Astronomical telescopes: refracting and reflecting

The refracting telescope

A telescope gives an angular magnification of distant objects. An astronomical refracting telescope has an objective of long focal length and large aperture, and an eyepiece of short focal length. Light from a distant star forms a real, inverted image in the focal plane of the objective. The eyepiece magnifies this image. In normal adjustment, the image lies at the focus of the eyepiece, so the final image is at infinity and the eye is relaxed. The magnifying power is m = β/α. Here α is the angle the object subtends at the objective, and β is the angle the final image subtends at the eye. The final image is inverted, which does not matter for stars. A terrestrial telescope adds a pair of inverting lenses to give an erect image.

Derivation: magnifying power of a refracting telescope

1. 1
   A distant object subtends a small angle α at the objective. The telescope is tiny compared with this distance, so α is also the angle at the unaided eye.
2. 2
   The objective forms an image of height h in its focal plane: α ≈ tan α = h/fₒ.
3. 3
   In normal adjustment this image lies at the focus of the eyepiece. The final image at infinity subtends β ≈ tan β = h/fₑ.
4. 4★
   Magnifying power m = β/α = (h/fₑ) ÷ (h/fₒ) = fₒ/fₑ.
5. 5
   The distance between the two lenses is the tube length, L = fₒ + fₑ.
6. 6
   If the final image is at the near point, the eyepiece works as a simple microscope, and m = (fₒ/fₑ)(1 + fₑ/D). (Your textbook does not derive this case, but its exercises use it.)
7. 7
   So a large m needs a long fₒ and a short fₑ.

★ key step

How to draw: refracting telescope in normal adjustment

1. 1
   Draw a horizontal axis. Draw a large convex lens on the left (objective) and a small convex lens on the right (eyepiece).
2. 2
   Keep them fₒ + fₑ apart. Mark a point F between them, fₒ from the objective and fₑ from the eyepiece: the two foci coincide here.
3. 3
   From the far left, draw three parallel rays from the top of a distant object, sloping down at a small angle α to the axis.
4. 4
   Draw the middle ray through the centre of the objective, undeviated. Bend the other two so that all three meet at one point below F, in the focal plane.
5. 5
   Draw the real, inverted image as a short arrow from F down to this point.
6. 6
   Continue the rays to the eyepiece. Draw the ray from the image tip through the centre of the eyepiece undeviated, and bend the others so they leave parallel to it.
7. 7
   Draw an eye on the right. Mark α at the objective and β at the eye, and write 'final image at infinity'.
8. 8
   Label objective, eyepiece, fₒ, fₑ, F and the image, and put arrows on all rays.

How to draw: Cassegrain reflecting telescope

1. 1
   Draw a large concave mirror on the right, facing left, with a small hole at its centre on the axis. This is the primary mirror (the objective).
2. 2
   Draw parallel rays from a distant object coming from the left and striking the primary mirror.
3. 3
   Reflect them towards the primary's focus. Before they reach it, draw a small convex mirror on the axis, facing the primary. This is the secondary mirror.
4. 4
   Reflect the rays from the secondary mirror back towards the primary, so that they pass through the hole.
5. 5
   Show the rays meeting at an image just behind the hole. Draw the eyepiece behind this image, and an eye beyond it.
6. 6
   Label: primary concave mirror (objective), secondary convex mirror, hole, eyepiece, eye, and light from a distant star.

Refracting versus reflecting telescope

One drawback of a reflecting telescope: the secondary mirror blocks a little of the incoming light.

Worked example: telescopes