Telescope — Refracting and Reflecting

What a telescope does

A distant object subtends a very small visual angle at the unaided eye, too small for its detail to be perceived. A telescope increases this visual angle and brings the image of the object effectively nearer to the eye. Two classes are in common use: the refracting telescope, whose objective is a lens, and the reflecting telescope, whose objective is a concave mirror.

For a telescope the figure of merit is its angular magnification, defined as the ratio of the angle subtended at the eye by the final image to the angle the object itself subtends. Light-gathering power and resolving power, both governed by the objective aperture, decide how faint and how closely spaced the objects it can separate.

The refracting telescope

The astronomical refracting telescope is a two-lens coaxial system. The lens facing the object — the objective — has a large aperture and a large focal length $f_o$. Light from a distant object enters the objective and a real, inverted image is formed in the tube at the second focal point of the objective. The eyepiece, of small aperture and short focal length $f_e$, then acts on this intermediate image as a simple magnifier, producing a final inverted image.

In normal adjustment the eyepiece is positioned so that the intermediate image lies at its focus and the final image is formed at infinity, where the relaxed eye views it most comfortably. Heavenly bodies being roughly round, the inverted final image does not impair astronomical observation; terrestrial telescopes add an erecting arrangement.

Magnifying power and tube length

Let $\alpha$ be the angle the distant object subtends at the objective and $\beta$ the angle the final image subtends at the eye. The magnifying power is $m = \beta/\alpha$. Since both angles are small, they may be replaced by their tangents. If the intermediate image has height $h$, then $\tan\alpha = h/f_o$ and $\tan\beta = h/f_e$, giving the normal-adjustment result:

$$m = \dfrac{\beta}{\alpha} \approx \dfrac{\tan\beta}{\tan\alpha} = \dfrac{h/f_e}{h/f_o} = \dfrac{f_o}{f_e}$$

The magnitude is large when the objective focal length is large and the eyepiece focal length is small. In this same normal adjustment, the separation between the two lenses — the tube length — is $L = f_o + f_e$, because the intermediate image lies at the common focus of both lenses. For an objective of $f_o = 100\ \text{cm}$ and eyepiece $f_e = 1\ \text{cm}$, NCERT gives $m = 100/1 = 100$, so a pair of stars actually $1'$ apart appear separated by $100' = 1.67^\circ$.

Image at the near point

The eyepiece may instead be adjusted so the final image forms at the least distance of distinct vision $D$ (taken as $25\ \text{cm}$ for a normal eye). Treating the eyepiece with the lens formula and the Cartesian sign convention, the magnifying power becomes

$$m = -\dfrac{f_o}{f_e}\left(1 + \dfrac{f_e}{D}\right)$$

The negative sign records that the final image is inverted. The extra factor $\left(1 + f_e/D\right)$ exceeds unity, so the magnitude at the near point is always larger than the normal-adjustment value $f_o/f_e$. In numerical work the magnitude is reported, the sign being a statement about image orientation rather than size.

Why large aperture and focal length

Two distinct considerations push the objective toward a large diameter. The light-gathering power depends on the area of the objective; with a larger diameter, fainter objects can be observed. The resolving power — the ability to observe two objects in nearly the same direction as distinct — also improves with the diameter. A large focal length, separately, raises the magnifying power $f_o/f_e$. Hence NCERT states the desirable aim in optical telescopes is an objective of large diameter.

The largest lens objective in use, at the Yerkes Observatory in Wisconsin, has a diameter of about $1.02\ \text{m}$ (40 inch). Such large lenses are heavy, hard to support at their rim, and expensive to make free of chromatic aberration and distortion — limitations that motivate the mirror objective.

The reflecting telescope

A reflecting telescope replaces the lens objective with a large concave mirror. Because a mirror does not refract, it introduces no chromatic aberration; a parabolic mirror is additionally free of spherical aberration, giving a sharp image. A mirror also weighs far less than a lens of equivalent optical quality and can be supported across its entire back surface rather than only at its rim, easing the mechanical problem of large apertures.

One difficulty is that the concave objective focuses light inside the tube, so the eyepiece and observer would obstruct incoming light. The Cassegrain design solves this by placing a convex secondary mirror in the converging beam; it reflects the light back through a small hole at the centre of the primary mirror, where the eyepiece is mounted. This yields a large focal length within a short tube.

The reflecting telescope's brightness is proportional to the area of the objective; the brightness ratio relative to the unaided eye is $B = D^2/d^2$, where $D$ is the objective diameter and $d$ the pupil diameter. Notable instruments include the $5.08\ \text{m}$ (200 inch) Mt. Palomar telescope in California and, in India, the $2.34\ \text{m}$ Cassegrain reflector at Kavalur, Tamil Nadu, operated by the Indian Institute of Astrophysics. The largest reflectors in the world are the pair of $10\ \text{m}$ Keck telescopes in Hawaii.

Refracting vs reflecting

Worked examples