Telescopes

Bringing a distant object closer

A microscope looks at a small object that can be brought near. A telescope looks at a large object that cannot be brought near: the Moon, a planet, a bird on a far tree. Its job is the same, to enlarge the angle at the eye. Here \(\alpha\) is the angle the object makes at the unaided eye where it stands, far away.

The astronomical telescope has two convex lenses. The objective has a long focal length \(f_o\) and a wide aperture. The eyepiece has a short focal length \(f_e\). Light from a distant object arrives as parallel rays, so the objective forms a real, inverted image \(I_1\) at its focus. The eyepiece then views \(I_1\) as a magnifying glass.

In normal adjustment the final image is at infinity, so the eye stays relaxed. The eyepiece sends out parallel rays only when \(I_1\) lies at its own focus. So the focus of the objective and the focus of the eyepiece coincide, as in Fig 5.1. The two lenses are then \(f_o + f_e\) apart.

Take \(f_o = 100\,\text{cm}\) and \(f_e = 5\,\text{cm}\). In normal adjustment \(M = 100/5 = 20\), and the telescope is \(105\,\text{cm}\) long. The Moon, which makes \(0.5°\) at the eye, then makes \(20 \times 0.5° = 10°\) through the telescope. Its image is inverted, which does not matter for the Moon. This book quotes a telescope's \(M\) as a size, as the NCERT does, and says in words that the image is inverted.

The final image can also be placed at \(D\), for the largest view. Then the eyepiece acts as a simple microscope with the image at \(D\), and it enlarges the angle \(1 + f_e/D\) times more. For the same telescope, \(M = 20 \times (1 + 5/25) = 24\). The eyepiece object distance is now \(|u_e| = 25 \times 5/30 = 4.17\,\text{cm}\), so the length is \(100 + 4.17 = 104.2\,\text{cm}\).

A large magnifying power needs a long objective and a short eyepiece. The objective is also made wide, for two reasons. A wider lens gathers more light, so faint stars become visible. A wider lens also shows finer detail: two close stars appear as two, not one. The chapter on diffraction measures this second gain as the resolving power.

An upright view on land

An upside-down Moon is no trouble, but an upside-down ship or bird is useless. A terrestrial telescope turns the image upright with a third convex lens, the erecting lens, between the objective and the eyepiece. It is placed so that \(I_1\) lies at twice its focal length. It then forms a real image of \(I_1\) at twice its focal length on the other side. That image has the same size as \(I_1\) but is upright, and the eyepiece views it.

The erecting lens does not change the magnifying power, because its magnification is \(-1\). It only turns the image over. But it adds \(4f\) to the length. With an erecting lens of \(5\,\text{cm}\), the telescope of \(105\,\text{cm}\) grows by \(20\,\text{cm}\), to \(125\,\text{cm}\).

A long tube is awkward to hold. Prism binoculars solve both problems with two right-angled glass prisms in each half. The prisms reflect the light by total internal reflection, as the chapter on Refraction showed. Each prism turns the light back on itself, so the long path folds into a short body. The two prisms together also turn the image upright and right way round. A pair marked "8 × 42" magnifies eight times, with objectives \(42\,\text{mm}\) wide.

Mirrors instead of lenses

Astronomers want wide objectives, but a very wide lens is hard to make and to use. Its glass must be flawless all through, and both faces must be ground true. It focuses colours apart, as the chapter on lenses showed. Worst, a lens can be held only by its rim, so a large one sags under its own weight and loses its shape.

A reflecting telescope uses a concave mirror as its objective, and each fault disappears. A mirror reflects all colours alike, so it has no chromatic aberration. A mirror ground to a paraboloid brings all parallel rays to one focus, with no spherical aberration. Light does not pass through the glass, so only one surface must be perfect. And the mirror can be supported over its whole back.

The mirror sends the light back towards the sky, so the focus lies inside the incoming beam. An observer's head there would block the light. The Cassegrain telescope solves this with a second, small convex mirror just in front of the focus. It reflects the light back through a hole in the centre of the large mirror, to an eyepiece or camera behind it. Fig 5.2 traces the path.

The folded path is the second gift of the Cassegrain design. The convex secondary makes the converging beam spread out more slowly. So the light behaves as if it came from a mirror of much longer focal length. In Fig 5.2 the primary has \(f = 40\,\text{cm}\), but the secondary stretches this three times, to an effective \(120\,\text{cm}\), in a tube shorter than \(40\,\text{cm}\).