Astronomy Principles and Practice Fourth Edition.pdf

326 Astronomical optical measurements
where t is the thickness of the base of the prism and dn/dλ is the dispersion of the glass. By considering
a prism with a base of 10 mm made from a typical glass which changes its refractive index by 0·02 over
a spectral range of 2000 Å (see figure 16.12), it can be seen, using equation (19.16), that the resolving
power is given by
10 × (0·02)
R =
2 × 10 −4
= 10 3 .
Prism spectrometers with resolving powers greater than this may be constructed by using prisms made
of glass with greater dispersions (flint glasses) and by using larger prisms or a train of prisms.
For the diffraction grating, the resolving power is given by
R = Nm (19.17)
where N is the total number of lines used across the grating and m is the order of interference. A
typical grating may have 500 lines mm −1 and, consequently, a 10 mm grating, used in second order,
would have a resolving power given by
R = 500 × 10 × 2
= 10 4 .
It will be seen immediately that, size for size, the grating gives a resolving power which is typically an
order of magnitude greater than that of the prism.
It is important to note that if full use is to be made of the dispersing element, its entrance face must
be fully illuminated by the beam from the collimator. The cone of light provided by the telescope must
also be accepted by the collimator. These conditions dictate the size and focal ratio of the collimator.
A spectrometer is, therefore, said to have a certain focal ratio and, in practice, its value should match
that of the telescope.
Any resolved spectral element within a spectrum, in effect, corresponds to an image of the
entrance aperture or slit of the spectrometer in the monochromatic colour of that region of the spectrum.
Thus, in order to achieve a given spectral resolution, it is important that the slit should be limited to a
particular size. This prevents the spectrometer accepting light which lies outside a defined small angle.
For a telescope–spectrometer system to be matched efficiently, it is important that the star image in
the focal plane of a telescope should be smaller than or equal in size to the slit width so that all the
light collected by the telescope is accepted by the spectrometer. Because of the practical limits to the
sizes of the conventional dispersing elements, high resolving power can only be maintained by having
small slits which, in some cases, are much smaller than the star images they are trying to accept. For
example, it has been estimated that the slit of the spectrograph at the coudé focus of the 200-in (5·08 m)
Hale telescope at Mt Palomar accepts only about 5–10% of the light in any star image.
There are, however, other spectrometric techniques which provide high spectral resolution with
the advantage of having more reasonable angular acceptances. Such an instrument is based on the
Fabry–Pérot interferometer. This comprises a pair of very reflective circular plates with a controlled
space, the plate surfaces being parallel. Multiple beam interference occurs within the cavity so that
only radiation of selected wavelengths is allowed to pass (see figure 19.13). Those allowed to pass
correspond to the gap optical path length being at an integer number of wavelengths, i.e.
mλ = 2nτ cos θ
where m is an integer (the order of interference), τ is the physical space between the plates, n its
refractive index and θ the angle of the beam inside the cavity (see figure 19.13).