THE CASSEGRAINIAN TELESCOPE WITH SPHERICAL SECONDARY MIRROR: "The customary curves for Cassegrainian telescopes are the paraboloidal primary and hyperboloidal secondary. As far as I know nobody has ever raised the question of why these curves are used instead of others which might be easier to produce, or even better. Probably the first Cassegrainian telescope was a revamped Newtonian, and the secondary was just naturally made to fit.

"Schwartzschild proposed the aplanatic curves in 1905 and these amount to overcorrecting both mirrors. Apart from increasing the difficulties of construction, these curves are a waste of time and effort, as far as visual instruments are concerned. As a matter of fact, there is no very good reason for not going the other way and removing some of the correction from both mirrors. In changing the figure of the secondary to a sphere, only a fraction of the correction must be removed from the primary mirror. In the usual method of testing a Cassegrainian telescope with a flat, when the secondary is yet uncorrected the entire telescope appears overcorrected. Instead of overcorrecting the telescope as a whole by correcting the primary right up to the full parabolic form, and then going backward by changing the secondary to a hyperboloid, why not stop the correction of the primary at just the right point so that no correction at all will be necessary on the secondary? It can be easily finished up spherical by means of the King test, or by polishing the tool enough to show fringes in monochromatic light, and it will generally be impossible to tell any difference in visual observation between a telescope with these curves and any other curves.

"In Figure 1, if we place a light source at F', the focus of a Cassegrainian telescope having a spherical secondary mirror, a marginal ray will diverge and fall upon the rim of the secondary at H', from whence it is reflected to the rim of the primary mirror at H, and making an angle with the axis. If the ray is traced backward along the dotted line it will cut the axis at a distance b from the apex of the secondary mirror. The longitudinal spherical aberration is b-p. We shall designate this spherical aberration by a, and it can quite easily be shown that

(1)

where R' is the radius of curvature of the secondary mirror, r' the radius of its marginal zone, and p' the distance of focus F' from the apex of the secondary mirror.

"Now, obviously, if the primary mirror has this same amount of aberration, then the telescope will be corrected as a whole. If we designate the radius of curvature of the primary mirror by R, the radius of its marginal zone by r, and compute a from the above formula, we may write

undercorrection (2)

and the percentage of correction necessary for the primary mirror is

percentage correction (3) 

The quantities a and e are computed only for the marginal zone, and the percentage found for (3) holds for all zones of the mirror.

"If we designate the radius of any zone of the primary mirror by y, then a parabolic mirror tested at center of curvature has the knife-edge or grating movement (assuming the pinhole remains fixed)

(4)

as every amateur knows. If we want to make a Cassegrainian telescope with a spherical secondary mirror, then the knife-edge or grating motion is simply the percentage of given by formula (3), or to state it all together

(5)

and one has only to apply this correction to the primary mirror, with the assurance that the secondary mirror will fit if left spherical.

"A WORD OF CAUTION TO NONMATHEMATICAL READERS: These formulas are derived according to the conventions of analytical geometry, which is coming more and more into practice, and it is necessary to bear in mind that, after making a drawing of the-telescope such as Figure 1, R and R' are positive if their center of curvature lies to the right of their surface, and negative if to the left. Likewise, p or p' will be positive or negative according as they lie to the right or left, respectively, of the surface. As an example, p' in the figure is negative.

"I have felt for a long time that amateurs have been overlooking a good bet with these 'elliptical Cassegrainians', as the chief difficulty in producing a compound telescope has always been the satisfactory figuring of the secondary mirror. As a matter of fact, this is about the only real difficulty given by Porter on page 62 of ATM as a basis of his 'and why not to'. I might add, however, that the idea was suggested to a Tacoma amateur some time after this theory was worked out by a member of the Lick observatory, and upon investigation, I understand further that Dr. J. A. Anderson suggested it two or three years ago, and expressed the thought that he would like to see it tried.

"There is another interesting possibility. The primary mirror formed by this formula will not be exactly an ellipsoid, but Schwartzschild has shown that mirrors of reasonable sizes conforming to these kinds of formulas differ from ellipsoids only by very small fractions of a wavelength. The direct focal test could therefore be applied, and a complete Cassegrainian telescope produced with no necessity for zonal measurements whatever, making the job on the whole much easier than the ordinary Newtonian job."

THE idea of a Cass of this unusual type turns out, on investigation, to be older than is mentioned above; evidently it is quite aged, in fact. Kirkham, however, hit on it independently. This happens again and again. With thousands of new and enthusiastic recruits to telescoptics, it is not even remarkable that many a man has a bright idea that someone had in his grandfather's day. Your scribe recalled that Dall, of England, proposed a similar idea in the Journal of the British Astronomical Association in 1932-a monthly which, by the way, often contains most interesting discussions of telescoptical subjects-and that he had also discussed it in private letters in the same year. But he never published the details. However, as an amenity, since Dall was known to have made three such Cassegrainians of this type, Kirkham's paper was shown to Dall before publication, and Dall at once voluntarily surrendered any special claim to ownership. Kirkham, when then shown Dall's early correspondence, asked that mention be made that the method suggested in his final paragraph is the same as the one Dall had described in that correspondence with your scribe. This kind of mutual courtesy is a pleasing relief, considering that many a telescoptician, on learning that another had been trespassing on one of his pet preserves without by yourleave, has growled and given the other fellow the dog-eye. Verily, your telescolumnist knows, because he has so often stood at the focal point of numerous tilts of this kind, this being the exact point at which the brickbats cross.

SECOND item by Kirkham is also introduced by informal comments from his letter of transmittal: "Let me join with others in the opinion that one of the best things that has yet hit amateur telescope making is the RFT and, in connection with this, altogether apart from the spherical Cass problem, equation 2 of the above Cass paper can be used for deriving the proper amount of overcorrection for an RFT mirror to compensate for spherical aberration of the eyepiece, which is very considerable at f/3 or f/4. A Huygenian ocular has an unpermissible amount of aberration when you get above f/10. A Ramsden begins to gum up the image at about f/5, and therefore an RFT cannot possibly work its best unless something is done about the spherical aberration of the eyepiece. It amounts in some cases to 50 percent or more overcorrection."

EYEPIECES FOR THE RFT: "In the past, it has seldom been necessary to apply eyepieces to objectives of greater aperture ratio than about f/8. A Huygenian eyepiece of 1" focal length has about .02" longitudinal spherical aberration at f/8-quite enough to interfere with the performance of a first class, fully corrected mirror. Most of us have avoided this difficulty simply by asserting that Huygenian eyepieces are no good with reflecting telescopes, and using Ramsden eyepieces. A characteristic Ramsden eyepiece has been found to have about .006" aberration at f/8, which in practice is altogether negligible.

"The commonly accepted aperture ratio for RFT seems to be f/4 (for reflectors), and it now becomes necessary to see what happens to the eyepiece aberrations when called upon to handle such wide cones of rays. While eyepieces can be designed to have no spherical aberration at all, it is never practical to do so. The whole beauty of the RFT plan is in obtaining a wide field as full of stars as possible. Eyepieces have other aberrations much greater than spherical aberration, which have to be kept within limits An ocular has quite a job to perform if it gives a moderately wide field of view free from excessive astigmatism, distortion, and curvature, and it is generally impossible to make the eyepiece free from spherical aberration and at the same time keep these other errors within workable limits. The things one would have to do to an ocular to free it from spherical aberration and some kinds of color errors are directly opposed to what is needed to keep the astigmatic and other color errors within the desired limits. When the objective is of very short focus, it becomes absolutely necessary for it to be overcorrected an amount corresponding to the eyepiece longitudinal aberration if most satisfactory results are to be obtained.

"It can be shown by a somewhat lengthy calculation that an ordinary Ramsden eyepiece of 1.14" focus when used with a fully corrected parabolic mirror of f/4 focal ratio, has almost exactly the maximum permissible amount of spherical aberration, so that a star right in the center of the field will not appear to be larger than if the error were absent. We must not jump to the conclusion, however, that the state of affairs is therefore satisfactory. There are six other major errors which when taken all together, will run the total error of the system way over the limit, especially in the edges of the field. It is easy to see that, if the one error uses up every bit of available 'tolerance' for a star most favorably situated, there is no hope at all for stars to look like more than dirty spots at a distance of 15 degrees to 20 degrees from the center. On the other hand, it is quite simple to overcorrect the primary mirror sufficiently to bring this large central error to zero, so that we have all the tolerances left to take care of the other aberrations about some of which absolutely nothing can be done.

"In most cases, this method gives a rather close approximation to an 'aplanatic telescope'. Since coma is the principal part of the aberrations of a reflector after spherical aberration is eliminated, the plan increases the general usefulness of the system as a whole far beyond what could be hoped for if it were possible or practical to obtain eyepieces free from spherical aberration. The faults of the eyepieces indeed turn out to be a blessing in disguise.

"In order to figure a mirror so that it will have a certain predetermined amount of spherical aberration, it is necessary to know how much the knife-edge must move in measuring the different zones. It is not difficult to show that, calling the radius of curvature of the mirror R, the desired amount of aberration a, and the diameter of the mirror d, the amount of overcorrection necessary is

If we denote the radius of the zone we wish to measure by y, then the knife-edge shift for this zone is

instead of the old formula

which corresponds to a paraboloid. In these formulas, the aberration a is considered positive when the marginal rays come to a focus closer to the eye than the axial rays; t that is, when the Ronchi bands behave, in testing the eyepiece by the method about to be given, like those of an oblate spheroid.

"In practice, a is the amount of aberration of the eyepiece to a bundle of rays which pass straight through it. If first class eyepieces are purchased from a reliable dealer (at a fair price!) he may in some cases be able to furnish the telescope maker with exact data as to the amount of aberration of his oculars. In case of oculars of known construction it can he found very exactly by ray tracing, using the formulas given in many optical books. For Ramsden eyepieces of about the usual form, having two lenses separated about three-quarters of the focal length of the entire ocular neither of which is a compound lens, a sufficiently exact working guess is where , is the focal length of the eyepiece, provided that the objective has a focal ratio of f/4. If the particular Ramsden eyepiece differs much from the standard ones described in ATM by Hastings and others, the formula will be quite worthless.

"The aberration can be found quite accurately by experimental methods requiring no special equipment not already possessed by the mirror maker. Figure 2 represents a Ramsden eyepiece being tested for a. A stop S₂ is made exactly the size of the exit pupil of the completed 'scope, and placed exactly in the center of the eye-lens as shown. It is then turned toward a distant streetlight, and a Ronchi grating placed at F, with the eye just behind. A round spot of light will be seen, just as in testing a mirror at center of curvature, and it will be crossed by the Ronchi bands in exactly the usual way. It is more than a safe bet that the eyepiece will have positive aberration, and the appearance will be that of an oblate spheroid. It is necessary only to measure the aberration by King's test, viz., by finding alternately where the grating must be placed to make V in B, Figure 2, equal to W in A, Figure 2. The longitudinal shift necessary is equal to a for that particular eyepiece and exit-pupil. If a couple of thin wires are stretched equidistant from the center, across the stop S₂, they will be plainly visible like ruled lines across the spot of light, and will be very convenient reference marks to go by in judging the position of the Ronchi bands. In the first place, adjust the grating to make the bands coincide with the wires in the center (V), and then move the grating longitudinally until they coincide at the edges (W). The shift will be equal to a. The wires should not be more than one-third the diameter of the stop S₂ apart, preferably somewhat less.

"What kind of eyepiece is recommended for RFT use? Nothing will be found to have much advantage, if any, over the Ramsden ocular of about the usual design, but by all means obtain a first class one. The only really outstanding fault of these oculars is the color error. This can be practically eliminated by several methods without changing the general design of the Ramsden ocular very much, since it is so satisfactory in other respects. The writer has found that the substitution of special glasses instead of the original crown glass can be made to reduce the error almost to the vanishing point. The Köning ocular is another entirely satisfactory revision of the Ramsden. It is sold under many names by different makers, and is often designated as the 'Achromatic Ramsden'. It has a compound eye-lens made of crown and flint glasses of almost the same refractive index but widely different dispersions. It is not to be confused with the ordinary Kellner ocular, which is not at all satisfactory. All these eyepieces have about the same central spherical aberrations and longitudinal chromatic aberration, however.

"Most first-rate manufacturers make oculars of truly wonderful character, not all of which are satisfactory for use with very short focus objectives."

LAST October this magazine published an outstanding article in which most human beings were divided into extraverts and introverts, the extravert being a practical-minded, direct-acting, go-getter of a fellow who wants results as quickly as possible and is impatient of theorizing that lacks application, while the introvert finds his fun in thinking. Amateur telescope makers can similarly be divided into extravert and introvert types. The extravert mainly wants a telescope and is impatient of any kind of foolishness like stopping to philosophize about the whichness of the why as he proceeds to make one, while the introvert, who is probably in the majority in this hobby, largely regards the whole job as a good excuse for all sorts of brain racking about such intrinsically interesting things as the following, by Kirkham:

"The amount of glass to be removed in paraboloidizing a mirror varies with the cube of the diameter of the mirror, if the focal ratio is the same in both cases. You have to tear out eight times as much glass in figuring a 12" mirror as you do in figuring a 6" mirror, provided that the figuring is done in the center zones of the mirror, and nothing is removed from the edge. Furthermore, the amount of glass removed varies inversely as the cube of the f number. For example, you have to remove 2x2x2 = 8 times as much glass in figuring an f/4 mirror, as in figuring an f/8 mirror. It's quite interesting to note that you have to scrape off 64 times as much glass to figure a f/4, 12" mirror as you do when you figure a 6" f/8 mirror.

"When you figure a 6" mirror, of 48" focus, you have to remove .000,108 cubic inch of glass. If all of this was in one chunk, it would make a cube .0477 inch square, or about 1/21", which looks like a lot more than one would imagine. There is another way to figure a mirror; that is, to leave the middle untouched and remove glass from the outer zones. This method involves removing exactly twice as much glass. The size of the representative cube of glass (1/21") varies directly as the diameter of the mirror, and inversely as the f number. Hence, a 6", f/4 mirror would have a cube of glass 2/21" thick removed in figuring. A 12", f/4 mirror would have a cube 4/21" removed."

With the above, "Kirk" included a formal proof, but this contains too many integration signs and too many formulas to present here; interested readers may, however, borrow it from ye scribe.

In a private letter Kirkham writes: "The average amateur will do well to forget about 'super-colossal-ultra-hyper' oculars and get a real, grade A, plain Ramsden. And then if someone feels he simply must go high hat, let him go after an achromatized Ramsden and, incidentally, prepare to part with some real kale. It takes a real optical designer just to pick out a suitable ocular for special purposes like RFT."

STELLAFANE convention, Sat Aug 6.

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