Minimum Obstruction for Newtonian Telescopes

By Santiago Roland - Los Molinos Observatory IAU CODE 844 (OALM) - Amateur Astronomers Association (AAA)

Montevideo - Uruguay

12 Dec, 2005.

Introduction - I've been planning my first home-made newtonian telescope and I started to read some. The most interesting book for me was the 2nd edition of "How to Make a Telescope" by Jean Texereau. In this book I found lot of material and well explained formulae for some design issues like size of secondary mirror, and stuff. Anyway I noticed that some statements made in this book relative to telescope design parameters, were not well justified. So I decided to make my own calculations an arrive to useful formulae for my understanding and for share with my amateur mates in the AAA.

The Problem - It's well known that secondary obstruction in the newtonian system causes little deterioration in the wave-front seen on the focal plane. This distortion is reduced when the size of obstruction is reduced too. I choose to try a planetary newtonian because I live in the city, so I want to make this wave-front deterioration, as minimum as possible. This distortion caused by the secondary obstruction acts like a "low pass filter" on the image. In fact, the image that we see in the focal plane is the Fourier transform of the telescope's aperture or (incoming pupil). One can think this Fourier transform like a very large sum of spatial frequencies. So this "low pass filter" effect, cuts down the higher frequencies, meaning that subtle detail on lunar landscapes and planetary views are lost. When the obstruction gets bigger, more fine detail is lost.

Concepts - The figure below is the schematic representation of the newtonian system. Parallel light goes to the primary mirror and after reflection ends up in the focal plane "d". To calculate the size of the secondary mirror we consider the distance "L". This distance L is the sum of a semi-diameter "D/2" of the primary mirror, the air gap between the mirror and the internal tube face (s), and the focuser distance (e).

Note: Tube thickness (not represented) is included in the focuser distance (e).

The key to minimize the secondary obstruction is to make "L" the minimum as possible. This can be done by minimizing "e" and "s". In the Texereau's book I didn't find the minimum possible distance "s", the air gap. There is a common rule that this distance should be "d/2", a half the distance "d" that represents the fully illuminated field of the telescope.

So, I calculated this distance "s" in order to cause the same vignetting (fully illuminated field), of the secondary mirror. This is because the secondary mirror acts like a baffle that allows to enter certain amount of rays in a given field of view. When this parallel rays reaches the entire primary mirror, we have in the focal plane a given fully illuminated field. On the other hand, if we make bigger the distance between the secondary mirror and the end of the tube (x), we restrict this field and start to loose illumination. So, the size of the secondary mirror and the telescope's end of tube acts like baffles. When we make this "baffles" in order to restrict the same illuminated field, we have done right. When the distance to end of tube (x) is reduced, then, the air gap (s) can be reduced too in order to have the same "baffle effect" of the secondary and the telescope's aperture.

The figure below explains the paragraph above.

In respect to the air gap, I didn't found on Texereau's any reference about the minimum air gap allowed. Commonly amateurs use to set this air gap in 1 inch or so. But I found that this quantity is more than needed. In fact the "d/2" criteria is also more than needed because this should be, if the tube length equals to the focal length. I think that usually the tube length is slightly shorter than the focal length of the primary mirror, and the focal plane lies outside the telescope's aperture.

With this concepts understood, we must make them the minimum in order to have the best performance and image quality.

Note: The justification of the 1 inch air gap is based on thermal behavior of the telescope. I think that this can be enhanced with a computer fan and adequately thermal equilibrium with the outdoor weather.

Calculations - Using trigonometric calculus, one can find (as shown in Texereau's book), that the size of obstruction is given by the following equation.

    (1)

is the focal ratio

is the fully illuminated field (degrees)

 s  is the air gap between mirror and tube

 e  is the focuser distance

 D  is the primary mirror diameter

for comfort, is the "illuminated field factor"

Here I substituted "d" as a function of eta (illuminated field factor), D (primary mirror diameter) and phi (focal ratio). I done this because everywhere, the fully illuminated field is treated like physical distance in the focal plane, in millimeters for example.

So, the question is, how much illuminated field do you want? Some say, 14 mm. I say 0.5º (equals to the moon's diameter), or 0.25º (entire view of Jupiter and it's  moons), for example.

Now consider the minimum distance for "s". Again, calculating this, we arrive to this other formula for "s"

    (2)

 x  is the secondary-to-front of tube distance

Now, if we plug-in the equation (2) in (1) eliminating "s", and making some arrangements in order to make factors of phi, the focal ratio, and dividing by D, we arrive to equation (3).

     (3)

Equation (3) is the obstruction function of the telescope design. a, b and c are constants that depends on the initial design parameters.

The interesting thing of this function, is that looking at the shape of it, one can notice that it has a minimum! . This minimum tells me in which focal ratio, the telescope has the minimum obstruction. This is, for a constant field illuminated field (theta), focuser distance (e), end of tube distance (x), and diameter (D).

Important Note: The constant value of the fully illuminated field (theta) means that I can see the moon's diameter (for example), fully illuminated, no matter the focal ratio I choose. The difference with most of the articles about obstruction is that fully illuminated field appears in terms of millimeters on the focal plane, but is not the same true field for different focal ratios (say in a 6 inch diameter mirror). The 14 mm of focal plane means 40 arcmin in a 6 inch f/8, and 30 arcmin in a 6 inch f/10. The true fully illuminated field (in degrees) is reduced as the focal ratio is increased.

So, making the fully illuminated field constant in degrees means that when you put the ocular lens in the telescope, you'll see the entire moon fully illuminated.

Conclusions - The fact that the obstruction reaches a minimum, for a set of parameters, including fully illuminated field in degrees, means that there is an optimum configuration, that minimizes the secondary obstruction. So, we cannot build a telescope with given parameters, better than the one who has the optimum focal ratio. If we are talking about planetary newtonians I think that this fact is pretty relevant. A thing to have on mind for those who are planning a home-made planetary newtonian. One can predict, setting design parameters first, in which focal ratio the telescopes maximizes it's preformance, or what's the same, minimizes it's obstruction.

The figure below represents the obstruction function for a given set of design parameters.

The optimum focal ratio can be found by differentiating the obstruction function and making it zero.

resulting on,

Thoughts - This fact makes me think if there's an ideal newtonian. Maybe there's no, but I feel I'm closer.

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