Telescopes and mathematics

Summary: Image clarity in telescopes is achieved through extremely precise measurements and mathematics.

In 1608, the Dutch lensmaker Hans Lippershey applied for a patent on what was soon named a “telescope.” It is not clear if Lippershey was the true inventor; at least two other Dutch lensmakers also claimed credit. The news of this new invention quickly spread. In 1609, Galileo Galilei in Italy started using telescopes to observe heavenly objects. Among other findings, he discovered the rotation of the sun, the phases of Venus, and the first four satellites of Jupiter. A mathematician as well as a physicist and astronomer, Galileo also used geometry to measure the heights of lunar mountains by determining how long they remained illuminated after the lunar sunset.

Other mathematicians and physicists helped develop the modern telescope.Isaac Newton determined that lenses acted like prisms in spreading out the spectrum of visible light (an aberration phenomenon known as chromatic aberration). Newton and the mathematician James Gregory independently invented the reflecting telescope, which does not have this problem. Leonhard Euler made a mathematical analysis of chromatic aberration, and in England so-called achromatic lenses (a combination of two lenses that together bring light of different colors to a focus) were invented in the early eighteenth century.

Optics

A telescope is an optical device for seeing objects that are either far away, or very dim, or both. Consider a typical magnifying glass, as shown in Figure 1, which is a piece of glass or other transparent substance shaped so that both sides are sections of spheres. Light rays from an object (such as a candle) come to a focus on Screen 1. In other words, light rays from any given point on the candle converge onto a single point of Screen 1, forming an image. Screen 2 is at the wrong distance, meaning the light rays do not converge properly on Screen 2. Screen 1 is said to be “in focus,” and Screen 2 is “out of focus.”

The first lens the light goes through is the called the “objective lens.” The plane (Screen 1) where the image is in focus is called the “focal plane.” The “focal length” is the distance to the focal plane for a source at infinity (incoming parallel rays).

Magnification is measured in diameters. If the image is twice as tall and also twice as wide as the original, then there is a magnification of two diameters. Some optical devices, however, are measured in power, which is the square of the magnification in diameters; for example, a microscope advertised as “100 power” actually magnifies 10 diameters.

Figure 1a shows the same configuration as Figure 1, but the focal length is twice as long. The image on the focal plane is thus twice as high and twice as wide—it is magnified twice as many diameters. Since the image is spread out over four times the area, it is only one-fourth as bright. Conversely, for a given focal length, doubling the size of the objective lens lets in four times as much light, hence the image is four times as bright.

In Figure 1, if one were to put a light-tight box around screen 1, set up a shutter to control when light enters the box, and replace screen 1 with photographic film, the result is a camera. Replace the photographic film with an electronic light-sensitive screen, and the result is a digital camera. If the camera is used to take pictures of far-away or dim objects, then it qualifies as a telescope.

Astronomical Telescopes

Since astronomers are interested in dim celestial objects, a big objective is necessary for astronomical telescopes. Amateur astronomers frequently use a 6-inch (15-cm) objective as a good compromise between light-gathering power and cost. Professional astronomers rarely use objectives less than about half a meter (1.5 feet) in diameter. The largest objective lens in the world as of 2010 is 40 inches (1.106 meters) at Yerkes Observatory in Wisconsin.

The eye has its own lens, and the telescope has two lenses (or sets of lenses): the objective and the eyepiece. Figure 3 shows how the two-lens telescope delivers a greatly magnified image to the eye. The magnification in diameters is equal to the focal length of the objective divided by the focal length of the eyepiece. For example, the 40-inch telescope at Yerkes has a focal length of 744 inches. With a one-inch eyepiece, this telescope magnifies 744 diameters.

A microscope operates in the same way, except that the object being viewed, instead of distant and dim, is well lit and close to the objective lens.

Diffraction and Refraction

The useful magnification of a telescope is limited by diffraction. Light rays at the edge of the objective lens are diffracted—they are bent around the edge of the lens. These diffracted light rays cause a pattern of light and dark circles around bright images, which will blur adjacent images together. An empirical formula traditionally used to specify the limit of useful magnification is the Dawes Limit (also called the Rayleigh Limit): the resolution in arc-seconds is 4.56/D, where D is the diameter of the objective in inches; or 11.6/D, where D is in centimeters. For example, the diameter of the pupil of the human eye when dark-adapted is approximately 8 mm. By the Dawes Limit, the eye can resolve 11.6/.8 (14.5 arc-seconds), or about 1/125 of the diameter of the full moon. The Yerkes telescope can resolve about 0.1 arc-seconds.

A telescope using a lens as its objective is called a refracting telescope, since light is “refracted” (bent) by the lens. As of 2010, the 40-inch Yerkes instrument is the largest refracting telescope. A lens that size has to be thick to stand up to gravity, and thick lenses absorb so much light that beyond the size of Yerkes, absorption begins to outweigh the increased light gathered by a wider lens. Hence all current telescopes with objectives greater than 40 inches are “reflecting telescopes” in which the objective is a mirror rather than a lens.

Observer Placement

Unlike a lens, an objective mirror has a parabolic rather than a spherical surface. There is also the mechanical problem of where to place the observer or camera. There are several possibilities, some of which are shown in Figure 5.

One method, called “prime focus,” places the photographic film (or other astronomical instrument) inside the path of the incoming light. A few very large reflecting telescopes, such as the 200-inch Hale Telescope at Mount Palomar, actually allow for a human observer to ride in a cage at the prime focus.

A more common arrangement, invented by Isaac Newton and called the “Newtonian,” consists of a small flat mirror at an angle, which moves the focal plane to the side of the telescope. Two other common arrangements have a convex mirror at the prime focus, reflecting the light back down the length of the incoming light and also increasing the focal length. In the Cassegrain arrangement, a hole is cut in the middle of the mirror for the light to pass through. In the coudé arrangement, the light is reflected one more time into the mounting of the telescope, allowing the use of stationary instruments too heavy to be loaded onto the tube of the telescope.

Bibliography

Alloin, D. M., and Jean-Marie Mariotti. Diffraction-Limited Imaging With Very Large Telescopes. Berlin: Springer, 1989.

Edgerton, Samuel. The Mirror, the Window, and the Telescope: How Renaissance Linear Perspective Changed Our Vision of the Universe. Ithaca, NY: Cornell University Press, 2009.

Gates, Evalyn. Einstein’s Telescope: The Hunt for Dark Matter and Dark Energy in the Universe. New York: W. W. Norton, 2009.

Maran, Stephen. Galileo’s New Universe: The Revolution in Our Understanding of the Cosmos. New York: BenBella Books, 2009.