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The Dall & Kirkham

Telescope…

By Charles Rydel. Société Astronomique de France.

Homage to Jacobs.

Introduction---

THE SHORT TELESCOPE FAMILLY constituted by two mirrors, the telescope proposed by Dall & Kirkham is the simplest to be achieved by the amateur if one excepts the Gregorian. Indeed,

whereas the Cassegrain requires a difficult to execute secondary, the Dall-Kirkam is content modestly with a spherical secondary mirror. One knows that such a surface is easy to test previously to the fringes in a concave calibre, or even passed to the Bath interferometer describes on this site. In fact the concave calibre will also be the tool that will serve to achieve the convex mirror.

In term of imagery, the price to pay is a stronger spot than the one that is finds on a Cassegrain out of axis, this defect destining this telescope a priori to planetary use and to the obscurity of weak opening like F/12 to F/15. Nevertheless, some commercial realizations exist, like the reputed Mewlon telescopes from Takahashi and a correction should be possible with a couple of lens close to the primary. Let's

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examine how compare the performances of the DK to the Cassegrain with the same opening and focal distance.

Recall on the two mirrors telescopes Schwarzschild established in 1905 the general theory of the two mirrors systems of which the Cassegrain with a parabolic primary mirror is just a peculiarity. We take here what is even better explained by J. Texereau in is well known book to which the reader will have advantage to turn.

Let's take the notations of is book. The γ coefficient is the magnification due to the secondary mirror which send back the bundle of rays through the primary and it is equal to the ratio p'/p.

Drawing of J.Texereau, extracted from the CTA.

If one chooses e, which is the distance that give the focale plane accessible and the magnification γ, one finds the position of the secondary mirror: p = (F1+e)/(γ +1) where F1 is the primary focal and therefore the distance p ' = γ.p. The radius of curvature of the secondary is given by: R2 = (2.p. γ)/(γ -1) or 2.p'/(γ -1). The diameter of the secondary for a null field of full light is given by : D2 = p. (D1/F1). To this value, it will be right to add some mm in order to cover a non null field of light, or even we will verify with OSLO the covered field. What about of the conic coefficients? One knows since Schwartchild that if one chooses the coefficient of one of the two mirrors, it is a priori possible to calculate the coefficient of the other in sort to annul spherical aberration. In this case, one says that stigmatism is achieved. So are born other combinations that the Cassegrain, for instance the Pressman-Camichel for which the primary is spherical but whose

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coma is phenomenal because of a very distorted secondary, the Ritchey-Christian telescope which annuls both spherical aberration and coma but of which the secondary mirror is more deformed that the Caseegrainian and primary is hyperbolic, and the Dall-Kirkam of which the secondary is a portion of a sphere. For this last, the aspheric coefficient of the primary is given by the expression of left and for the Cassegrain by the right-hand expression. The coma will be, relatively to the Cassegrain, (γ²+1)/2 higher. It implies that secondary magnification will be kept advantageously in the order of 3 to 3.5, with alas an increase of the central obstruction. This will damage the picture in a homogeneous way, contrary to the coma. This evidently if one wishes to pull the DK toward a generalist instrument. That says, with a gain equal to 3, the coma will only be of 4 times the coma of the Cassegrain for a same distance to the axis. With a gain of 4 times on the other hand, it will nearly be height times which is excessive for a generique use but unimportant for planetary where the field always is very small, except evidently in the case of the Moon.

Calculations of b1 for the DK & b2 for the Cassegrain (on the right)

Cassegrain versus D&K __ Primary mirror, D1 =250mm and F1 = 875mm either F/D=3.5. Distance of clearing e = 150mm. magnification γ = 4.5, F/D resulting 15.75 and focal distance 3937mm. To note that the coma of the DK is (4.5²+1)/2 times bigger, either 10,6 times. From there we calculate p= 186mm and therefore p ' = 839mm, d = 839-150 = 689mm which is the distance of the secondary to the primary. R2 = 479mm which is the radius of curvature of the secondary. The minimum diameter will be equal to 186/(F1/D1) around 54mm. One will take 60mm to cover a more ample field. The obstruction is of 60/250 = 24%. Parameters here are common to the two instruments.

The conic coefficient b1 of the primary for the DK will be -0,753 and -1 for the Cassegrainian. The conic coefficient b2 of the cassegrainian secondary will be -2.47 (0 for the DK). Let's judge the realization difficulty.

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The gap to the sphere is maximum for the point ε=(b/32)*(h4/r3). The DK will be

grossly 30% less deformed than the parabola. As the difficulty is in R3, one can consider that a mirror, shorter of 10% (F/D=3.15), has the same difficulty of realization that a parabola at F/D=3,5. Let's note that the value of ε in the case of the DK is the order of the µm. The parabola is underneath in dotted line and the DK in red. On the right, we can see the slopes on the glass. They are proportional to the conic coefficient b.

Gap between the sphere of reference on the one hand and the DK ellipsis and the Cass. parabola

Variation of the slope according to the ray.

Let's examine what these two instruments give on a à.25° half field.

On top, very close to the field of the DK, a spot of 95µ due to the coma, but the curvature of field is weaker (-450mm) and play weakly at this level on the quality of the picture. If one knew how to correct the coma (without introducing astigmatism), then the performances of the DK could be of the order of the the Cassegrain, or even better because the astigmatism is about two times lower. It is the case of the Wynne-Rosin telescope, with its parabolic primary, its spherical secondary and two correction lenses, perfectly feasible by an amateur. But let’s see what the Cassegrain will give.

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The Cassegrain essentially presents a spot due to the curvature of field that is of -200mm or 5 Dioptres. There is 100% of the energy in 12µ on the side of the field where the eye will adjust automatically if it is not the one of a retired guy, whereas the circle of diffraction is of 11µ. The side of the field is then to λ/3 P/V. The coma is not visible, remains only the astigmatism. In photographic on plane field, the spot essentially makes 35µ to the side due to field curvature. While finalizing on the middle of the field, the spot is better distributed and the result will be better too. It is as truly for the DK where the aberrations spot could be around 40 to 45µ, in fact 30-35µ because a part of the coma tail (about 1/3) is not visible. Is the telescope of Dall and Kirkham more difficult to collimater that a Cassegrain? Let's repair an injustice. In is book, J.Texereau answers yes to this question. Yet it is clear that the secondary of a Cassegrain that is hyperbolic must be at the same time coaxial to the primary and the bundle must be centred too. In other words, it is necessary to centre and adjust the tilt. In the case of a spherical secondary, a decentre can always be recovered by a tilt and vis versa because the sphere doesn't have a symmetry centre. What tell us then for example the simulations for a two mm decentring of the secondary?

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One notes that the Cassegrain is very sensitive to a secondary decentre, the coma appearing on the whole field, whereas the DK is little affected at the centre in term of imagery by this shift, contrary to what is affirmed in the Texereau book. What now for a tilt of 1°? One finds the same results globally on the Cassegrain and on the DK. The following simulation shows that a tilt of 1° of the secondary can be compensated by a decentring of 8.7mm on the axis on the DK. So if it is not sufficient, another way to judge the compared robustness of these two types of telescopes is to have a look to how varies the Strehl ratio for a decentre and a tilt of the secondary. The report of Strehl is often considered the best way to measure synthetically the quality of an optical system.

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But first, what is the Strehl ratio? Strehl was a physicist of the beginning of the XXe century. He noted that the diffraction figure changed in presence of a central obstruction or aberrations in the optical system, a part of the energy situated in the core passing in the rings. The Strehl ratio measures the reduction of the intensity in the centre of diffraction figure, relatively to the perfect diffraction figure. A perfect objective has a Strehl ratio of 1. An λ/4 objective will sees 20% of the core energy passing in the diffraction rings. Its Strehl ratio will be 80%.

The OSLO software will permits the calculation of the Strehl ratio, let's establish how it varies on the axis, when vary the shift and the tilt of the secondary. The results are on line with the previous considerations. The tilt has the same effect on the DK and the Cassegrain. On the other hand the sensitivity of the DK to the shift of the secondary is much lower. In these conditions, an important conclusion is that the DK with its spherical secondary asks only for a tilt adjustment of the keg, feasible with three screws, whereas the Cassegrain and generally all systems for which the secondary is not spherical, ask besides a lateral six screws adjustment, this complicates the realization and the adjustments of the collimation of the

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telescope. It takes out again of it that the adjustments of a DK is at a the same time simpler and more perennial than the one of the Cassegrain and generally than all other telescope of which the secondary is aspheric. An example: Let's suppose that in the keg, three screws of adjustment of Ø 3mm x 0.5 are put 23mm of centre of the secondary on 120°, a turn of screw displaces the secondary by an angle equal to Atg(0.5/35.68) either 0.8° by turn. As a simple calculation demonstrates it, 35.68mm is the distance between the adjusting screw and the line uniting the two other screws on which rest the secondary. This calculation in order to show that with a little fingering and if one is not parkinsonian, one reaches the 20th of turn or 0.04° of displacement without too much pain, what is sufficient to drive the system toward the λ/10 on the wave.

About baffling.

One uses here the program optic MODAS. One notices that the baffling increases the central obstruction to 36%. In fact, one can be content with the baffling of the primary.

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Complementary results.

Answer in intensity and picture of the DK for the side of the field (1/4°)

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Idem Cassegrain for the side of the field (1/4°) to the same scale.

One sees here to what point the picture of the Cassegrain is better in theory at field side - under a perfects sky - that the one of a DK. But in practice this quality gets itself through an extremely precise collimation as we saw it before, requiring a translation and a tilt of the secondary. The question is to know, in practice, if the installation is robust enough and if it is indeed susceptible to make benefit us without issues and during a sufficiently long time of all its Cassegrain qualities, in particular for planetary, where the DK is more robust and faster to put at work in terms of collimation. That says, the simulations show that a mistake of 0,34mm in translation puts us to λ/4 in the case of the Cassegrain but lets us again to λ/10 p/v in the case of the DK! Considerations to which it is right to add the uncertainties proper to the correct realization and without zones of a hyperbolic secondary, correctly measured by an amateur! These considerations are not foreign to the Takahashi choice of the DK for his telescopes of the Mewlon kind, regardless of the fact that an excellent spherical secondary is easier to achieve than a good hyperbolic. In the first case, one will be to λ/10 on the wave in the centre. In the other only to λ/4 and therefore more sensitive to the turbulence with results that risk not to be so different in practice on the sky. In particular, the tests that one finds on Internet seem to show, supporting photos for a Mewlon 250mm opened to F/12, that the results are excellent. Considering the fact that the tube by itself is sold $6000 and as much in Euro in Europe, one would be disappointed that it will not! It arranges in addition to various accessories that permit to decrease the focal distance and probably to correct the field abit. One finds on this site an analysis of the Mewlon 250. The twisted mind will object that the primary of the Cassegrain is usable in Newton, what is well convenient and what not the case of the DK. That is right. But the usable visual field is weak, not only because of the huge coma important in the case of a parabolic opened at F/3.5, but of the astigmatism of the oculars for which F/3.5 is a dangerous test to which their parents (Plössl, Abbé…) didn't prepare them. Only some oculars with large field permit to go to F/4, as the Pentax for instance, in these conditions. On the other hand the big opening is an interesting partner for a CCD camera or a Webcam to photograph deep sky

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objects, presenting a much reduced field. Deconvolution software can also improve the things. But one would be maybe better inspired in this case to push the asphérisation farther a little and to pass the Cassegrain to the Ritchey-Christian telescope. The hyperbolic primary associated to a Ross two lenses corrector will permit to assure the stigmatism and approximately plane field, useful for the photograph to very extended field with a very open telescope. Associated to the secondary, one would have an a lot more extended field then than the Cassegrain, but at the cost of an obstruction of more than 40%. It is the formula dedicated to the very big telescopes. But we come here out of the working drawing! We had left to describe an easy telescope to achieve by an amateur and we end up now with the most difficult!

Conclusions The DK is rather a telescope easy to achieve. In counterpart of what his field is weaker, what makes a telescope rather specialized for the planetary surfaces. Otherwise, the fact to have a spherical surface to the secondary makes more easily this one feasible and measurable. Besides, the DK is a robust optical system in the sense that, contrary to a widespread idea, it is less sensitive to the lateral shift, in any case well less that his near cousin, the Cassegrain. One is also in right to think that it is better to have a good spherical secondary that a passable hyperbolic secondary, in particular on a bad sky. The primary mirror that is less deformed than a parabolic could be easier to achieve, in particular if this mirror is opened, but it entails the impossibility to use it as a Newtonian. Of this point of view, if it has been possible to have a parabolic primary conjugated to a spherical secondary, it would have been a real advantage adaptable to open Newtonian as soon as they have a big diameter. It is possible if one accepts to add a corrector like in the Rosin. And if it could be close to the secondary, it would be even better… It is certainly always possible to improve the field while choosing a less open primary, a lower magnification coefficient, but with the price of a bigger central obstruction (around 30-35%) and a superior clutter, what ended up putting the system in direct competition with a F/6 less obstructed Newtonian with a good x3 Barlow. As always, everybody will see noon at is door! In the telescopes as elsewhere, there is no free lunch and the results are a function of the effort that one will be ready to provide. As always, the improvement is asymptotic whereas the effort is exponential ! That says the ratio efforts/results of the DK remains attractive. Another starting point is the DK Mewlon, an amateur can reproduce. It will have various accessories of the market that will widen his field of action. It is quite a manageable contest for an amateur already having two or three mirrors to his asset.

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References Astronomical Optics, D.J.Schroder, Academic Press.1990, Optics telescope, Rutten & Venrooij sieve, William Bell, 5th edition. The CTA, J.Texereau, Vuibert, 2004.