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

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### #26 Cotts

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Posted 15 March 2015 - 08:57 PM

So, in the final analysis, scope designs differ.  The variety of designs each have their pluses and minuses.  It is almost impossible to quantify these differences with a single number or factor....

who'd a thunk it?

Dave

### #27 Traveler

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Posted 16 March 2015 - 04:03 AM

Interesting thread Jim! Do you think the local seeing conditions is somehow part of your rule of thumb?

### #28 davyboy07

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Posted 16 March 2015 - 06:42 AM

A great original idea gone a bit silly. I may be mistaken, but I believe Jim's "rule" was never meant to be a rule.

### #29 Jon Isaacs

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Posted 16 March 2015 - 07:27 AM

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).

OK....     To compute this example:

Throughput:  The throughput is equal to the transmission of the corrector times the two reflectivities times the proportional area of the secondary mirror.

TP = 0.9975 x .95 x .95 x (1-.362) = 0.784

For the C-6, about 78.4% of the light entering the scope reaches the eyepiece.

What is the aperture of a perfect scope that will transmit 78.4% of a 6 inch aperture?

The area of a 6 inch scope is 6 x 6 x Pi/4 = 28.3 square inches.   The effective area is 28.3 sq-in x 0.784 = 22.2 square inches.

What is the aperture of the scope whose area is 22.2 square inches:

A = 5.317 inches.  You can verify this yourself..

For the C-6, the ratio between the effective aperture and the actual aperture is not 0.777 but rather 0.886 (5.317/6)  which is 6 inches times the square root of 0.784.

The premise that this thread is founded on is faulty, it may be that the average transmission of a reflector or CAT is about 78% but that does not mean the equivalent effective aperture is 78% of the scope's aperture, it means it is 88% of the scope's aperture.

If one wants a rule, the rule of the Crazy 8s seems reasonable.

Jon

### #30 Diego

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Posted 16 March 2015 - 07:40 AM

Hi!

Interesting post. I don't want to deviate from the original topic but how does CO affect resolution?

For comparisons sake, lets say we have a 90 mm mak with the equivalent un-obstructed aperture in the 60-70mm range. Would this scope have the resolving power of a 90 mm aperture or of a smaller 60 or 70mm?

Let's take the extreme case of a 50% CO, would it still resolve up to its total aperture?

For grab and go planetary viewing, I wouldn't mind having a 90-100 mm obstructed scope with a smaller true aperture, but with the resolution of a larger scope, with slightly less contrast and dimmer view.

### #31 bandazar

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Posted 16 March 2015 - 08:35 AM

The formula is too simple, imo.  But it does appeal to us lazy/convenience thinkers.  You're not factoring coatings, apo, acros, diagonals, vignetting caused by improper baffelings, quality control, etc.  You're sort of comparing oranges and apples.  Although there is some rough correspondence, the optics are not 100% comparable.  If I was a planetary viewer, I would put more emphasis on quality optics.  If I was deep sky, for example, I would put a little more emphasis on aperture.  But both qualities affect performance of both objects in different ways.

But there is a little bit of truth to the rough formula, even though it is not the truth.  A 90mm apo does perform somewhat like a 100mm achro, which does perform somewhat like a 5" sct on many objects.  But some objects are not really comparable, like Jupiter/Saturn for example, which probably will look much better in the refractors than in the SCT.  Ulness you get a very high quality cassegrain and everything is cooled, coliminated, etc.

### #32 Mark Costello

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Posted 16 March 2015 - 08:41 AM

My understanding is that the light gathering power is directly related to the effective area of an object (accounts for a central obstruction) and the throughput fraction for the elements in the objective - reflectivity for mirrors and transmissivity for lens.  Given this and that the throughput fraction is dimensionless, it appears to me that Jon and EJN are correct.

### #33 schang

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Posted 16 March 2015 - 08:56 AM

While rule helps in selecting a scope of similar apertures (say within two inches), there are other things that are more pertinent and critical than that...I called object based equipment.  I would not nitpicking to death about the merits of one from the other, when both may not be the right tool for certain objects. One simple example would be for DSOs, that 4-6" scopes, no matter what design or coating they have, are simply not quite suitable...likewise for double stars...

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Posted 16 March 2015 - 08:57 AM

Since when does a "rule of thumb" get confused with hard mathematics? Oh yeah, we're on CN.

### #35 jrbarnett

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Posted 16 March 2015 - 09:37 AM

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

Jon:

I did mine long hand.  That is computed the area in mm(2) of the primary and subtracted the area in mm(2) of the secondary obstruction.  I then took the resulting clear area and applied the per-surface transmission data to each surface (corrector at 99.25% x2 surfaces and primary and secondary at 87% (it's not only Celestron that rates standard coatings in the 86-88% range, though a key question is "over what wavelengths").  The resulting "area" I then divided by pi and then took the square root, yielding radius of the equivalent theoretically perfect unobstructed optic.

Regards,

Jim

### #36 jrbarnett

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Posted 16 March 2015 - 09:39 AM

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

Yes, but not based on absorption; merely based on coating type on each surface.

- Jim

### #37 Abhat

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Posted 16 March 2015 - 09:44 AM

If one needs to be reallly spot on with calculation then an assumption about what diagonal is used should be added. Typical diagonals have 92% reflectivity.  That loss should be applied to Cats and Refractors.

### #38 jrbarnett

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Posted 16 March 2015 - 09:46 AM

One simple example would be for DSOs, that 4-6" scopes, no matter what design or coating they have, are simply not quite suitable...likewise for double stars...

It really depends where you observe from for DSOs and which DSOs you chose to observe.  A 5.5" from NELM 7.2 desert is fantastic on brighter members of every class of DSO, showing structure in galaxies, substantially resolving bright globulars, etc.

As for double stars, you are in error IMO.  A 4" to 6" refractor will perform much closer to its theoretical limits than will a 10" or 12" obstructed scope on the vast majority of nights.  Refractors also manage light much better than obstructed designs, improving contrast at a given aperture.  Many, many nights one can pick out "tough" unequal doubles like Sirius with a 3" to 6" refractor, but be unable to do so with 6" and larger obstructed scopes.  Lastly once you factor in atmospherics the number of nights a large refractor (5" or 6") will resolve sub-1 arc second splits will be greater than the number of nights a 10" or 12" light bucket will do the same.

Refractors are the best tools for double stars.  The limit being their high cost per unit of aperture in the 6" range and larger.

- Jim

### #39 jrbarnett

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Posted 16 March 2015 - 09:51 AM

If one needs to be reallly spot on with calculation then an assumption about what diagonal is used should be added. Typical diagonals have 92% reflectivity.  That loss should be applied to Cats and Refractors.

That's a fair point.  Though I don't think 92% is correct for "most".  It's hard to find a diagonal that isn't 97% or better these days due to the dielectric coating fad.  But then if we wanted to model the entire system and not merely the instrument, we'd also have to factor in eyepieces and surfaces, and then the users glasses if he or she wears them.

- Jim

### #40 csrlice12

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Posted 16 March 2015 - 10:08 AM

While y'all were up all night discussing mathematics, I had my eye up to an eyepiece last night, and didn't think about math at all......

Did I win?

### #41 Mark Costello

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Posted 16 March 2015 - 10:15 AM

While y'all were up all night discussing mathematics, I had my eye up to an eyepiece last night, and didn't think about math at all......

Did I win?

### #42 Jon Isaacs

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Posted 16 March 2015 - 10:25 AM

Jon:

I did mine long hand.  That is computed the area in mm(2) of the primary and subtracted the area in mm(2) of the secondary obstruction.  I then took the resulting clear area and applied the per-surface transmission data to each surface (corrector at 99.25% x2 surfaces and primary and secondary at 87% (it's not only Celestron that rates standard coatings in the 86-88% range, though a key question is "over what wavelengths").  The resulting "area" I then divided by pi and then took the square root, yielding radius of the equivalent theoretically perfect unobstructed optic.

Regards,

Jim

Jim:

That is the long way around but it does result in the same numbers.  The results here are very dependent on the assumptions about the coatings.

If you use the coating numbers that Celestron advertises for the C-6, 0.95, the effective aperture is 5.32 inches  if you use the numbers for their standard coatings, the effective aperture is 4.87 inches.  If you use your rule of 7s, it's 4.62 inches.  These are fairly large differences.

This is what Celestron has to say:  Starbright XLT coatings.

Reading that, I see two things, they say the XLT corrector is less efficient than you assumed, less efficient than I assumed.  Secondly, I see no mention of the effect of central obstruction.  They say that the system transmission is 83.5% from 400nm to 750nm but I see no mention of the secondary area loss.  If one bases the effective aperture on that throughput and accounts for the size of the secondary, then the effective aperture of the C-6 is 5.11 inches... or about 85% of the original aperture.

These variations are why I suggested that it is important to actually make the calculation and not rely on a general rule of thumb.  Are the fresh coatings enhanced coatings on my 16 inch Dob really 96.5% average reflectivity, I doubt it, I doubt they are that high.  Are they 87%, I doubt they are that low.  That puts the bounds on that particular scope at somewhere between 13.5 inches and 15.0 inches..  at some point, it might have the light gathering of a perfect 12 inch aperture as you suggested but that would require the coatings to be about 78% reflective.

Jon

### #43 austin.grant

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Posted 16 March 2015 - 10:25 AM

While y'all were up all night discussing mathematics, I had my eye up to an eyepiece last night, and didn't think about math at all......

Did I win?

That depends on the shape and throughput of your cornea, coupled with the efficiency of your optic nerves. I'll have to create a rule of thumb so we can calculate what place you actually came in.

### #44 mikey cee

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Posted 16 March 2015 - 10:35 AM

What about a layer of dust on both the primary and secondary in reflectors? In cats the primary and secondary remain dust free. Just a light dusting on the front of refractors and cats is all that is needed to stay ahead of build up. Not likely with mirrors that most would never clean with any dry method only a bath.  Mike

### #45 Jon Isaacs

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Posted 16 March 2015 - 11:19 AM

What about a layer of dust on both the primary and secondary in reflectors? In cats the primary and secondary remain dust free. Just a light dusting on the front of refractors and cats is all that is needed to stay ahead of build up. Not likely with mirrors that most would never clean with any dry method only a bath.  Mike

I personally believe the dust build up is inconsequential, generally one washes the mirrors before it is a factor.  But the long term degradation of Newtonian coating is a real thing and I have seen coatings that have really suffered.

But in the big picture, the differences in telescopes are much bigger than just the light through put.... How would a 16 inch F/4.5 Newtonian with enhanced coatings and a 25% secondary compare to a say a 13.5 inch F/4.5 refractor?  I have to think that even with the most complex designs and the fanciest glasses, a color free 13.5 inch F/4.5 refractor is beyond the realm of possibility.

Jon

### #46 jrbarnett

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Posted 16 March 2015 - 11:33 AM

Interesting thread Jim! Do you think the local seeing conditions is somehow part of your rule of thumb?

Hard to say.  Probably not. If what you're using to differentiate is something like "dimmest star seen" which is, I think, the likely way you'd empirically (if casually) test the figure (0.777x).

- Jim

### #47 Abhat

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Posted 16 March 2015 - 11:59 AM

It is hard to nail this factor and validate its accuracy due to many variable involved. But it sure is easy to remember. 777 reminds me of Herman Cain's 999 campaign in 2012 presidential primary.

### #48 DJCalma

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Posted 16 March 2015 - 12:03 PM

My personal experience has shown that .777 seems closer to the mark than .88, and maybe the true answer is somewhere in between. However, I don't care what pages and pages of math say on the matter. The weather man's multitude of charts and computer simulations showed that there was no chance of precipitation last night, but I stepped outside and felt the raindrops myself.

### #49 schang

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Posted 16 March 2015 - 01:51 PM

One simple example would be for DSOs, that 4-6" scopes, no matter what design or coating they have, are simply not quite suitable...likewise for double stars...

It really depends where you observe from for DSOs and which DSOs you chose to observe.  A 5.5" from NELM 7.2 desert is fantastic on brighter members of every class of DSO, showing structure in galaxies, substantially resolving bright globulars, etc.

As for double stars, you are in error IMO.  A 4" to 6" refractor will perform much closer to its theoretical limits than will a 10" or 12" obstructed scope on the vast majority of nights.  Refractors also manage light much better than obstructed designs, improving contrast at a given aperture.  Many, many nights one can pick out "tough" unequal doubles like Sirius with a 3" to 6" refractor, but be unable to do so with 6" and larger obstructed scopes.  Lastly once you factor in atmospherics the number of nights a large refractor (5" or 6") will resolve sub-1 arc second splits will be greater than the number of nights a 10" or 12" light bucket will do the same.

Refractors are the best tools for double stars.  The limit being their high cost per unit of aperture in the 6" range and larger.

- Jim

There are always special cases to my general statement, but they are by no means correct across the board...

### #50 DJCalma

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Posted 16 March 2015 - 02:26 PM

However, I don't care what pages and pages of math say on the matter.

The weather man's multitude of charts and computer simulations showed that there was no chance of precipitation last night, but I stepped outside and felt the raindrops myself.

One of the most annoying things about CloudyNights is that anecdote and opinion
often count for more than well established science & engineering, ...and reality.

Optics are not like predicting weather, which is highly non-linear. The science

of optics is well established and matches empirical measurements to a high

degree of accuracy; that is an extremely flawed analogy.

Or do you think you can see the flag on the moon with a 60mm refractor if you

crank the magnification way up and try hard enough?

Please show me where my opinion differs from "well established science & engineering...and reality"?  The point is simply that I'd rather trust the empirical evidence of what is actually observed through the eyepiece than a mathematical formula, void of any empirical evidence whatsoever. I'm sure nobody who is debating the math on this issue would dare suggest that their .777 or .88 theory is based on "well established science & engineering." What annoys me is when someone sets up a straw man argument and then proceeds to knock it down.

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