The numerator has the same form as Eq. (3.129), derived in connection with the form of the primary in a certain simple limiting geometry of the DK telescope. The roots of this numerator are the magic number t = +1.618034 in the case of m2 positive (Gregory), or —1/t = —(t — 1) = —0.618034 in the case of m2 negative (Cassegrain). With m2 = t or —1/t in the Gregory or Cassegrain cases respectively, Eq. (3.382) gives (Ra)p = 0, corresponding to a telescope in which the secondary has shrunk to a point at the prime focus. In the Gregory case, the range 0 < m2 < t gives only virtual solutions with Ha positive; while real solutions exist for t < m2 < ro. This function has a minimum of (Ra)p = —0.357 at m2 = +2.2695. In the Cassegrain case, there are two ranges with real solutions: the range —^/2 > m2 > —ro covering normal Cassegrain telescopes and the range 0 > m2 > —t/2 covering Schwarzschild-Couder types with a concave secondary without intermediate image.
If the paraxial relations referred to above are substituted in (3.382), one can derive the relation
for the condition that the final image distance Lp from the secondary shall give agreement between the CFP and the exit pupil for a given value of m2. The numerator is again the magic number equation. In the Cassegrain case, with m2 negative, it is clear that the bracket of (3.383) is greater than unity for the normal range of Cassegrain solutions: the final image must be behind the primary. This is, of course, the case for normal Cassegrain telescopes. For example, if m2 = -4, then Lp = - 1.357d1 and (Ra)p = + 0.253. These are quite typical values and validate the assumption that the CFP for RC telescopes is near the exit pupil. However, for m2 = -2, the values are Lp = - 2.5d1 and (Ra)p = + 0.556; while for m2 ^ -ro, the values are LP ^ -d1 and (RA)P ^ 0.
In the aplanatic Gregory case, Eq. (3.383) gives the factor of the bracket less than unity, resulting in positions of the final image inside the primary for real solutions. With the normal image position behind the primary, there is therefore a poorer agreement between the CFP and the exit pupil than in the Cassegrain RC case.
Finally, we consider the case of the spherical 'primary (SP) telescope. Combining (3.366) and (3.375) with (3.377) gives
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