The magnifi cation of a telescope
5.5 The magnifi cation of a telescope
The objective of a telescope, be it lens or mirror, produces an image in its focal plane. Here a charge-coupled device (CCD) array might be placed to produce an image of the object or the image might be viewed by using an eyepiece. The eyepiece works like a magnifying glass enabling one to view the image in detail. Eyepieces have focal lengths ranging from ∼2.5 up to ∼55 mm. The magnifi cation of a telescope is simply obtained by dividing the focal length of the objective by the focal length of the eyepiece, so a 12 mm focal length eyepiece coupled with a 1200 mm focal length objective gives a magnifi cation of ⫻100. It is also possible to place a concave lens in the light path close to the eyepiece. This is referred to as a Barlow lens, and has the effect of diverging the light cone and giving the effect of having an objec- tive of greater focal length, so providing an effective increase in magnifi cation for a given primary focal length. The effective focal length is usually doubled (called ⫻2 Barlows), but can be obtained with effective magnifi cation ratios of 2.5 or even 4.
Eyepieces generally give excellent performance on axis, but image quality drops off the further off axis one observes. The designer of an eyepiece will determine the diameter of the area that can be observed with the eyepiece without signifi cant loss of detail and use a ‘fi eld stop’ to limit the observed fi eld. This then determines the area of sky that can be observed when used with a given objective.
Suppose that there is an object that lies away from the centre of the fi eld of view by an angle of θ/2 that appears right at the edge of the image encompassed by the fi eld
Observing the Universe
Figure 5.10 The fi eld of view of a telescope.
stop of the eyepiece. As Figure 5.10 shows, the image of this object will lie a distance of (f ⫻ θ/2), where θ is in radians, from the centre of the image. If the diameter of
the fi eld stop of the eyepiece is D, then D ⫽ 2 ⫻ f ⫻ θ/2 ⫽ f ⫻ θ. So, the diameter of the fi eld of view, θ, of a telescope whose focal length is f mm coupled with an eyepiece whose fi eld stop has a diameter D mm, is given by D/f rad. As an example, consider
a 1200 mm objective when used with an eyepiece whose fi eld stop is 12 mm across. Then θ is 1200/12 ⫽ 1/100 rad or 57.3/100° ⫽ 0.573°. This combination would thus nicely encompass an image of the Moon which is ∼0.5° across.
The diameter of the fi eld stop is approximately proportional to the focal length when eyepieces of similar design are used, so using a shorter focal length eyepiece gives greater magnifi cation at the expense of observing a smaller fi eld of view. The Plossl eyepiece, made up of two achromatic doublets, is very widely used as
a ‘standard’ eyepiece. Eyepieces using fi ve or more elements can allow wider fi eld stops for a given focal length and so allow a larger fi eld of view to be observed. These are called ‘wide fi eld’ eyepieces and can be very expensive.
One might well think that as the magnifi cation is increased more detail will be seen. This is not necessarily so and it is very rare that magnifi cations greater than
⫻200 will be useful. Both the eye itself and the atmosphere play a part. The detail in an image observed by the eye is limited by two things: the diffraction effects
due to the limited aperture of the eye and the density of the rods and cones in the retina. The achievable resolution is limited by the aperture of the pupil, say 3 mm, which gives an Airy disc whose size, ∆θ, is given by:
∆θ ⫽ 1.22 λ/D
⫽ 1.22 ⫻ 5.5 ⫻ 10 ⫺7 /0.003 rad ⫽ 2.24 ⫻ 10 ⫺4 rad ⫽ 2.24 ⫻ 10 ⫺4 ⫻ 57.3 ⫻ 60 arcmin ⫽ 0.77 arcmin
Introduction to Astronomy and Cosmology
Thus the eye has a theoretical resolution of about 1 arcmin (1/60th of a degree). This image falls upon the retina where it is sampled by the rods and cones. If there were an insuffi cient number of these, the resolution would be compro- mised, but evolution has ensured that this is not a limiting factor! If we have a telescope with a magnifi cation of ⫻60 it means that we could see detail about 1 arcsec across. It is rare that the atmosphere will allow us to see detail fi ner than this so one might say that ⫻60 is all that you need. In fact, spreading out the image on the retina helps so a magnifi cation of ⫻120 – ⫻200 will usually let us see more detail.