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Spherical Vs. Parabolic mirrors


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I've just read a thread where a newcomer asked what to get and in the course of the discussion two different mirror types, spherical and parabolic, were mentioned without any firm indication of how they are different, why one might be preferred over the other and for what reason this might be.  Yes, I could go off and 'Google' it, but it occurred to me that to run a thread might provide some information in the beginners area that might be to someone's advantage.

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Spherical is an ideal figure of resolution. Its easier to test and produce to a high standard of accuracy. Unfortunately it produces severe spherical aberration in short focal length single components unless corrected by additional optics. Maksutovs and Schmidt-Cassegrain telescopes have spherical primaries but have optical correctors. Most normal focal length Newtonian mirrors are parabolic, the parabola corrects the spherical aberration on axis but introduces coma off axis, there is nearly always an optical compromise. The difference in surface shape between a parabola and a sphere diminishes as the focal length increases, for a 6" aperture mirror at F10 the difference is almost negligible, at this length a good sphere would probably outperform a poorly figured parabola.

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ive a bird/jones in the loft think its a five inch one, I havnt  used since my daughter was born that's 27 years ago, as I remember it was quite good on the moon but not very good on anything else. charl.

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So we are talking about the physical shape of the mirrors - is a spherical one then as though you have sliced the edge off of a perfect sphere where as a 'parabolic' one being one with either less or more curvature than the side of a perfect sphere would have?   I must admit when I peer into the back of my reflector I am not altogether aware that the mirror I look at is even curved - to the naked eye it looks, well....., flat!

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If you draw a shallow spherical arc and rays impacting it from infinity you will notice that the ray central to the arc will be reflected back to infinity, this is the on axis ray. Rays further from the centre will be focused to the focal point gradually further from the focal point due to the angle of incidence equaling the angle of reflection. It soon becomes obvious that rays from the edge are not focusing at the same spot. This is spherical aberration. The obvious single arc solution is to bend the outer portion outwards so that the rays from this position reach a common focal point. The figure that satisfies this condition is the parabola.  

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Many centuries ago, I attempted to make a parabolic mirror (8" f8), using the then-standard textbook. To make the spherical step was easy, the next part (parabolising) was not. Gave me a lifelong newtophobia and has left me suspicious about the quality of parabolic mirrors unless a lot of money is spent.....

Chris

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1 hour ago, JOC said:

So we are talking about the physical shape of the mirrors - is a spherical one then as though you have sliced the edge off of a perfect sphere where as a 'parabolic' one being one with either less or more curvature than the side of a perfect sphere would have?   I must admit when I peer into the back of my reflector I am not altogether aware that the mirror I look at is even curved - to the naked eye it looks, well....., flat! 



Hi,

So a spherical mirror is a Spherical section (hollow circle chopped through near its edge), its gradient's gradient (second derivative, f " [x]) doesn't change (I'm pretty sure it stays at 1, someone please correct me if I'm wrong ;)). Whereas, a parabolic mirror is a parabola section--a cone cut parallel to its centre line. As a general rule of thumb, I think you'll find it's taller at the edges. The parabola focuses rays at the same distance, (no spherical aberration) but "splats" the image when it hits a non-parabolic surface like the secondary, producing coma.

I think a hyperbolic will act as something between the two--but usually only spherical and parabolic are used in newts.

EDIT: After some research, it appears that a hyperbolic is used for very fast systems, like cats and astrographs.

John

Edited by JohnSadlerAstro
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2 hours ago, JohnSadlerAstro said:

Just to throw this in as well--I have heard of hyperbolic mirrors. I assume these would act like a compromise between S and P?

John

A parabolic mirror and hyperbolic mirror have very different curves. A parabolic curve is simply represented as "x^2" (x squared). So moving twice as far away from the center of the curve gives you 4x as much height.

image.thumb.png.86676ee85cfc822d92417387f3ba0176.png

Meanwhile a hyperbolic curve has a sharp bend near 0, but higher than 1 and lower than -1 the curve "flattens". Unlike parabolic curves, hyperbolic curves vary in "eccentricity" (shape) as their definition allows multiple curve shapes depending on the conic constant used.

image.thumb.png.3073fad016258e88015f87e90e6beb20.png

 

Now, I can't claim to understand the maths fully. But at K = 0 we have a circle:

image.thumb.png.4f919b99e46f5f8d6c960718d62c0ceb.png

At k = 0.1 to k = 0.9 we have an ellipsis:

image.thumb.png.17572d5362bf4425b1fdae4a37348298.png

At K == 1 we have a parabola (where the curve becomes open-ended):

image.thumb.png.7bec61bded069ce5c48e5cffce9778e0.png

Where K is greater than 1, we have a hyperbolic curve:

image.thumb.png.96bf1dc949101e04af326d71a83be794.png

We can see that even at K = 1.1, the curve has "flattened" compared to the parabolic curve.

At a higher K value:

image.thumb.png.63737f28d7037184b636bde6b0a5b9b5.png

 

 

Unfortunately I can't say what the result would be if a newt's primary were hyperbolic, but I presume there are reasons (besides cost!) why the only designs utilizing hyperbolic mirrors either have a second curved mirror in the optical path. Hopefully this was interesting even if it doesn't answer the question exactly.

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1 hour ago, JohnSadlerAstro said:

Hi,

So a spherical mirror is a Spherical section (hollow circle chopped through near its edge), its gradient's gradient (second derivative, f " [x]) doesn't change (I'm pretty sure it stays at 1, someone please correct me if I'm wrong ;)). Whereas, a parabolic mirror is a parabola section--a cone cut parallel to its centre line. As a general rule of thumb, I think you'll find it's taller at the edges. The parabola focuses rays at the same distance, (no spherical aberration) but "splats" the image when it hits a non-parabolic surface like the secondary, producing coma.

I think a hyperbolic will act as something between the two--but usually only spherical and parabolic are used in newts.

EDIT: After some research, it appears that a hyperbolic is used for very fast systems, like cats and astrographs.

John

It is not the secondary that introduces coma, it is the paraboloidal surface itself. Hyperboloidal surfaces actually have the reverse spherical aberration of a spherical surface, and two hyperbolic surfaces are used in Ritchey-Chretien scopes to eliminate both spherical aberration and coma. Elliptical curves would yield a compromise between parabolic and spherical shapes.

 

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I have been led to believe that parabolic mirrors give better views because of the curve on them. They gather and concentrate more light towards the secondary mirror than a plain flat spherical mirror. 

Radio telescopes are parabolic. Must say something.

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Spherical mirrors in addition to on axis spherical aberration have off axis aberrations as well. As discussd above a lot depends on the focal ratio but also the field of view you want to cover. This almost inevitably means the need for corrector of one sort or another for wide field imaging.

Regards Andrew 

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16 minutes ago, JohnSadlerAstro said:

Surely spherical aberration could be corrected using a lens very similar to glasses? Isn't it roughly equivalent to astigmatism, except that it is predictable and symmetrical across the fov?

John

I think all the main types of telescope need a corrector for off-axis aberrations. Newts (Parabolic or Spherical), RC, DK they all need some kind of glass to cover the maximum field.

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Here's a nice diagram showing how rays parallel to the optical axis focus differently in spherical vs parabolic mirrors:

spherical.gif

However, not all bundles enter parallel to the optical axis causing off-axis coma in a paraboloidal mirror:

paraboloid-5deg-offaxis-coma.gif

Edited by Louis D
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Great thread and answers many thanks.  There was me sitting on the fence this morning thinking - 'shall I post the thread - they'll laugh me off the forum for not knowing the difference'.  Instead it looks an interesting subject ?

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1 hour ago, JohnSadlerAstro said:

Surely spherical aberration could be corrected using a lens very similar to glasses? Isn't it roughly equivalent to astigmatism, except that it is predictable and symmetrical across the fov?

John

Spherical aberration in refractors can be corrected - the Aries Chromacor did this as well as correcting around 80%-90% of the CA in achromats. Not easy to design and make though - it needed very special glass types and cost around £700 per unit when available. The objective needed to be matched with the corrector as well to ensure that the right sort of SA correction was being applied.

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Thank you for starting the thread--its great to learn more about this topic, I'm slightly ashamed I knew so little to be honest! :embarrassed::D 

Really helpful and interesting diagrams @Louis D and @pipnina, thank you for sharing! :) I don't think my 130PDS mirror is 5 degrees off axis, although I did once go for 5 months without noticing the primary had slipped one of its shoes. I thought it a bit odd that I had to collimate at about 10 degrees of straight! :D 

John

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3 minutes ago, John said:

£700 per unit when available.

That explains why mirrors went to Parabolic shapes, instead of being corrected at the eyepiece. I'm guessing the only reason why this would be worth it would be in extremely large mirrors, when making it parabolic would cost more than the corrector.

How simple are the MPCC 4s optically speaking? They seem to just magnify the centre, but there must be more going on! ?

John

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39 minutes ago, JohnSadlerAstro said:

I don't think my 130PDS mirror is 5 degrees off axis

No, it's not the mirror but the object that is 5 degrees off axis.  Admittedly, this might be pusing it for your 130PDS, but the result is similar (and of a smaller magnitude) for objects one or two degrees off-axis.

Think about a large star cluster like the Pleiades.  Only the single star in the center of the field will be coma free.  Stars elsewhere in the field will have increasing degrees of coma depending on how far off axis they are.  Referring to the coma diagram, light ray bundles from off-axis stars will be focused off to the side of an on-axis star's focused image.  Your eyepiece then takes that field at best focus in the focuser and manipulates and magnifies it for your eye to view.

Wider apparent field of view eyepieces will show stars farther off axis than narrower apparent field of view eyepieces.  Coma will be more evident in these wider field of view eyepieces than in the narrower field view ones.

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5 hours ago, 25585 said:

What type mirrors do conventional cassegrain scopes use, the ones with just mirrors?

Dall-Kirkham telescopes use a concave elliptical primary and a convex spherical secondary, Ritchey Chretien telescopes have a concave hyperbolic primary and a convex hyperbolic secondary, the "classic cassegrain" has a parabolic primary and a hyperbolic secondary.

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