Physics Bootcamp

Subsection 48.6.1 Refracting Telescope

Historically, refracting telescope was the first telescope to be used for observing astronomical objects. Although the accounts vary as to who invented the first refracting telescope, but records suggest that Galileo Galelei built his own refracting telescope and was the first one to make observations in astronomy using a telescope.

Refracting telescope is similar to a microscpe. It also uses two lenses - the first of these lenses facing the object is called the objective or objective lens, and the second lens near the eye is called the eyepiece. While in microscope, both lenses were used to magnify, the task of the objective in a telescope is to bring the far away object near and then eyepiece is used to magnify this real image.

Figure 48.6.2 shows the optical setup of a telescope. The objective forms a real image \(\text{Q}_1\) of a star that lies in the direction of \(\theta_\text{o}\text{.}\) This image forms in the focal plane of the second focal plane of the objective lens. The first focal plane of the eyepiece is very close to it and left of it sso that when we use the real image as the object for the eyepiece, we get a virtual image in the direction \(\theta_\text{i}\text{.}\)

The magnification or power of a telescope is given by the ratio of the angle subtended by the image to that subtended by the object.

We can identify a triangle between the two lenses using the chief rays to the image \(\text{Q}_1\) from the objective and the ray in the direction of the final virtual image as shown in Figure 48.6.3. Then, angles \(\theta_\text{o}\) and \(\theta_\text{i}\) can be related to the focal lengths of the two lenses since their separation is very close to the sum of the foci.

By using tthe geometry in Figure 48.6.3 and assuming small angles, we can rewrite the magnification of a refractive telescope as

where I have introduced a minus sign to take care of the negative direction of the direction to the final image. Thus, magnifying power of a telescope simply depends on the foci of the objective and the eyepiece. Since power of a lens is inverse of their focal length, we can also write the magnification as

Therefore, higher power eyepiece and lower power objective will give you more magnification.

Typical eyepieces have focal lengths of \(2.5\text{ cm}\text{,}\) and \(1.25\text{ cm}\text{.}\) If the objective of the telescope has a focal length of \(1\text{ meter}\text{,}\) then these eyepieces would result in magnifications of 40-times, and 80-times respectively. The angular magnifications would make the image appear 40-times, or 80-times closer.

In a refracting telescope objective lens collects light from the source. For fainter objects in the sky you need large diameter objective lens. The largest refracting telescope in the world is the 40 inch diameter Yerkes telescope operated by the University of Chicago and located at Lake Geneva, Wisconsin (Figure 48.6.4).

It is very difficult and expensive to build large refracting telescopes. You need large defect free lenses, which in itself is a technically demanding task. Refracting telescope basically looks like a tube with support structure to rotate it in different directions.

A refracting telescope suffers from a number of defects. The aberration of lenses causes the image to be blurred. Also as the lenses become thicker for larger lenses more light is absorbed making the faint stars more difficult to observe. Large lenses are also very heavy and deform under their own weight. Some of these problems with refracting telescope is addressed by avoiding refraction for collecting light. Instead a curved mirror is used for the objective. Such telescopes were first designed by Isaac Newton. These telescopes are called the reflecting telescopes which we discuss next.

Subsection 48.6.2 Reflecting Telescope

In the reflecting telescope, also known as a reflector, light rays from a distant source fall upon the surface of a concave mirror fixed at the bottom end of the tube. The concave mirror focuses the rays on its focal plane. To observe the focused image Newton used a design in which the focused light from the concave mirror was reflected to one side of the tube into an eyepiece as shown in Figure 48.6.5. This arrangement is common in many amateur telescopes and is called Newtonian design.

Some telescopes reflect the light back towards the middle of the concave mirror by using a convex mirror as shwon in Figure 48.6.6. In this arrangement the light-gathering concave mirror has a hole in the middle through which light emerges to be incident on the eyepiece lens. This arrangement of the objective and eyepiece is called Cassegrainian design.\ Most big telescopes, including Hubble telescope (see Figure 48.6.7), are of this design. There are other arrangements also possible. Sometimes one puts a light detector right at the spot where light is focused by the curved mirror.

Hubble telescope shown in Figure 48.6.7, put in the orbit around earth in 1990, is another large reflecting telescope with the primary mirror of diameter 2.4 meter. That makes \(f_\text{o}= 2.4/4 = 0.6\text{m}\text{.}\)

The formula for the angular magnification \(M\) of a reflecting telescope is same as that of the refracting telescope.

where \(f_\text{o}\) is half of the radius of curvature, \(R_\text{o}\text{,}\) of the objective mirror and \(f_\text{e}\) is the focal length of the eyepiece. You can see that making large objective not only helps out with collecting more light into the telescope it also helps with the magnification of the image.

Subsection 48.6.3 Resolving Power of a Telescope

An important use of a telescope is to be able to resolve images of two nearby stars. The image seen through a telescope is affected by a number of factors, such as poor optics and blurring by atmosphere. When a telescope, such as Hubble, is placed above the atmosphere and optical aberrations discussed below are eliminated, resolving two point-like objects is still limited by diffraction effects, which cannot be eliminated since it is fundamental aspect of wave nature of light.

We will discuss diffraction in a later chapter. Here, we summarize some of the important features of resolution of two stars in approximately same direction. Suppose we project the image of a single star on a CCD detector. What we will observe is that there is a central bright disk in the direction of the star, called the Airy disk, surrounded by dark ring which is alternately surrounded by less and less bright and dark rings. This is due to the diffraction effect.

Quantitatively, sizes of the rings depend upon the objective's aperture \(D\) and the wavelength \(\lambda\) of light. From the center of the aperture, which will be at the center of the lens of the CCD camera if telescope is equipped with a CCD detector, the central bright disk will be spread over an angle \(\Delta\theta\) given by

\begin{equation*} \Delta\theta = \frac{1.22\lambda}{D}. \end{equation*}

Each star will form its blob of this size on the screen. If the angular separation of the two stars was less thin this angle, it will difficult to tell whther you have two stars or one - they are said to be not resolved. This is called the Raleigh criterion. This is a rule of thumb that works well. We call this Raleigh limit of resolution of the telescope and loosely speak of this as a limit of resolution.

Although Raleigh criterion provides a useful metric of resolution of images of two nearby stars, one can often go beyond it. There is another criteria, called Sparrow criterion, which looks at the presence of any dip in the image and use that dip to separate the images of the two stars. We will not go into that here.