Telescopes

A microscope looks at small things close up; a telescope looks at things far away, like a planet, a ship or a cricketer across the ground. The trick is the same pair of lenses, used the other way round.

The astronomical telescope#

Two converging lenses share an axis. The objective faces the distant object; it has a large aperture (to gather light) and a long focal length f₀. The eyepiece has a small aperture and short focal length fₑ. The eyepiece tube slides inside the objective tube to change their separation.

Light from a distant object forms a real, inverted image in the objective’s focal plane. The eyepiece magnifies it. In normal adjustment, that first image sits at the eyepiece’s focus and the final image is at infinity; push the eyepiece closer and the final image comes to the near point, for maximum magnification.

Magnifying power

The object is so far away that it subtends practically the same angle α at the objective as at the unaided eye. If h′ is the height of the first image, α ≈ h′/f₀, and in normal adjustment the final image subtends β ≈ h′/fₑ. β and α are on opposite sides of the axis (the image is inverted), so

With the final image at the near point, the lens formula puts the first image fₑD/(fₑ + D) from the eyepiece, and

The terrestrial telescope#

An upside-down view is fine for stars but odd for a cricket match: the players would seem to hang from the field. A terrestrial telescope adds a converging erecting lens of focal length f between the objective and the eyepiece, placed 2f beyond the first image. It forms an inverted, same-size copy of that image a further 2f along (magnification −1), which the eyepiece then views.

The Galilean telescope#

Galileo used a diverging eyepiece. It is placed so that it intercepts the converging rays from the objective before they form the first image; that would-be image acts as a virtual object for the eyepiece. The final image is erect.

With the final image at the near point instead, $m = \dfrac{f_0}{|f_e|}\left(1 - \dfrac{|f_e|}{D}\right)$ and the tube is a little longer. Galilean telescopes are short and give upright images, which is why opera glasses use them.