1.7 Summary of Section 1 and questions
Converging lenses or mirrors cause parallel beams of light to be brought to a focus at the focal point, situated at a distance of one focal length beyond the lens or one focal length in front of the mirror. Diverging lenses or mirrors cause parallel beams of light to diverge as if emanating from the focal point of the lens or mirror. Light paths are reversible, so a converging lens or mirror may also act as a collimator and produce a parallel beam of light.
The simplest astronomical telescopes are refracting telescopes comprising either one converging lens and one diverging lens (Galilean telescope), or two converging lenses (Keplerian telescope). The effectiveness of refracting telescopes is limited by the problems involved in constructing large lenses, and their spherical and chromatic aberrations which are, to some extent, unavoidable.
Reflecting telescopes, such as the Newtonian and Cassegrain designs, make use of a curved (concave) objective (primary) mirror to focus the incoming light. Reflecting telescopes are free from chromatic aberrations. Spherical aberrations can also be greatly reduced by using a paraboloidal mirror or a Schmidt correcting plate.
Large-diameter reflecting telescopes are easier to construct than similar sized refractors. Also, by using the Cassegrain design, a long focal length (and hence high angular magnification) can be contained in a relatively short instrument.
When reflecting telescopes are used with photographic or electronic detectors, the eyepiece is removed, and sometimes so also is the secondary mirror. This removes the aberrations and absorption losses that are due to these components and allows a real image to fall directly onto the light-sensitive surface of the detector.
The main parameters of an optical telescope are its light-gathering power, its field-of-view, its angular magnification or image scale and its limit of angular resolution.
The angular size of the point spread function of a telescope can be used to quantify the astronomical seeing. The technique of adaptive optics can compensate for the effects of atmospheric turbulence and produce images whose PSFs are close to being diffraction-limited.
A telescope may have an alt-azimuth or equatorial mounting. The former is less complex to construct, but with the latter it is simpler to point and drive a telescope.
Summarise how the following characteristics of a visual telescope
(i) light-gathering power,
(iii) angular magnification,
(iv) limit of angular resolution,
depend on the aperture Do and the focal length fo of its objective lens (for a given eyepiece of focal length fe).
(i) The light-gathering power is proportional to Do2, but independent of fo.
(ii) The field-of-view is inversely proportional to fo, and independent of Do, although it does depend on the diameter of the eyepiece field stop.
(iii) The angular magnification is proportional to fo, but independent of Do.
(iv) The limit of angular resolution is inversely proportional to Do and independent of fo, but note that it varies with wavelength.
(a) Calculate the ratio of the light-gathering power of a reflecting telescope of diameter Do = 5.0 m to that of a refractor of diameter 1.0 m (neglect losses of light, mentioned in the text).
(b) Compare the (theoretical) limits of angular resolution of these two telescopes (at the same wavelength).
(a) Light-gathering power is proportional to (aperture diameter)2, so the ratio of light-gathering powers for the two telescopes is (5.0/1.0)2 = 25.
(b) The (theoretical) limit of angular resolution is inversely proportional to the aperture of the objective lens or objective mirror. Thus, a telescope with Do = 5 m can theoretically resolve two stars with an angular separation five times smaller than a telescope with Do = 1 m (neglecting air turbulence and aberrations). In practice, of course, for ground-based telescopes, atmospheric seeing is usually the limiting factor.
(a) The atmospheric seeing at a particular observatory site is 1 arcsecond (1″). What is the aperture of a diffraction-limited telescope (at a wavelength of 485 nm) which would have a resolving power equivalent to this seeing?
(b) Why then do you think that astronomers build such large and expensive telescopes for use in ground-based observations?
(a) The aperture of a diffraction-limited telescope that would have a resolving power of 1″ at a wavelength of 485 nm is given by
Now 1″ is equivalent to /(180 × 3600) radians = 4.85 × 10−6 radians. So
A diffraction-limited telescope with an aperture diameter of only 12 cm would therefore have an angular resolution of 1″ when operating at a wavelength of 485 nm (i.e. in the blue part of the visible spectrum).
(b) Although a large (and expensive) telescope will have a better (theoretical) limit of angular resolution than one only 12 cm in diameter, in practice its resolution is limited by atmospheric seeing. The main reason that very large ground-based telescopes are built is to increase the available light-gathering power.
List the important advantages and disadvantages of reflecting telescopes compared to refracting telescopes.
Some advantages of reflecting telescopes over refractors are:
(i) larger apertures are possible and hence higher light-gathering power and better angular resolution are achievable;
(ii) a Cassegrain telescope can have a relatively long focal length within a short tube and hence higher angular magnification can be achieved;
(iii) reflectors are easier to manufacture;
(iv) reflectors are not subject to chromatic aberration and we may reduce spherical aberration by using a paraboloidal mirror or a Schmidt correcting plate.
Some disadvantages of reflecting telescopes are:
(i) they have higher losses of intensity through absorption;
(ii) there is a gradual deterioration of the reflecting surfaces with age.
(a) What is the Schmidt correcting plate and how does it improve the performance of a reflecting telescope?
(b) Draw a diagram illustrating how a Cassegrain telescope equipped with a Schmidt correcting plate focuses light from a distant object.
(a) The Schmidt correcting plate is a transparent glass plate built into the ‘mouth’ of a reflecting telescope. Its shape is calculated so as to introduce some differential refraction to various parts of the incoming wavefront, in such a way as to compensate for the spherical aberration of the main mirror. The result is good resolution over a significantly wider field-of-view.
(b) The important point to note for a Schmidt–Cassegrain telescope is that the light passes through the Schmidt plate first, and then reflects off the primary (objective) mirror. The remainder of the telescope is the same as the normal Cassegrain type. The overall design is shown in Figure 14.
(a) Calculate the image scale in the focal plane of a 300 mm diameter telescope whose optical system is stated as F/10.
(b) The angular diameter of the planet Mars varies from about 14″ to 25″ depending on its distance from the Earth. Calculate how large the image would be in the focal plane of a 300 mm diameter, F/10 telescope at its closest and furthest.
(a) The image scale is determined by just one value: the focal length, f, of the telescope. The f-number, 10, is the focal length divided by the diameter, i.e. f/300 mm = 10, so the focal length must be f = 3000 mm. The image scale is then I/(arcsec mm−1) = 206 265/(f/mm) = 206 265/3000 = 69 (to two significant figures), i.e. the image scale is 69 arcsec mm−1.
(b) If 69 arcsec covers 1 mm, then 14 arcsec will cover
14 arcsec/69 arcsec mm−1 = 0.20 mm. Similarly, 25 arcsec will cover
25 arcsec/69 arcsec mm−1 = 0.36 mm.