Evolution of the Telescope

Abstract

Conic sections were first “discovered” by Menaechmus (380–320 bc) and fully studied by Apollonius of Perga (died ca. 190 bc), a contemporary of Archimedes. He introduced the common names ellipse, parabola and hyperbola for the three conic sections that serve as the defining curves for reflectors of electromagnetic radiation. Pappus of Alexandria (~290–350 CE) proved that a point on a conic exhibits a constant ratio of its distance to a given point and to a given line. This constant is known as the eccentricity of the conic; the fixed point is the focus and the line is called directrix.

Addendum: Geometry of Optical Configuration

In this addendum, we present the geometry of the Cassegrain and Gregory reflector antenna with an emphasis on the optical parameters that dominate the structural and mechanical design (see also Baars 2007) . Once the diameter has been selected, the focal length is the only remaining free parameter for primary focus operation. The great majority of radio telescopes have a primary f-ratio between 0.3 and 0.5 with extremes of 0.25 and 0.8. Since the early 1970s, preference has been given to dual-reflector geometries in the form of a Cassegrain or Gregorian optical layout. The dual-reflector configuration offers flexibility in the choice of the defining parameters and the resulting geometry of the telescope. In addition to the primary diameter, three parameters fully determine the geometry of the system. Normally, these are chosen to be the primary focal length, the position of the secondary focus with respect to the vertex of the primary and the magnification , denoted m, of the dual-reflector system. This term originates with the magnification of the observed object in the first simple optical telescopes . The virtual equivalent parabola has a focal length of m times the primary focal length as illustrated in Fig. 2.11. Thus, the magnification is the ratio of the effective focal length of the Cassegrain /Gregorian system to the focal length of the primary. An important aspect of the dual-reflector geometries (Cassegrain , Gregory ) is the improved optical quality in the secondary focal plane due to the large effective focal length.

Fig. 2.11

Illustrating the concept of the equivalent paraboloid E-P. The focal length of E-P is m, the magnification , times that of the primary paraboloid f. Lines from F _C past the edge of the secondary reflector S project E-P at a distance m · f from F _C

The geometry of the dual-reflector systems is shown in Fig. 2.12. The Cassegrain employs a hyperbolic secondary reflector with one of its foci coinciding with the primary focus . The second focus of the hyperbola provides the focal point of the telescope system. Normally, the final secondary focus is located behind the vertex of the primary reflector ; the distance from the primary vertex to the secondary focus is called the back focal distance , which we denote by the symbol b. The dual-reflector system’s focal length is the distance between the two foci of the secondary reflector, denoted here by the symbol f. With F denoting the primary focal length, we have f = F + b, where b is reckoned positive if the focus is behind the primary vertex. Using the variables for the geometry given in Fig. 2.12, we summarise the geometrical relations in Table 2.1. The defining constants of the secondary reflector are parameters c and a; their ratio is called the eccentricity e of the reflector. The eccentricity is determined by the chosen magnification m (Eq. 2.9). When the Cassegrain focus is in the vertex of the parabola, we have f = F(b = 0), and it follows that m = d _P/d _S, where d _P and d _S are the diameter of the primary and secondary mirror, respectively. This simple relation does not apply to the Gregory system.

Fig. 2.12

Geometry of the Cassegrain Reflector antenna

Table 2.1 Geometrical relations of the Cassegrain /Gregory telescope

Some recent radio telescope proposals apply a Ritchey–Chrétien optical layout to exploit the coma -free focal plane that allows the use of large two-dimensional detector arrays over a large field of view. The Ritchey–Chrétien employs two hyperboloidal mirrors. The expressions for the conical constants K _P and K _S of the coma-free optical system are also given in Table 2.1. The eccentricity of the mirror is equal to the square root of the absolute value of its conic constant. From Eqs. (2.15) and (2.16), we see that the eccentricity is larger than one; hence, both surfaces are hyperboloids. The full theory of the RC optics can be found in Wilson (1996) .

The electromagnetic treatment of the reflector antenna defines the collection area of the primary reflector as the area of the plane aperture through the outer edge of the reflector. The physical area of the reflector (Eq. 2.14) is larger by an amount depending on the focal ratio, shown in Fig. 2.13. The physical area must be fabricated and hence a deep reflector will be more expensive. In practice, based on electromagnetic, structural and manufacturing arguments, most antennas and radio telescopes have a primary f-ratio between 0.3 and 0.5, with a surface “surplus” between 16% and 6%.

Fig. 2.13

Ratio of the true physical area of the reflector to the aperture area as function of the focal ratio

The choice of the primary focal ratio is a compromise between structural and electromagnetic requirements. The f-ratio determines the opening angle of the reflector (Eq. 2.7). This angle must be matched by the beam width of the feed horn in the focus. Input from the receiver engineer is necessary here. There exist a wide variety of feeds. A useful rule of thumb is that the angular width of the feed beam at the −10 dB level must equal the opening angle of the reflector (see Eq. 8.1). For a basic horn feed , this means an aperture width of about 0.5–1.0 wavelengths for f-ratios from 0.25 to 0.60. It is obvious that the horn aperture will increase with decreasing opening angle because a larger aperture creates a narrower beam, just as a larger reflector will produce a narrower antenna beam. In a dual-reflector system, the opening angle of the secondary reflector is much smaller than that of the primary reflector and the feed will be significantly larger.

The choice of location of the secondary focus determines the back focal distance . It is typically between 0.2 and 0.4 times the primary diameter to accommodate large receiver units in an equipment room behind the backup structure at for instance a Nasmyth focus . This sets the distance between the primary and secondary focus (Eq. 2.12), which we call the focal distance of the dual-reflector system. We can now choose either the diameter d _S or the opening angle Φ₀ of the subreflector with Eq. (2.10) to finalise the geometry of the telescope. The resulting full opening angle of the secondary reflector is normally of the order of 10°.