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# Reflector/Refractor equivalence formula

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### #1 CollinofAlabama

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Posted 29 March 2013 - 12:41 PM

Folks,

I decided to post this in the Refractor forum, too, to get people's input here. Jarad Schiffer, in the thread "Long focus Newtonian Vs refractor" in the Reflector forum wrote the oft quoted formula that to determine a Newtonian's performance to an equivalent refractor, do this ...

Reflector Primary - Secondary = Refractor Primary Equivalent

I noted Jon Isaacs many times writing that the real difference is more like

Reflector Primary - 1" = Refractor Primary Equivalent

I propose, to quantify Jon's assertion, perhaps the formula should look like this ...

Reflector Primary - (0.5 * Secondary) = Refractor Primary Equivalent

As one got above a 40% obstruction, this formula might not apply, but I'm not as concerned about astrographs functioning as visual instruments

I take the example of the Celestron Omni XLT 150, which has a 150mm objective with a 46.5mm CO. Now according to Jarad's formula, that should make the Omni XLT equivalent to a 103.5mm refractor for planetary performance. But in my proposed reworked formula, the "Refractor Equivalence" of the Omni XLT should be a 126.75mm scope. Now, people may say this is wishful thinking, and I cannot say with any certainty that it isn't, but it would be nice to see how an Omni XLT 150 compares with 4" ED, 110mm ED, and 120mm ED (as well as the equivalent achromatic) scopes. I own neither the 110 nor 120mm ED scopes, nor any achromat, though there's a STRONG temptation for me to buy the XLT and compare it to my 102mm ED scope. According to Jarad's formula, it should still surpass it, but really, just barely, and the planetary performance should really be about the same. Stay tuned, but anyone else wishing to evaluate my recalculation of that old Reflector ~ Refractor Equivalence formula is more than welcome to educate me and the rest of the CN Brotherhood.

For the record, the Omni XLT delivers the same number of photons to your retina as a 142.6mm refractor -- (Pi * r squared of the Primary) minus (Pi * r squared of the Central Obstruction). Obviously, those photons are maligned a bit for human vision by the CO, but how much is the question.

### #2 Eddgie

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Posted 29 March 2013 - 01:18 PM

The old "Primary minus Secondary obstruction) offers at very best, a very crude approximation of how the instruments will perform visually, and only visually.

And if you want to make the comparison and not get an approximation, try using Abberator 3.0 to generate your own MTF plots. It is easy to use.

The MTF plot tells the truth about how much contrast is lost.

Here is an MTF for an 8" Newtonian with 23% obstruction (Dashed red) vs a perfect 8" aperture (solid red) and a perfect 6" unobstructed aperture. I have included the spider vanes (which means the scope started with a 21% secondary mirror and I added 2% to that for the spider vanes).

The Green line is for a perfect 6" Aperture with the assumption that the color correction is perfect (which is rarely the case by the way, but I am assuming a best case.

As you can see, the 8" scope offers contrast that is on par with the perfect 6" aperture for larger details, but for the smallest details, the reflector pulls away, still showing small detail after the perfect 6" aperture has run out of steam.

So, no need to apply coarse formulas when you can plot it pretty closely and know a more accurate picture.

Abberator will only allow you to plot one scope.

I plot the larger instrument, then hand draw the curve for the smaller perfect instrument.

In this case, the 6" has .75 of the maximum spatial response of the 8" scope, so it is plotted to the .75 index on the frequency line at the bottom of the chart.

Also, MTF ignores the higher illumination you get out of the larger aperture for a given exit pupil. A brighter image helps the eye better see the lowest contrast detail on the target.

### #3 hfjacinto

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Posted 29 March 2013 - 01:33 PM

I think it should be this:

Ha!

### #4 Eddgie

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Posted 29 March 2013 - 01:37 PM

Here you go..

I did it for you.

This is a plot showing a perfect 150mm aperture (red line), and 150mm aperture with a 33% obstruction (the secondary obstruction is only 31%, but the spider vanes add about 2% and dude, you have to count that too, though it is often not mentioned in these dialogs).

The green line shows the equivalent aperture for a scope that is a perfect 90mm aperture.

Now remember, just like most people don't count the spider vane, refractor people think refractors are all perfect, but fast ED scopes do indeed loose some contrast because of chromatic aberration, but at 90mm, it is usually not enough to matter (though at larger apertures it can indeed be meaningful).

As you can see, for the larger details (the left side of the graph, your scope will only equal about a 90mm unobstructed scope.

As the detail gets smaller though, your scope draws even, but at the important "Visual frequencies" that are generally represented by the left half of the graph, the scope is not really performing any better than a 90mm unobstructed scope.

For the right part of the graph (past about .5 on the frequency scale), the 6" easily pulls ahead.

The problem is that this represents detail that is very small, and better caught by a CCD camera. Visuallly, the exit pupil gets so small by the time the image gets large ehough that the brightness falloff causes the image to suffer.

A camera would show the difference in performance, but visually, the contras would be no better.

Ah, but once again, the larger scope has a huge brightness advantage. The human eye finds it easier to see low contrast detail when the image is better illuminated.

This means that when you magnified the image to 150x in both scopes, it would be much brighter in the 6" making the lowest contrast detail easier to see. Both scopes will show the detail with the same contrast, but all of the detail will be brighter in the larger scope.

I personally find that illumination is an important factor when planetary observing.

The last issue is quality. Don't expect your scope to have optics of the same quality as is common in even inexpensive ED scopes made today. I gave your instrument the benefit of the doubt and made it perfect.

Bottom line? It is very complicated, but I think you are being optimistic.

Expect planetary performance below what the best 4" APOs can deliver.

### #5 junomike

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Posted 29 March 2013 - 01:49 PM

My main issue with all of this (formula's and graph's) is It's only in "theory"! In the field the main contributor to how my C11 is gonna compare to my Apo (in this case an AT111EDT ~ 4.37") is.........SEEING! IME the Central Obstruction in a Reflector is much more harmful in regards to Contrast in poor seeing.

Last year I had both in the field and on a few nights the much smaller Apo brought in more detail than the larger scopes (my C11 and an excellent 12" Dob).

When seeing did permit the use of higher magnification, the smaller Apo was easily bested by the larger scopes. This occurred less than 50% of the time however!

This is the main reason I believe there is so much of the
"this scope beat that much larger scope" and Refractor vs. Reflector controversy.

It's all about the four rights: The right scope of the right size used on the right target in the right Seeing Conditions.

This is also why one scope is not usually sufficient for everything.

Mike

### #6 Sean Puett

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Posted 29 March 2013 - 01:50 PM

That formula is supposed to be for planetary performance. Jon correct me if I am wrong, I think the -1" is for everything else.
My personal opinion is that the CO affect is a bit overstated if it is under 25% dso or 20% planetary. I was considering lengthening my ota and using a low profile (moonlite) focuser to allow me to get my secondary under 20%. Jon Isaacs told me to make secondary masks larger by the same percentage to see if it would be worth all the time, money, and effort. What I noticed on Jupiter was that the 5% make very little difference and so I used an even larger one to make the difference more obvious. I decided that it wasn't worth it.
Later I found out that the center of the refractor lens does very little if anything anyway. Now that I typed this, I think I should have answered in the reflector forum... Go easy on me.

### #7 Eddgie

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Posted 29 March 2013 - 01:54 PM

You know, I consider myself to be a pretty balanced kind of guy.

A 6" APO is indeed a very difficult scope to beat.

Even the Ultimate Newtonian 8" (which is what I plotted the first time) is only on par with a perfect 6" APO from a raw MTF perspective.

To get a clearly superior image, you would need to go to perhaps a 10" reflector.

That actually speaks volumes about the excellence of a high quality 6" APO.

I own one and can attest to their astonishingly good performance.

But they can be beat. It just takes more than an inch of aperture in most cases.

And often it takes more than even a few inches to equal the contrast transfer. I would put my 6" APO up against a good quality C11 and expect it to hold its own.

And the difference in the quality of the view between my C14 and my 6" APO is not "Breath-taking." Yes, better in the C14. I routinely observe detail that is out of reach of the 6" APO, but only with great patience and good seeing.

Many people think I am a refractor hater and nothing could be further from the truth. I admire them greatly, Though I have no use for small ones.

They can be beat though, but below about 8", and they really do have the upper hand unless you go to something like the MN 78 which is highly specialized and of superb optical quality.

### #8 Sean Puett

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Posted 29 March 2013 - 03:25 PM

The above described situation from my last post came as a result of getting my first decent refractor and wanting my 12" do more "refractor like". I started reading about secondary sizes and effects on contrast and since I have the two scopes I want at this time, I thought seriously about trying to improve the reflector. The project may still happen at some point just because I may rebuild my OTA and dob base. It is hard to have near perfection in a refractor and not want it for your big scope.
After seeing that 10" apo video (with seeing issues) and comparing it to a memory of a d14 at a low angle, conditions limited both too much for any real comparison. I haven't ever used an apo big enough to compare to my 12" dob on a good night. I have had better views than that video, but what does that mean? So I guess in a very long way, I have said that I cannot confirm or dispute the formula.

### #9 GlennLeDrew

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Posted 29 March 2013 - 06:02 PM

Such 'formulae' are exit pupil dependent. At larger exit pupils, resolving power and contrast transfer are not impacted by a central obstruction.

### #10 Eddgie

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Posted 29 March 2013 - 06:32 PM

This is a point that I strongly concur with.

While much of the attention is focused on the contrast transfer the instruments, (and it is so much more than "Theory") people often omit the role of illumination.

In a thread going on right now, someone said that they looked through a 400mm f/15 achromat and had one of the best views that they have ever had (though they did not do a side by side comparison with a similarly good 400mm reflector at the 5000 foot elevation of the observatory).
With 400mm of aperture, the brightness from Jupiter is probably so high at 300x that the eye is moving well into mesoptic mode, where the eye's contrast sensitivity threshold is much better than in scotopic mode that is common in much smaller instruments.

I almost always say that the bigger aperture will always have the advantage of having a brighter image at a given magnification, and that this added illumination can make low contrast details easier to see.

My own experience has been that the amount of detail I cold see on planets was almost dead linear to the clear aperture of the scopes I have owned. In almost every case, the more aperture I used, the more detail I was able to see.

Until I owned the C14, I never really was able to resolve any detail on the Jovian moons for example. Ganymede would show variations in shading but not really show specific details with specific shapes.

Only when I moved to the C14 did I start to resolve this kind of detail.

The smallest refractor that I have heard a similar report of resolving Osiris and Galilee Regio where from an 8" APO.

Clear aperture rules.

Anyone wanting better planetary performance should not be looking at "Planetary eyepieces." They should be looking for more good quality clear aperture.

And the illumination that comes with aperture is in itself an important advantage.

### #11 timps

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Posted 29 March 2013 - 07:14 PM

So should I buy the 152mm Explore Scientific Apo or a 14" Meade/Celestron?

### #12 Scott Beith

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Posted 29 March 2013 - 07:32 PM

So should I buy the 152mm Explore Scientific Apo or a 14" Meade/Celestron?

Both!

### #13 Jon Isaacs

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Posted 29 March 2013 - 08:52 PM

I noted Jon Isaacs many times writing that the real difference is more like

Reflector Primary - 1" = Refractor Primary Equivalent

Honestly, I don't remember writing this. What I remember are calculations for light through put based on reflectivity and surface area measurements and mentions of rule of thumb estimations of planetary contrast based on "clear aperture", the aperture minus the secondary diameter.

Jon

### #14 CollinofAlabama

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Posted 29 March 2013 - 09:41 PM

First, Jon, I'm pretty sure you recently wrote this, but I'm on an iPhone right now, not the tool to find your posts on reflectors' performance (you have quite a few in the regard, mind you!)

Ed, first, thanks for doing the graph. This kind of information is enjoyable to look at. Something tangible, and yet ...

Sean made the point about "in the field". And here's where your graphs run afoul of reality. Joe Bergeron wrote this review in 2007, comparing his 92mm apo, 150 XLT and AP 155. In his estimation they lined up as one would expect, with the AP 155 best, the 92mm the worst, and the 150 XLT performing in between. Of course, this sheds little light on where and to what degree the 150 XLT would fall -- above a 102mm? above a 110mm? above a 120mm? -- since all those certainly fall between a 155mm and a 92mm scope. But Bergeron specifically notes it performing better than the 92mm, thus implying the graph you supply, Ed, may not apply in the field. "In overall performance, the \$400 Omni was midway between these two fabled refractors, and closer to the big one than the little one." You may say this proves nothing, and you may be right. But you could be wrong, too.

I'd just like to hear from someone who has used the Omni XLT and one (or more) of the 4", 110mm or 120mm refractors.

I have always liked refractors, but can't forget the way I saw an 8" LightBridge put an Orion 120 ED to shame on Saturn one evening.

### #15 timps

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Posted 29 March 2013 - 10:57 PM

That would be ideal but if you could only have one, which would it be?

### #16 JKoelman

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Posted 30 March 2013 - 12:20 AM

A single formula will not capture all performance aspects. Assuming diffraction-limited optics and perfect seeing, for planetary detail the aperture-based formula:

NewtApert - ObstructDiam = EquivApertAPO

seems reasonable. For faint fuzzies, I'd go with a straightforward light gathering-based equation:

NewtApert^2 - ObstructDiam^2 = EquivApertAPO^2

which is more favorable towards Newtonians.

Purely from a theoretical perspective, a formula like

NewtApert - ObstructDiam/2 = EquivApertAPO

can't be correct for the full range of secondary obstructions, as for secondaries approaching the size of the primary the equivalent APO aperture should drop to zero. For small percentage obstructions, however, the equation with the squares is mathematically equivalent to

EquivApertAPO = NewtApert - ObstructDiam^2/2NewtApert

Therefore, as a rough rule:

NewtApert - ObstructDiam < EquivApertAPO < NewtApert - ObstructDiam^2 / 2NewtApert

where the leftmost expression is more relevant for planetary, and the rightmost expression more for DSO.

### #17 azure1961p

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Posted 30 March 2013 - 07:59 AM

I agree the rule of thumb about subtracting CO size to arrive at a clear aperture visual observation equivalent is a very very rough approximation. This had always been a rather nebulous notion among observers and where there were voids in proof it was - and still is - a place where ego filled in. Now we have MTF. I think I was one of the last people to actually refer to it or use it but alas its about as real as you're ever going to get. It might seem like an abstraction of logic but once I got in the hang of it then it all made sense and with logic clarity.

I know what you are looking for - a simple equation somewhat like the CO subtraction - and in lieu of MTF - its a fair if rough call, but the graphic mapping out of it all sais it so handedly (and ego free) its probably the uncontested as the most revealing way to compare - though the author could tweak things a but to be quite frank. Too there are other variables Id like to see addressed in the graph. Environmental effects like fan versus no fan, cool down versus insufficient cool down. Being able to model these into the MTF graphic would be engaging stuff. Adding Pickerings scale of seeing effect as well could produce some great comparisons - along with the ability to overlay multiple scope MTF curves.

Anyway its worth downloading Aberrator even if at first it might seem a little wonky. Its all to a good end. And strangely like some other top notch programs - its free.

Pete

### #18 Cotts

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Posted 30 March 2013 - 09:06 AM

The trouble with a rule of thumb is that people's thumbs vary greatly in length.....

Dave

### #19 Eddgie

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Posted 30 March 2013 - 09:13 AM

Sean made the point about "in the field". And here's where your graphs run afoul of reality.

Clearly you missed the part where I said that Illumination is an important characteristic of the telescopes.

The MTF is concrete. It is defined by diffraction and makes the outcome absollute in terms of performance at the focal plane.

But in the same post, I very clearly said that illumination is usually ignored in these dialogs (I never ignore it) and said that while the obstructed apeture might not have any better contrast transfer, that the better illumination would make whatever detail was present easier to see.

So while the instruments are peformeing the same at the focal plane, the observer's eye has a stronger signal to work with, which again, is an advantage to apeture that is often not included in these converstaions.

In other words, I agree with you in one sense, that the field result will be a bit different, but the MTF is absolute. Nothing violates it.

And that is why I have said that clear aperture has for me been the most important attribute of how two telescopes would perform on planets and extended targets. Even if the contrast is the same, the observer always benefits from the brighter image in the larger scope.

### #20 Eddgie

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Posted 30 March 2013 - 09:23 AM

It might seem like an abstraction of logic but once I got in the hang of it then it all made sense and with logic clarity.

Yes, once one really understands MTF, the clarity it brings to how an instrument will perform is very satifying.

And it is no more of an empty threory than the theories on the wave nature of light.

It is the wave nature of light that causes contrast loss, and it is absolute and easy to model, and can predict with very high accuracy how the instrument will behave.

That is why almost all professionals will use MTF to define instrument performance

It is all inclusive from design, quality, contrast transfer, and angular resolution.

One graph tells you the truth, the whole truth, and nothing but the truth of how the image will be formed at the focal plane.

But as in my previous post, it is important to say that the human eye's contrast sensitivity has to be factored in when using the instrument visuallly, and illumination of a larger aperture aids in detecting and resolveing the smallest detail on the target.

Even when seeing limits my to 250x, the image is so much brighter in my C14 that all of the details are much easie to see than in my 6" APO.

Illumiantion is a huge key to our ability to resolve detail.

Take a black and white picture from a newspaper out at night and see what happens. The contrast on the target doesn't change but the illumination changes.

And the more illumination you put on that picture, the more details will emerge.

Simple experiment and conclusive. Increase the illumination 50% and you will see more on the target.

### #21 Eddgie

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Posted 30 March 2013 - 10:00 AM

So should I buy the 152mm Explore Scientific Apo or a 14" Meade/Celestron?

Two very different instruments.

I own both a 6" APO and a C14.

I use the 6" APO mostly for wide field observing during the summer and winter Milky Way, and for doubles during this time of the year.

It works well for planets and if it is already set up, I will use it for planets, but if there is no scope set up and I want to look at planets, I bring out the C14.

But I get the appeal of the big refractor. For wide field work, the view is the best I have ever had. At all powers, stars are sharp right to the edge of the field stop.

And I get the appeal there. I love the pinpoint stars offered by the 6" APO.

But for must general use, if it fits into the field of the C14, I use that.

If it doesn't fit into the field of the C14, but fits into the field of the EdgeHD 8", I use that.

Only if it doesn't fit into the field of the EdgeHD 8" am I inclined to pull out the big refractor.

So, to me, they are different scopes for different uses.

What do you want to do most?

If it is deep sky, the C14 is clearly a better choice even with the off axis curvature and coma. For most deep sky targets, you don't notice the off axis abberations of the C14.

So, I cannot advise you really on what you should get. Think about what you want to observer and how important off axis performacne is to you.

Good luck.

### #22 GlennLeDrew

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Posted 30 March 2013 - 03:01 PM

Eddie,
One must be careful of treating the greater aperture as delivering a brighter image, this being the main reason for seeing more detail. Aperture, image brightness and resolving are inextricably intertwined. The larger instrument delivers better resolution because of the diameter of the entrance pupil, extra light is part of the equation.

To assess performance differences most objectively, and to minimize the variables, comparisons must be conducted at identical exit pupil. The exit pupil is the arbiter of the visual appearance of diffraction effects. A 1" and 100" aperture, both working at the same exit pupil, will provide identical appearances for a star (the stars chosen being of suitable brightness for the aperture, of course.) And for extended objects, like planets, at the same exit pupil the object surface brightness is the same, and the degree of resolution *as perceived on the retina* will be the same. It's just that the larger scope delivers a commensurately larger image and a concomitantly more detailed view. But on the retina, the *apparent* angular resolving power is the same.

### #23 Eddgie

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Posted 30 March 2013 - 04:06 PM

This does not differ at all from what I have said.

I have said that a larger aperture will enjoy a better illuminated image for a given magnification.

And of course that means that at the same exit pupil, the image will be larger, but just as bright.

And the consequence of that is that you can magnify the image more in the larger aperture which makes it easier to resolve the detail.

And I have repeatedly stated here that these dialogs usually only talk about what happens at the focal plane and ignore the working of the human eye.

It would appear that we are not in dis-agreement here.

But I disagree that at the retina there will be no difference.

If the larger aperture is allowed by seeing to transfer more contrast, that contrast will show in more detail at all powers in the scope with the larger clear aperture. Angular resolution is quite a bit different than contrast ttransfer. You can resolve 192 line pair per millimeter in even a very poor f/10 telescope.

One observer might see a particular detail rendered with 6% contrast (right at the edge of perception for the dark adapted eye) while the other might see it with 15% contrast. The one that sees the higher contrast will say that the view is "Sharper". After all, that is what contrast transfer is. It is image sharpness. But the loss of contrast for mid-frequency detail does indeed make it less crisp in the eyepiece for that same instrument,

The MTF of the instrument allows the instrument to simply transfer more of the low contrast detail to the focal plane.

If there is more on the focal plane to see, then the observer will see it.

But once again, in my very first post on this topic, I said it was very complex. We are far more in agreement than we differ I think,

### #24 t.r.

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Posted 30 March 2013 - 05:28 PM

First, Jon, I'm pretty sure you recently wrote this, but I'm on an iPhone right now, not the tool to find your posts on reflectors' performance (you have quite a few in the regard, mind you!)

Ed, first, thanks for doing the graph. This kind of information is enjoyable to look at. Something tangible, and yet ...

Sean made the point about "in the field". And here's where your graphs run afoul of reality. Joe Bergeron wrote this review in 2007, comparing his 92mm apo, 150 XLT and AP 155. In his estimation they lined up as one would expect, with the AP 155 best, the 92mm the worst, and the 150 XLT performing in between. Of course, this sheds little light on where and to what degree the 150 XLT would fall -- above a 102mm? above a 110mm? above a 120mm? -- since all those certainly fall between a 155mm and a 92mm scope. But Bergeron specifically notes it performing better than the 92mm, thus implying the graph you supply, Ed, may not apply in the field. "In overall performance, the \$400 Omni was midway between these two fabled refractors, and closer to the big one than the little one." You may say this proves nothing, and you may be right. But you could be wrong, too.

I'd just like to hear from someone who has used the Omni XLT and one (or more) of the 4", 110mm or 120mm refractors.

I have always liked refractors, but can't forget the way I saw an 8" LightBridge put an Orion 120 ED to shame on Saturn one evening.

I can't speak about a 150XLT...but I can assure you that my 1/6-1/7th wave C-6XLT handily beat out my old Tak Sky 90, Megrez 90 and TMB 92L! Take it to the bank. So I have concluded that 6" of sct aperture trumps 3.5" of unobstructed aperture consistently. In addition, for a season I compared a 4" apo to a C5, and the C5 fell short. Bracketing would put a 3.5" apo in line with a C5.

### #25 timps

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Posted 30 March 2013 - 06:18 PM

I would like to use the telescope for most things. (Nebula, star clusters, galaxy, planetary and lunar).Visual and imaging. I believe that a 14" SCT, be it Meade or Celestron, is the best telescope for this. I think they are the best "all rounder". They are good value for money too when you consider the price of even a "cheap" 6" Apo.
However, I would still like a 6" Apo. Maybe I should start off with the SCT & get the Apo further down the track.

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