DALL OF ENGLAND was the first to make the special type of Cassegrainian telescope having a spherical secondary mirror instead of the usual hyperboloid, the primary figured to fit it. He made several, the first in 1931. But he did not stop to publish the design details. Independently, the late Alan R. Kirkham in 1938 published the design data in these columns. George P. Arnold, 519 Holmes St., State College, Pa., now contributes the following article on the testing and performance of this type.

THOSE desiring to study the Moon and planets from a comfortable and safe position would do well to consider the Cassegrainian. Properly constructed, it is the equal of any ordinary refractor of the same aperture, and is practically as good as the equivalent Newtonian. Despite its desirable features, many may have hesitated to make Cassegrainians after reading the warnings by various authorities on the difficulties involved, which are probably real enough with the conventional type. By modifying the design, however, construction is greatly simplified and testing becomes much easier.

The biggest step toward simplifying Cassegrainian was made by Dall and Kirkham, who made practical the substitution of a spherical secondary for the customary hyperboloid, thus, eliminating the need for another large optica1 surface for testing, and at the same time avoiding the difficult task of figuring the hyperboloid. To compensate for the aberration introduced by the secondary mirror, the primary is left undercorrected by an amount depending on the dimensions of the system, in accordance with the equations given by Kirkham (Sci. Am., June 1938). These equations may be combined and rearranged into a form possibly more suitable for calculation:

(1)

where N is the fractional correction, R the radius of curvature of the primary R' the radii of curvature of the secondary, p and p' the lengths from the secondary mirror to the primary and secondary foci, respectively. All quantities in the above and following equations are to be considered positive.

The primary is thus figured to an amount N instead of the usual . With mirrors of short f-ratio, testing in the usual way is still rather difficult, as small zonal irregularities may easily be overlooked and the knife-edge must be set and read with considerable accuracy. Kirkham suggested that the undercorrected primary could be considered to be an ellipse, in which case the source is at one focus, and the mirror, with the knife-edge at the other focus, should present the appearance of a sphere at the center of curvature. Dall mentioned (Sci. Am., May 1939) that he tested his 15" primary with the knife-edge at the remote focus some 120' from the mirror. Neither, however, gave a formula for finding the position of the foci. Referring to Figure 1, f and f' are the foci of the ellipse, and it can be shown that the distances a and c are very nearly

(2)

where R is the radius of curvature of the primary, N the fractional correction given by Kirkham's formula.

In practice, the source may be placed either at f at a distance of a + c from the mirror, or at f', distant a-c from it, the image will be found at the other focus. As an example, consider a 10" f/5 primary, to be 73 percent corrected:

The correction of an under-corrected mirror can be found by measuring the distances of f and f' at which the mirror appears flat, and using the first of equations (2):

(3)

Equations (2) can also be used to find the minimum distance of an artificial star for testing a Newtonian primary, by letting 100(1 - N) equal the maximum percentage tolerance of figure. For example, a 6" f/8 could be figured flat to the knife-edge with the source as close as 30' from the mirror, and would still give excellent performance.

A word should be said about the accuracy of equations (1) and (2), and the tolerances in figuring the mirrors. Kirkham's equation (1) is probably accurate enough for any system of reasonable dimensions. Trigonometrical ray tracing shows it to be correct to less than 1 percent of N for an 8" f/12 system with an f/3 primary, and again for a 10" f/19 with an f/5 primary. Equations (2), also owing their simplicity to binomial expansions, hold to the same order of precision. The tolerance of figure for a Cassegrainian primary is the same as if it were to be used for a Newtonian, and can be found from the table given by Wright in "A.T.M."; that is, the correction N, expressed in percent, may vary by the amount given in the table. The secondary may deviate from a true sphere by a quarter of a wavelength or half a fringe, provided the primary is perfectly figured, and should be tested by interference against its polished and figured tool*, or by the King test if optical glass is used. The above tolerances are for half of the Rayleigh quarter-wave limit, since the optical path error can be reduced to a quarter of its extreme value by proper focusing. Even the keenest observer can detect no departure from perfection of the image produced by a system corrected to within half of the Rayleigh limit.

If the optical parts have been figured correctly, and the system properly collimated, the only reason for any relatively inferior performance is the diffraction effect produced by the secondary. Many writers have expressed alarm at the amount of light thrown by such obstructions into the diffraction rings surrounding the central disk. Few have considered the quantitative aspect of the situation.

Two important effects are produced by the obstruction from the secondary. The most noticeable, of course, is the loss of light from the central disk and the enhancement of the first few bright rings. Assuming perfect optics, no spider diffraction (which is negligible, anyway), and that the obstruction is centered and circular, Figure 2 shows the amount of the total image light remaining in the central disk for various sizes of obstruction. (Strictly speaking this holds only when the obstruction is directly over the objective, but the effect of placing it several feet in front of the objective is practically the same.) Note that, with no obstruction, about 84 percent of the light is in the central disk. Now Conrady ("Applied Optics and Optical Design") has pointed out that, in the average case, at the Rayleigh limit this figure drops to about 68 percent. At double the limit, only about 40 percent of the light remains, but up to this point the size of the disk has not increased appreciably. Thus, considering Figure 2, a good reflector with a 25-percent-diameter obstruction would be at least as good as a refractor corrected to the Rayleigh limit. However, beyond apertures of about 5", the secondary spectrum of the ordinary f/15 refractor exceeds the Rayleigh limit for chromatic aberration. Even allowing for the decreased sensitivity of the eye for the scattered colors, it is hard to see how a large refractor could perform as well as any reflector of the same aperture; a long-focus Newtonian with its very small flat should be far superior.

The second effect is a favorable one. As the size of the obstruction increases the actual diameter of the central disk decreases, becoming about 80 percent of the normal size for a half-diameter obstruction, and about 75 percent normal size for a three-quarter obstruction. This effect serves to compensate for the increased brightness of the rings by increasing the resolving power of the instrument. In fact, with a three-quarter-diameter disk over the center of the objective, it is quite possible that the Dawes limit could be exceeded.

The net result of the diffraction is, of course, to obscure detail of very low contrast which is near the limit of resolution of the telescope, the bright rings from points at the edge of a bright area overlapping the disks from a relatively darker area. Since, for moderate sizes of obstruction, it is the first bright ring which will cause trouble, a rough indication of its effect may be obtained by considering intensity of the brightest part of the ring as compared to the intensity at the center of the disk. This is shown in Figure 3. It is seen that the secondary may be almost a third of the diameter of the primary before the ring has more than 5 percent of the intensity of the disk. At this point in the discussion physiological factors enter, but it would seem unlikely that the average person could detect any difference, in changing to a perfect refractor where the ring is about 2 percent as bright as the disk. All in all, one may conclude that the diffraction rings, so noticeable around bright stars, are not nearly so harmful as they would appear.

Thus the modified Cassegrainian, while admittedly somewhat harder to make than a long-focus Newtonian, is not a very formidable project. The secondary involves only a sphere, the easiest of optical surfaces to make; the primary, since testing involves no zonal measurements, should be actually easier than the common Newtonian primary. Combined, the two make a compact and powerful instrument whose performance will leave little to be desired.

GARAGE door becomes an alt-azimuth mounting for a Newtonian reflecting telescope in the yard of F. L. Frazine, 1016 Seventeenth Avenue North St. Petersburg, Fla. Through the door, near the outer edge, is a horizontal axle, the telescope's horizontal axis. Vertical axis then is the hinges of the door, which may be swung as a door usually is swung.

* A fluorescent lamp makes a satisfactory light source for interference testing, although the fringes are not so dark as those from a sodium or mercury lamp.

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