## F

giving with (3.147)

Consider the first value Ra = —0.4142 from the negative root of (3.150) and (3.153). This is normal in that |Ra| < 1, implying that the secondary is smaller than the primary and that the primary is concave. From (3.150)

Since f is negative, f must be positive and from (3.147)

As must be the case, d 1 is negative like f and ^| > |f |. The telescope is then a Gregory form with a real primary image. However, the secondary must be convex with the same curvature as the primary to satisfy Siv = 0, giving a virtual final image and the positive final f required by (3.154). The form is therefore useless for a 2-mirror telescope since it does not form a real image.

Consider now the positive value Ra = +2.4142 arising from the positive root of (3.150) and (3.152). Ra > 1 implies that the secondary is larger than the primary, which is now convex with f positive, i.e. no intermediate real image. From (3.150)

implying that f is positive with f positive. From (3.147), it follows that d 1 = — 2f' = — f 72 (3.157)

and is the required negative quantity. Figure 3.8 shows the resulting system. To give Siv = 0, the secondary must be concave with the same curvature as the primary. The magnification m2 of the secondary is given from (3.148) and (3.157) by m2 = +0.7071 (3.158)

Fig. 3.8. Karl Schwarzschild's first impractical telescope solution fulfilling four Seidel conditions [3.1]

The "axial obstruction ratio" was given by (3.152) for the positive root as

meaning that, without field and with the stop at the primary, the hole in M2 produces a linear obstruction of 1/2.4142 on M2 or (Ra)2 = 0.4142. Since the final image is formed behind M1, there is also a linear obstruction ratio (Ra) 1 from the hole in M1 given by

giving from (2.75)

the same value as for (Ra)2. Since Schwarzschild was interested in an angular field of several degrees, in order to profit from the excellent theoretical correction potential, the actual obstruction seemed prohibitive. Also, a secondary larger than the primary at once excludes the solution in practice for telescopes of significant size. Its physical length L = 2.4142/ is also unfavourable. However, it does have the characteristic, essential for Schwarzschild at the time of being "fast"; a merit which explains its use in spectrograph cameras.

Fig. 3.9. Karl Schwarzschild's original aplanatic telescope (1905) [3.1][3.13]

Schwarzschild's conclusion that no practical 2-mirror telescope solution existed correcting four Seidel conditions was of fundamental importance in the development of reflecting telescope theory.

Having abandoned the aim of a 2-mirror telescope system correcting four Seidel aberrations, Schwarzschild decided to seek a solution based on the Schwarzschild theorem (§ 3.2.6.1) in which 2 aspheric mirrors could always correct the two conditions E S/ and E S//. He accepted the presence of finite astigmatism and field curvature, but imposed the condition that their optimum combination should give a flat field. He sought a system of a "speed" (f/3.0) which was quite revolutionary at the time, bearing in mind that a primary of f/5 was considered "fast" and that Cassegrain foci were rarely faster than f/15. He therefore considered a number of Cassegrain systems using a concave secondary 3 to achieve the desired high relative aperture of f/3.0, a speed becoming available in photographic objectives. This had two consequences: the final image position lay between the mirrors (Fig. 3.9) and the

3 Because of the concave secondary, the Schwarzschild aplanatic telescope is sometimes referred to as a Gregory form. This is quite wrong as there is no real intermediate image: it is simply a Cassegrain telescope with |m2| < 1.

magnification of the secondary m2 was in the range 0 > m2 > —1. The relatively short focal length implied that existing plate sizes could cover a field of several degrees and thereby fully exploit optimum field correction. After a careful analysis of the obstruction and vignetting aspects, including those of the plateholder, Schwarzschild proposed the system of Fig. 3.9 which bears his name.

Complete data, as given by Schwarzschild and normalized to f = +1, are given in Table 3.7. The aplanatic condition is defined from the given geometry by Eqs. (3.109) and (3.114) for (b^Apian and (bs 1 )Apian respectively. The secondary magnification m2 is +1/ — 2.5 = —0.4 from (2.55), while L = +0.5 from (2.75) and Ra = +0.5 from (2.72). The power of the secondary is defined by (2.90) as +0.83333.

 f ' = +1 Di =