an upright image and are used in spyglasses. (b) Most simple telescopes have two convex lenses. The objective forms a case 1 image that is the object for the eyepiece. The eyepiece forms a case 2 final image that is magnified.

The most common two-lens telescope, like the simple microscope, uses two convex lenses and is shown in Figure 26.23(b). The object is so far away from the telescope that it is essentially at infinity compared with the focal lengths of the lenses ( $d_{o} \approx \infty$ ). The first image is thus produced at $d_{i} = f_{o}$ , as shown in the figure. To prove this, note that

\frac{1}{d_{i}} = \frac{1}{f_{o}} - \frac{1}{d_{o}} = \frac{1}{f_{o}} - \frac{1}{\infty} .

26.25

Because $1 / \infty = 0$ , this simplifies to

\frac{1}{d_{i}} = \frac{1}{f_{o}},

26.26

which implies that $d_{i} = f_{o}$ , as claimed. It is true that for any distant object and any lens or mirror, the image is at the focal length.

The first image formed by a telescope objective as seen in Figure 26.23(b) will not be large compared with what you might see by looking at the object directly. For example, the spot formed by sunlight focused on a piece of paper by a magnifying glass is the image of the Sun, and it is small. The telescope eyepiece (like the microscope eyepiece) magnifies this first image. The distance between the eyepiece and the objective lens is made slightly less than the sum of their focal lengths so that the first image is closer to the eyepiece than its focal length. That is, $d_{o}'$ is less than $f_{e}$ , and so the eyepiece forms a case 2 image that is large and to the left for easy viewing. If the angle subtended by an object as viewed by the unaided eye is $θ$ , and the angle subtended by the telescope image is $θ'$ , then the angular magnification $M$ is defined to be their ratio. That is, $M = θ' / θ$ . It can be shown that the angular magnification of a telescope is related to the focal lengths of the objective and eyepiece; and is given by

M = \frac{θ'}{θ} = - \frac{f_{o}}{f_{e}} .

26.27

The minus sign indicates the image is inverted. To obtain the greatest angular magnification, it is best to have a long focal length objective and a short focal length eyepiece. The greater the angular magnification $M$ , the larger an object will appear when viewed through a telescope, making more details visible. Limits to observable details are imposed by many factors, including lens quality and atmospheric disturbance.

The image in most telescopes is inverted, which is unimportant for observing the stars but a real problem for other applications, such as telescopes on ships or telescopic gun sights. If an upright image is needed, Galileo’s arrangement in Figure 26.23(a) can be used. But a more common arrangement is to use a third convex lens as an eyepiece, increasing the distance between the first two and inverting the image once again as seen in Figure 26.24.

A ray diagram from left to right depicts a concave objective lens, a small inverted image of a tree, a magnified upright final image of tree, an erecting concave lens, a small upright image of a tree, concave lens as an eyepiece, and an eye to view on the same optical axis. Rays from a distant object strike the edges of the objective lens, converge at the focus of the focal point, form a small inverted image of the object and pass through the erecting lens, again forming the upright small image of the object, and finally, the rays pass through the eyepiece to the eye. Dotted lines joined backwards from the rays striking the eyepiece meet at a point where the final enlarged upright image of the object is formed.

Figure 26.24 This arrangement of three lenses in a telescope produces an upright final image. The first two lenses are far enough apart that the second lens inverts the image of the first one more time. The third lens acts as a magnifier and keeps the image upright and in a location that is easy to view.

A telescope can also be made with a concave mirror as its first element or objective, since a concave mirror acts like a convex lens as seen in Figure 26.25. Flat mirrors are often employed in optical instruments to make them more compact or to send light to cameras and other sensing devices. There are many advantages to using mirrors rather than lenses for telescope objectives. Mirrors can be constructed much larger than lenses and can, thus, gather large amounts of light, as needed to view distant galaxies, for example. Large and relatively flat mirrors have very long focal lengths, so that great angular magnification is possible.

A ray diagram from left to right depicts a small diagonal mirror and a concave lens eyepiece placed parallel to each other. A large curved objective mirror is placed in front of the diagonal mirror. Parallel rays of light are falling at the edges of the objective mirror, which is curved just at the right amount to bounce all the light onto the diagonal mirror. From there, the light rays pass through the eyepiece lens, which bends the light into the eye.

Figure 26.25 A two-element telescope composed of a mirror as the objective and a lens for the eyepiece is shown. This telescope forms an image in the same manner as the two-convex-lens telescope already discussed, but it does not suffer from chromatic aberrations. Such telescopes can gather more light, since larger mirrors than lenses can be constructed.

Telescopes, like microscopes, can utilize a range of frequencies from the electromagnetic spectrum. Figure 26.26(a) shows the Australia Telescope Compact Array, which uses six 22-m antennas for mapping the southern skies using radio waves. Figure 26.26(b) shows the focusing of x rays on the Chandra X-ray Observatory—a satellite orbiting earth since 1999 and looking at high temperature events as exploding stars, quasars, and black holes. X rays, with much more energy and shorter wavelengths than RF and light, are mainly absorbed and not reflected when incident perpendicular to the medium. But they can be reflected when incident at small glancing angles, much like a rock will skip on a lake if thrown at a small angle. The mirrors for the Chandra consist of a long barrelled pathway and 4 pairs of mirrors to focus the rays at a point 10 meters away from the entrance. The mirrors are extremely smooth and consist of a glass ceramic base with a thin coating of metal (iridium). Four pairs of precision manufactured mirrors are exquisitely shaped and aligned so that x rays ricochet off the mirrors like bullets off a wall, focusing on a spot.

Image a is a photograph one of the antennas from the Australia Telescope Compact Array. Image b is a cutaway diagram showing 4 nested sets of hard x-ray mirrors of the Chandra X-ray observatory.

Figure 26.26 (a) The Australia Telescope Compact Array at Narrabri (500 km NW of Sydney). (credit: Ian Bailey) (b) The focusing of x rays on the Chandra Observatory, a satellite orbiting earth. X rays ricochet off 4 pairs of mirrors forming a barrelled pathway leading to the focus point. (credit: NASA)

A current exciting development is a collaborative effort involving 17 countries to construct a Square Kilometre Array (SKA) of telescopes capable of covering from 80 MHz to 2 GHz. The initial stage of the project is the construction of the Australian Square Kilometre Array Pathfinder in Western Australia (see Figure 26.27). The project will use cutting-edge technologies such as adaptive optics in which the lens or mirror is constructed from lots of carefully aligned tiny lenses and mirrors that can be manipulated using computers. A range of rapidly changing distortions can be minimized by deforming or tilting the tiny lenses and mirrors. The use of adaptive optics in vision correction is a current area of research.

An aerial overview of the central region of the Square Kilometre Array with the five kilometer diameter cores of antennas or dishes is seen. S K A-low array and S K A-mid array, which are phased arrays of simple dipole antennas to cover the frequency range from seventy to two hundred megahertz and two hundred to five hundred megahertz in circular stations, are also displayed.

Figure 26.27 An artist’s impression of the Australian Square Kilometre Array Pathfinder in Western Australia is displayed. (credit: SPDO, XILOSTUDIOS)