The telescope
The refracting telescope
The Moon is huge, but it is so far away that it makes an angle of only about half a degree at your eye. A telescope makes that angle bigger.
A refracting telescope has two convex lenses: a large objective of long focal length fₒ, and a small eyepiece of short focal length fₑ.
Light from a distant object arrives as parallel rays, at a small angle α to the axis. The objective brings them to a focus in its focal plane, forming a real, inverted image A′B′.
The eyepiece is placed so that this image sits at its focus. The rays then leave the eyepiece parallel, and the relaxed eye sees the final image at infinity. This is normal adjustment.
The image height is h = fₒ tan α, roughly fₒ α. Seen through the eyepiece, it makes an angle β = h/fₑ. So the magnifying power m = β/α = fₒ/fₑ.
The length of the telescope in normal adjustment is fₒ + fₑ. Example: fₒ = 100 centimetres and fₑ = 5 centimetres give m = 20, and the tube is 105 centimetres long.
If the eyepiece is moved in so the final image is at the near point, 25 centimetres, the magnification rises slightly, to (fₒ/fₑ)(1 + fₑ/D).
For high magnification, the objective needs a long focal length, and for bright images a large diameter, to collect more light. A wide objective also resolves finer detail.
But big lenses are heavy, hard to make without flaws, and sag under their own weight. The largest refracting telescope, at Yerkes in the USA, has an objective just over a metre across.
Reflecting telescopes
Example: a giant refracting telescope has an objective of focal length 15 metres. With an eyepiece of 1 centimetre, its magnifying power is 15 metres ÷ 0.01 metre = 1500.
Point it at the Moon, 3.48 × 10⁶ metres across and 3.8 × 10⁸ metres away. The Moon makes an angle α = 3.48 × 10⁶ ÷ 3.8 × 10⁸, about 9.2 × 10⁻³ radian.
The objective forms an image of diameter fₒ α = 15 × 9.2 × 10⁻³, about 0.137 metre. A 13.7 centimetre picture of the Moon, which the eyepiece then magnifies.
Modern large telescopes use a concave mirror instead of a lens as the objective. These are reflecting telescopes.
A mirror has no chromatic aberration: all colours reflect the same way. And if the mirror is made parabolic instead of spherical, it also has no spherical aberration.
A mirror can be supported over its whole back, so it does not sag. Huge mirrors are easier to make and hold than huge lenses.
One problem: the image forms in front of the mirror, inside the tube, where the observer would block the light. The Cassegrain design solves this.
A small convex secondary mirror sends the light back through a hole in the centre of the main mirror. The eyepiece sits behind the main mirror, out of the way.
The largest telescope in India, at Kavalur in Tamil Nadu, is a reflecting telescope with a mirror 2.34 metres across. The Keck telescopes in Hawaii have mirrors 10 metres wide.
Try it: a refracting telescope
- Parallel rays from a distant object at angle α meet at the objective’s focus, forming I₁, then leave the eyepiece parallel at a larger angle β.
- In normal adjustment, β/α equals fₒ/fₑ = 20 and the length is fₒ + fₑ = 105 cm.
- Try a shorter eyepiece: the magnification grows. A shorter objective: it falls.
- Press “Final image at 25 cm”. The eyepiece moves a little closer and m rises to (fₒ/fₑ)(1 + fₑ/D) = 24.
Normal adjustment
An astronomical telescope has an objective of focal length 100 cm and an eyepiece of focal length 5 cm. In normal adjustment, what are its magnifying power and length?
- m = 20, length 105 cm
- m = 20, length 95 cm
- m = 500, length 105 cm
- m = 0.05, length 105 cm
Answer
A. In normal adjustment the image by the objective is at the focus of the eyepiece, so the length is fₒ + fₑ = 100 + 5 = 105 cm, and m = fₒ/fₑ = 100/5 = 20. (0.05 is fₑ/fₒ, upside down; 500 multiplies the focal lengths.)
The Moon’s image
A telescope’s objective has a focal length of 15 m. The Moon (diameter 3.48 × 10⁶ m) is 3.8 × 10⁸ m away. What is the diameter of the image of the Moon formed by the objective?
- 13.7 cm
- 1.37 cm
- 1.37 m
- 9.2 mm
Answer
A. Angle α = 3.48 × 10⁶ / 3.8 × 10⁸ ≈ 9.16 × 10⁻³ rad. The image is in the focal plane, so its size is fₒ α = 15 × 9.16 × 10⁻³ ≈ 0.137 m = 13.7 cm. (9.2 mm is the size for a 1 m focal length.)
Summary
- Magnifying power (normal adjustment):
m = β/α = fₒ/fₑ - Length (normal adjustment):
L = fₒ + fₑ - Final image at the near point:
m = (fₒ/fₑ)(1 + fₑ/D)
- A telescope increases the angle a distant object makes at the eye.
- The objective (long fₒ, large diameter) forms a real image in its focal plane; the eyepiece (short fₑ) magnifies it.
- A large objective gathers more light and resolves finer detail.
- Reflecting telescopes use a concave (parabolic) mirror: no chromatic aberration, less spherical aberration, easier to support. The Cassegrain design uses a convex secondary mirror.