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# The "rule" of three 7s...

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### #1 jrbarnett

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Posted 15 March 2015 - 10:04 AM

Ever wonder how much larger an obstructed system needs to be to match a refractor in light grasp?

I have.  And many times have researched coating types, central obstruction diameters and the like, and done comparative transmission/reflection/occlusion analyses of specific scopes.  But that's a lot of work.  So looking at some of the past analyses I've done, I've come up with a proposed rule of thumb - the "Rule of three 7s".  Here's how it works.  To determine (roughly) the how a given obstructed scope (SCT, MCT, Newt, etc.) matches in light grasp/throughput to a refractor in terms of equivalent aperture, multiply the aperture of the obstructed scope by 0.777 to derive the aperture of an "equivalent" refractor.

Examples:

A 90mm MCT would have throughput equivalent to a ~72mm refractor.

A 16" Dob behaves like a ~12" refractor.

An 8" SCT operates like a ~6" refractor.

A C5 works like a 100mm refractor.

Why would the rule of thumb work for both catadioptrics and Dob/Newts?  Sure the latter generally have smaller COs and two fewer surfaces, but they also generally have standard mirror coatings whereas most modern SCTs have highly enhanced coatings on both mirrors and correctors.  Despite greater occlusion by the secondary, SCTs generally put through more light than equal aperture Dobs with "standard" mirror coatings.

Anyway as a rough rule of thumb what do you think?  It works in reverse too.  Just divide the aperture of your refractor by 0.777 to get it's peer obstructed aperture in throughput terms.

A TEC 140ED works like a 180mm obstructed scope.  In terms of dimmest star seen this is correct in my experience.  I had a 180mm Russian MCT, obtained the TEC later, compared them, and found that at similar magnification the TEC actually went as deep if not a hair deeper than the 7" MCT.  I offed the MCT since it lacked the TEC's other benefits of rapid cool down, wide field capability and collimation robustness.

Clearly throughput isn't the only relevant consideration when choosing telescope designs, but it is one important criteria when comparing scopes of different designs with different apertures.

Regards,

Jim

Edited by jrbarnett, 17 March 2015 - 10:40 AM.

### #2 Simoes Pedro

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Posted 15 March 2015 - 10:24 AM

Informative post.

### #3 DJCalma

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Posted 15 March 2015 - 11:00 AM

I like it! I do a lot of mental comparisons of various scopes and although there are obviously many variables, it's nice to be able to plug in a simple formula to get a rough estimate.

### #4 sg6

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Posted 15 March 2015 - 11:07 AM

Unfortunately trying to compare apeture equivalents between scopes just leads to arguements and ultimately leads nowhere. Is my 105 Mak "better" then my 90 Refractor? I honestly don't care. They are different, they produce different results and are used for different purposes.

If someone tried to explain their 127 Mak was better then my 90mm Megrez they would politely get told to stick the Mak where the sun don't shine, second attempt a bit less polite and third time I would do the hiding of the Mak.

Why do astronomers seem to want to compare equipment and have this A verses B contest?

### #5 Dakota1

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Posted 15 March 2015 - 11:10 AM

OH OH

### #6 osbourne one-nil

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Posted 15 March 2015 - 11:16 AM

I don't think it's a case of comparing and one-upmanship, I think it's more a very useful factor to consider when choosing a new telescope? I also just find it really quite interesting!

### #7 jrbarnett

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Posted 15 March 2015 - 11:44 AM

I don't think it's a case of comparing and one-upmanship, I think it's more a very useful factor to consider when choosing a new telescope? I also just find it really quite interesting!

Precisely.  It's a way of estimating limiting magnitude as a function of aperture among two different telescope designs.  Let's say you really like globulars and want to resolve them as well as you can with a lightweight, driven-mount telescope.

You figure "I need a scope that will reach mid-12 aperture from my suburban backyard to adequately resolve brighter globulars to my taste."  You know that light grasp is a function of aperture and figure a 6" SCT would work well enough on your iOptron Mini Tower Pro mount.  So you decide to order one, but skimming through Classifieds you see a great deal on a used William Optics Megrez 110ED.  It's a nice lightweight little scope, but it's "little" right?  Much "littler" than a 6-incher.  No contest for your intended purpose...or is it?

Turns out it is every bit a contest between the two scope; the "little" 110mm refractor will go just as deep as the "bigger" 6" SCT, and for your purposes be a reasonable alternative.  Thank heavens for the Rule of Three 7s!  Now you can pick up that "little" 4.4" refractor you've always wanted, enjoy its wide field views, and when it comes to the task of globulars, push up the magnification and resolve them to your heart's content.

- Jim

Edited by jrbarnett, 15 March 2015 - 11:47 AM.

### #8 junomike

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Posted 15 March 2015 - 11:46 AM

Interesting post Jim.  You 7Â³ seems about right from my experiences.

Mike

### #9 GJJim

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Posted 15 March 2015 - 11:56 AM

Well the "old" rule of thumb was based on subtraction. Subtract the diameter of the CO from the aperture to get the equivalent unobstructed aperture. Granted most COs are around 35%, and using .777 would give a reasonably valid result. However in practice the ratio of CO diameter to aperture can be as low as 20% for a long FL MCT, and as high as 45% for some of the newer SCTs.

### #10 junomike

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Posted 15 March 2015 - 11:59 AM

Well the "old" rule of thumb was based on subtraction. Subtract the diameter of the CO from the aperture to get the equivalent unobstructed aperture. Granted most COs are around 35%, and using .777 would give a reasonably valid result. However in practice the ratio of CO diameter to aperture can be as low as 20% for a long FL MCT, and as high as 45% for some of the newer SCTs.

I believe this was for Refractor Contrast equivalence, not light grasp? (or so I thought?).

Both do seem relatively close however.

Mike

### #11 JonNPR

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Posted 15 March 2015 - 12:42 PM

Can the amount of light grasp in various scopes be empirically measured? Rather than estimated? Aren't there instruments capable of measuring a standard "candle" at a standard distance, whether terrestrial/artificial or astronomical, say the full moon or Jupiter or Polaris, etc?

Jon

### #12 jrbarnett

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Posted 15 March 2015 - 12:54 PM

Well the "old" rule of thumb was based on subtraction. Subtract the diameter of the CO from the aperture to get the equivalent unobstructed aperture. Granted most COs are around 35%, and using .777 would give a reasonably valid result. However in practice the ratio of CO diameter to aperture can be as low as 20% for a long FL MCT, and as high as 45% for some of the newer SCTs.

Actually the 0.777 figure is based on the following assumptions.

All scopes with secondaries obstruct some of the area of the primary.

SCTs and MCTs generally lose more primary area to obstruction than Dob/Newts.

SCTs and MCTs have enhanced mirror coatings and fully multi-coated correctors.

Dob/Newts have standard aluminized primaries with quartz overcoat.

Light loss in an obstructed system has as much to do with reflection losses of the mirrors (transmission losses of the multi-coated correctors are negligible) as primary obstruction losses.

Example:  The C6 has a 36% CO by diameter.  The XLT corrector transmits 99.75% per surface; the XLT primary and secondary transmit 95% each; total system throughput is 89% before reduction for the big secondary.  That's 11% light loss due to coating inefficiencies, wholly apart from the losses due to the large obstruction.  And Dob/Newts using standard coatings are even worse (about 79-80% efficiency).

No effort to calculate light loss due to vignetting in SCTs and MCTs was made.

The figure was derived by actually subtracting secondary area from the primary area, and applying the actual mirror and, if applicable, corrector transmission data for several obstructed scopes and comparing them to fully multi-coated triplets and doublets of various apertures.  And then verifying the prediction with field comparisons of scopes obstructed and not which "by the predicted numbers" ought to have similar limiting magnitude.

It's a ballpark, not a precise tool, but in practice I think you'll find it pretty close to accurate in the field.

- Jim

Edited by jrbarnett, 15 March 2015 - 01:02 PM.

### #13 jrcrilly

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Posted 15 March 2015 - 01:24 PM

Well the "old" rule of thumb was based on subtraction. Subtract the diameter of the CO from the aperture to get the equivalent unobstructed aperture

That wasn't it. One would divide the obstruction into the aperture to obtain the percent obstructed. Then that figure would be squared to reflect the fact that light grasp varies with area rather than with diameter.  Then the aperture would be reduced by that percentage to obtain equivalent unobstructed aperture. This yielded figures that were optimistic (and much larger than Jim's method attains) because they didn't take into account losses in the reflective coatings of two surfaces. Jim's figure includes both factors so it has potential to be much more realistic. It seems pessimistic to me, but he has done comparisons and I haven't.

### #14 jrbarnett

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Posted 15 March 2015 - 02:03 PM

Well the "old" rule of thumb was based on subtraction. Subtract the diameter of the CO from the aperture to get the equivalent unobstructed aperture

That wasn't it. One would divide the obstruction into the aperture to obtain the percent obstructed. Then that figure would be squared to reflect the fact that light grasp varies with area rather than with diameter.  Then the aperture would be reduced by that percentage to obtain equivalent unobstructed aperture. This yielded figures that were optimistic (and much larger than Jim's method atains) because they didn't take into account losses in the reflective coatings of two surfaces. Jim's figure includes both factors so it has potential to be much more realistic. It seems pessimistic to me, but he has done comparisons and I haven't.

I did the numbers based mostly on MCTs with multicoated correctors and standard coated mirrors.  You're probably right about it being conservative when applied to a modern SCT with premium coatings on the mirrors and corrector (all surfaces).  You can see where the reflectivity of the mirrors or lack thereof plays a very big role.  There's a big difference between 89% reflectivity mirrors and 94.7% reflectivity mirrors (Celestron's claim for XLT mirror coatings).

In fact the error if the catadioptric uses the highest reflectivity mirror coatings could be as much as 10% (i.e., the right "rule" for an XLT scope could be 0.877 rather than 0.777).

On the other hand, there are additional light losses manifest as contrast reduction (scatter) and internal vignetting that are not captured by the model.  So "seat of the pants" in the drivers seat, matching magnifications, the rule of three 7s seems to work at the eyepiece.  I realize of course that eyes have trouble detecting even a 10% difference in illumination, so there's certainly some slop in the rule.

- Jim

### #15 Jon Isaacs

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Posted 15 March 2015 - 02:21 PM

I think you need to actually make the calculation.. I think you forgot a square root. When computing the effective aperture, the you need to take the square root of the throughput multiply it by the aperture to calculate the effective aperture. If the transmission is 78%, then the equivalent aperture would be reduced to about 88%..  To use you 78% rule, the through put would have to 60%.

But I think you need to make the actual calculation., Spectrum claims their enhanced coatings have an average throughput of 96.5% from 450nm to 650nm. There is no reason to believe a custom coating house can' do as well as Meade or Celestron.

So, I do the calculation for a 16 inch Newtonian with a 25% CO with  the enhanced spectrum coatings:

TP = .965 x .965 x (1-.25^2) = 0.873

Since light gathering is proportional to area.

The equivalent aperture is

Aeq = 16 x .873^.5 = 15.0 inches.

Using 0.90 reflectivity, I calculate an equivalent aperture of 13.5 inches.

This is for light throughput. It assumes no transmission losses in the refractors glass, probably a reasonable assumption in a small refraction, probably unrealistic is a large refractor, 10, 12, 14 inches.

In terms of resolution, that's independent of aperture.

Bottom line:  The rule three sevens might better be stated as the rule of two eights.  A scope whose thoughput is 77.7% will have the effective light gathering of a scope whose aperture is 88% of its actual aperture.

Something to consider anyway..

Jon

### #16 jgraham

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Posted 15 March 2015 - 02:58 PM

That is pretty close to my experience. I always thought that my 6" achromat compared well with my 8" SCT. I also ran the numbers and concluded that they should be very similar.

### #17 Erik Bakker

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Posted 15 March 2015 - 02:59 PM

You're an optimistic bunch

From what I've seen in smaller apertures, where the indicated equivalents are readily available, I'd deduct 10 % off Jim's rule of 7's.  I have not seen an 127mm SCT show a brighter or more detailed image than 102mm apo's, but it does better in some respects than an 80mm apo. So that suggests that a 90mm would be the rough equivalent, i.e. 0,7x factor. And that's on the verge of being optimistic in my book

### #18 Erik Bakker

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Posted 15 March 2015 - 03:11 PM

My Questar 7 with BB and special coatings also seemed to obey the 0.7x rule

### #19 austin.grant

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Posted 15 March 2015 - 05:02 PM

I think your simplified "rule" is fine for comparing light gathering ability. You hit the nail on the head, however, when you mentioned that there are certainly other factors. I'm more interested in contrast and resolving power between the different types. I found this website to be quite informative, but didn't notice any credentials or peer-review so take it with a grain of salt!

### #20 areyoukiddingme

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Posted 15 March 2015 - 05:09 PM

My seat of the pants comparison between my 101.6mm apo and c6 shows that the c6 goes just a bit deeper than the apo. . . if the seat of my pants isn't far off, then I'd say a .7 is a closer approximation.

### #21 jrbarnett

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Posted 15 March 2015 - 05:13 PM

So 0.777 and 0.81 seem to agree pretty well, particularly when "standard" coatings may not be 90% reflectivity, especially with some mileage on them.    Celestron gives standard aluminum coatings ad 86-88% for example.  And then, when you factor in other issues affecting obstructed systems in addition to blocking light, the 0.777 might even look optimistic for a Dob.  The obstruction also displaces some percentage of light from the airy disc to the first diffraction ring; this is in addition to the light it blocks.

- Jim

### #22 Jon Isaacs

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Posted 15 March 2015 - 06:52 PM

So 0.777 and 0.81 seem to agree pretty well, particularly when "standard" coatings may not be 90% reflectivity, especially with some mileage on them.    Celestron gives standard aluminum coatings ad 86-88% for example.  And then, when you factor in other issues affecting obstructed systems in addition to blocking light, the 0.777 might even look optimistic for a Dob.  The obstruction also displaces some percentage of light from the airy disc to the first diffraction ring; this is in addition to the light it blocks.

- Jim

0.777 by aperture means that the total throughput is only 60%.  Celestron gives standard coatings as 86%-88% because they are trying to sell you their fancy coatings...

Both EJN and I showed our calculations.  I would like to see yours...

An easy formulation:

Effective Aperture = Aperture x Reflectivity x (1-CO2).5

This takes into account the fact that the area is proportional to the square of the diameter. That is why the reflectivity not squared and the CO is taken to the 1/2 power.

The obstruction: that's a red- herring.. First, light throughput is a calculation that is important for faint deep space objects, but the size of the Airy disk is not, the eye is not able to resolve fine contrast at those low light levels. The size of the Airy disk is irrelevant.  Consider the size of the Airy Disk of a 16 inch Newtonian. It's a fraction of an arc-second. But secondly, even if it were important, the smaller scope has the larger Airy disk, by area, a 12 inch scope has a disk whose area is 80% greater than that of a 16 inch.. Talk about spreading out energy....

In any event, these are calculations that one can do relatively easily, there is no need for a rule of thumb.  A scope with a small central obstruction and fresh, enhanced coatings might have an effective aperture of 95%, a 10 inch would have the through-put of a perfect 9.5 inch scope.  A scope with dirty mirrors, even damaged mirrors and a larger central obstruction might have very poor transmission, I calculate that an SCT with coatings at 80% with a 35% CO would be at 75%, an 8 inch SCT might have the equivalent through-put of a perfect 6 inch.

Jon

### #23 Laika

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Posted 15 March 2015 - 07:36 PM

So 0.777 and 0.81 seem to agree pretty well, particularly when "standard" coatings may not be 90% reflectivity, especially with some mileage on them.    Celestron gives standard aluminum coatings ad 86-88% for example.  And then, when you factor in other issues affecting obstructed systems in addition to blocking light, the 0.777 might even look optimistic for a Dob.  The obstruction also displaces some percentage of light from the airy disc to the first diffraction ring; this is in addition to the light it blocks.

- Jim

Are we talking APOs or Achros?

While Im not an optics geek, doesn't the CA remove light from the airy disc?

### #24 GJJim

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Posted 15 March 2015 - 08:19 PM

So 0.777 and 0.81 seem to agree pretty well, particularly when "standard" coatings may not be 90% reflectivity, especially with some mileage on them.    Celestron gives standard aluminum coatings ad 86-88% for example.  And then, when you factor in other issues affecting obstructed systems in addition to blocking light, the 0.777 might even look optimistic for a Dob.  The obstruction also displaces some percentage of light from the airy disc to the first diffraction ring; this is in addition to the light it blocks.

- Jim

Are we talking APOs or Achros?

While Im not an optics geek, doesn't the CA remove light from the airy disc?

We've reached the point in this discussion where a "rule of thumb" requires an Excel spreadsheet and calibrated, three decimal place reflectivity measurements -- just to get started.

### #25 mogur

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Posted 15 March 2015 - 08:36 PM

I may have missed it, but was the light loss of the corrector-plate/meniscus taken into account for those scopes that have them?

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