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Telescopes

Module by: OpenStax College. E-mail the author

Summary:

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

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 1(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.

Figure 1: (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.
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.

The most common two-lens telescope, like the simple microscope, uses two convex lenses and is shown in Figure 1(b). The object is so far away from the telescope that it is essentially at infinity compared with the focal lengths of the lenses (dodo). The first image is thus produced at di=fodi=fo, as shown in the figure. To prove this, note that

1di=1fo1do=1fo1.1di=1fo1do=1fo1. size 12{ { {1} over {d rSub { size 8{i} } } } = { {1} over {f rSub { size 8{o} } } } - { {1} over {d rSub { size 8{o} } } } = { {1} over {f rSub { size 8{o} } } } - { {1} over { infinity } } } {}
(1)

Because 1/=01/=0 size 12{ {1} slash { infinity =0} } {}, this simplifies to

1di=1fo,1di=1fo, size 12{ { {1} over {d rSub { size 8{i} } } } = { {1} over {f rSub { size 8{o} } } } } {}
(2)

which implies that di=fodi=fo, 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 1(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, dodo is less than fefe, 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 θθ, and the angle subtended by the telescope image is θθ, then the angular magnification MM is defined to be their ratio. That is, M=θ/θM=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=θθ=fofe.M=θθ=fofe.
(3)

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 MM size 12{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 1(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 2.

Figure 2: 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 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.

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 3. 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.

Figure 3: 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.
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.

Telescopes, like microscopes, can utilize a range of frequencies from the electromagnetic spectrum. Figure 4(a) shows the Australia Telescope Compact Array, which uses six 22-m antennas for mapping the southern skies using radio waves. Figure 4(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.

Figure 4: (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)
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.

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 5). 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.

Figure 5: An artist’s impression of the Australian Square Kilometre Array Pathfinder in Western Australia is displayed. (credit: SPDO, XILOSTUDIOS)
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.

Section Summary

  • Simple telescopes can be made with two lenses. They are used for viewing objects at large distances and utilize the entire range of the electromagnetic spectrum.
  • The angular magnification M for a telescope is given by
    M=θθ=fofe,M=θθ=fofe,
    (4)
    where θθ is the angle subtended by an object viewed by the unaided eye, θθ is the angle subtended by a magnified image, and fofo size 12{f rSub { size 8{o} } } {} and fefe size 12{f rSub { size 8{e} } } {} are the focal lengths of the objective and the eyepiece.

Conceptual Questions

Exercise 1

If you want your microscope or telescope to project a real image onto a screen, how would you change the placement of the eyepiece relative to the objective?

Problem Exercises

Unless otherwise stated, the lens-to-retina distance is 2.00 cm.

Exercise 1

What is the angular magnification of a telescope that has a 100 cm focal length objective and a 2.50 cm focal length eyepiece?

Solution

40 . 0 40 . 0 size 12{ - {underline {"40" "." 0}} } {}

Exercise 2

Find the distance between the objective and eyepiece lenses in the telescope in the above problem needed to produce a final image very far from the observer, where vision is most relaxed. Note that a telescope is normally used to view very distant objects.

Exercise 3

A large reflecting telescope has an objective mirror with a 10.0 m10.0 m size 12{"10" "." 0`m} {} radius of curvature. What angular magnification does it produce when a 3.00 m3.00 m size 12{3 "." "00"`m} {} focal length eyepiece is used?

Solution

1 . 67 1 . 67 size 12{ - 1 "." "67"} {}

Exercise 4

A small telescope has a concave mirror with a 2.00 m radius of curvature for its objective. Its eyepiece is a 4.00 cm focal length lens. (a) What is the telescope’s angular magnification? (b) What angle is subtended by a 25,000 km diameter sunspot? (c) What is the angle of its telescopic image?

Exercise 5

A 7.5×7.5× size 12{7 "." 5 times } {} binocular produces an angular magnification of 7.507.50 size 12{ - 7 "." "50"} {}, acting like a telescope. (Mirrors are used to make the image upright.) If the binoculars have objective lenses with a 75.0 cm focal length, what is the focal length of the eyepiece lenses?

Solution

+ 10.0 cm + 10.0 cm size 12{+"10" "." 0`"cm"} {}

Exercise 6

Construct Your Own Problem

Consider a telescope of the type used by Galileo, having a convex objective and a concave eyepiece as illustrated in Figure 1(a). Construct a problem in which you calculate the location and size of the image produced. Among the things to be considered are the focal lengths of the lenses and their relative placements as well as the size and location of the object. Verify that the angular magnification is greater than one. That is, the angle subtended at the eye by the image is greater than the angle subtended by the object.

Glossary

adaptive optics:
optical technology in which computers adjust the lenses and mirrors in a device to correct for image distortions
angular magnification:
a ratio related to the focal lengths of the objective and eyepiece and given as M = fo fe M = fo fe

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

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Definition of a lens

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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What are tags? tag icon

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