Use Nominal measures to make calculations easier
First, we are not concerned here that the area of the lens is Pi R squared. That actually is not even a useful measure by itself. It is not significant to know the area of an 8” lens is 50.26 sq inches. What we are concerned with is the ratio of the area of one size lens to another lens. That gives us the difference in light gathering in ratio or in percent. That's the comparison that is always made throughout these discussions. Lens comparisons are simple mathematical proportion. So it is sufficient to calculate a nominal area for any size lens to compare with any other, as long as the nominal calculation is the same for every lens. The following short comparison of precise formulas and nominal formulas shows this is true.
Using the exact area foumula, pi r squared
A 8" lens has pi x 4x4 = 50.2655 sq in area
A 6" lens has Pi x 3x3 = 28.2743 sq in area
Then the ratio between them is
50.2655 / 28.2743 = 1.7778
Performing the same calculations using nominal measures gives
64 / 36 = 1.7778
To get percent difference in lens areas, not only is comparing nominal area accurate, but it is precise to the fifth decimal place, far greater precision than can be applied to this sort of problem. This can be done for any lens comparisons.
To What should Binocular Summation Factors be Applied
To allay misunderstanding at the outset, binocular summation factors are not applied to aperture diameter, they are applied on the area of the aperture being used to deliver light. Further, these factors are applied on the aperture delivering light to each eye, not the total area of the two apertures delivering light to both eyes. What I mean is this; you would not add the area of two 70mm lenses to get 4900 + 4900 = 9800, then take the sqrt 9800 to find total light delivered from a total 99mm aperture. The light is delivered from a 70mm aperture to each eye. The binocular summation factors are applied to that 70mm aperture.
What is Binocular Summation
Binocular Summation is the end result of a neural process. Studies have been performed and published in this regard. Using two eyes to observe produces gains in the neural processing of information. In a binocular or binoviewer/telecope there are three changes that can be measured; light gathering or limiting magnitude, contrast (one way to measure contrast is the ability to see extended object surface brightness) and resolution. Actually, there are two kinds of resolution.
A binocular delivering light to two eyes produces an equivalent gain in light gathering of about 20%-40% over that of the single aperture of one barrel to one eye. It produces a contrast gain of up to 40%. The portion of the gain that is from very slight increase in brightness, a gain in LM or a gain in contrast cannot be separated. But you can try to test at least for LM and contrast. There are increases to resolution due to several of the other attributes that realize gain. Resolution may be seen to show a gain of one line on a Snellen chart.
The brain perceives anywhere from 20% to 40% more information when viewing an object with two eyes (binocular summation) rather than one. I base my statements on all the articles written over the years on binocular summation. In addition, several times I have tested the gain in limiting magnitude and I have found it to be 20% to 35%. Also I personally did record a gain of one line on the Snellen chart using two eyes vs one.
A good source for reference is
I have linked here to one article on binocular summation, but there are others, and they make a clear distinction between light gathering and contrast. Observing extended objects is a good way to address contrast, but not at all easy to quantify. Observing limiting magnitude is an easy way to quantify light gathering.
Binocular Vision has been found to follow the rules of quadratic summation.
The links I provided include two papers that give some good explanation of binocular vision. One a college course module, the other a scientific study.
From that study:
"One common form of the probability summation model is one that assumes that binocular sensitivity can be predicted by the square root of the summed squares of the two monocular sensitivities (quadratic summation)—i.e.,
Sqrt [(SL)sqrd + (SR)sqrd]
where SL and SR are the monocular sensitivities of the left and right eyes, respectively, for corresponding visual field locations. We refer to this as the BINOCULAR SUMMATION model, and selected this particular form of binocular summation because it accurately predicts binocular contrast detection and other binocular visual tasks.
This model predicts that binocular sensitivity is approximately 1.41 times (40%) better than individual monocular sensitivities, assuming that the monocular sensitivities are equal. The larger the difference in sensitivity between eyes, the more the predicted binocular sensitivity approximates the value of the most sensitive eye. The lower the sensitivity of the worst eye, the less it contributes to binocular sensitivity. "
This is the most common predictive model used in Binocular Vision studies, and it is this model that results in the Sqrt 2 reference or more precisely 1.41x gain.
The gain factor is calculated by comparing (the the sum of the light to two eyes) versus (the amount of light reaching one eye). The the sum of the light to two eyes is Sqrt [(SL)sqrd + (SR)sqrd].
Using an example, the effective gain from light delivered to two eyes is
for 100mm binoculars
Sqrt [(100)sqrd + (100)sqrd ] = Sqrt 20000 = 141.
Then, 141/100 = gain factor, = 1.41
The gain factor is applied to the information reaching the dominant eye. Assume in this case it is getting the full 100mm or 10,000sqmm of light. The gain is then 10,000 x 1.41 = 14000. That represents an aperture of Sqrt 14,000 = 119mm.
for 100mm scope with binoviewer
first scope splits light, therefore (100 sqrd / 2) = 10000/2 = 5000 sqmm
therefore each eye receives only sqrt 5000 or 70.7mm aperture of light
then binocular vision takes place
Sqrt [(70.7)sqrd + (70.7)sqrd ] = Sqrt 10000 = 100.
Then, 100/70.7 = gain factor, = 1.41
(The Binocular Vision Gain factor will always be 1.41)
The gain factor is applied to the information reaching the dominant eye. Assume in this case it is getting the full 70.7 or 5,000sqmm of light. The gain is then 5,000 x 1.41 = 7000sqmm. That represents an aperture of Sqrt 7,000 = 84mm.
Testing LM for One Eyed Viewing vs. Two Eyed Viewing
I've done similar tests in the past to prove this for myself. See the archives “Two Eyes vs One”, where I tested by using a Fujinon 16x70 first by looking thru both lenses and then by looking thru just one lens, with my good eye. I also tested for limiting magnitude by observing a fixed field of stars first with a 16x70 Fujinon. Then I observed the same field using both a Stellervue AT1010 and a TV85. The AT1010 at 78mm aperture is slightly smaller than the effective 16x70 and the TV85 is slightly larger. I used good eyepieces, a 30mm Ultima to get 16x in the AT1010, a 26mm TV plossl to get 18x in the AT1010 and a 40mm TV plossl to get 15x in the TV85.
In every instance I found a gain of magnitude with the 16x70 Fujinons over the single eyed view thru the scopes. Overall I got a gain estimated at about 0.2 mag with the Fujinons and two eyes. The slight increase in magnification from 16x to 18x in the AT1010 did give a slight gain about 0.1 magnitude, but that did not exceed the 16x70 Fujinons. This test was performed under mag 4.4 skies.
The slightly higher magnifications with slightly lower effective aperture of the AT1010 and the slightly lower magnification on the TV85 which has a somewhat higher effective aperture, I think bracketed the target comparable equivalent very well. A 40% gain in limiting magnitude would be expected with an equivalent aperture gain in area of 40% or 70x70x1.4 = 6860. Sqrt of 6860 = 82.8mm.
In another test, observing from the Cr399 chart (see gallery for LM chart) on a very good night under my best mag 5.7 skies when the faintest star that could be observed in the Fujinon 16x70 was mag 10.83, several attempts were made with the same Fujinon 16x70 to record a difference in two-eyed vs. one-eyed views. The same mag 10.83 star could not be seen with the Fujinon 16x70 when attempted using either eye individually. Care was taken to eliminate any eye strain. The star E6-10.4 could still be seen with the Fujinon when attempted with my right eye. Star E8-10.5e was seen with the Fujinon when using only the dominant left eye. This test gave a 0.33mag gain from the single view dominant eye and a 0.43 mag gain from the less dominant eye. That’s 35% to 48% gains. At least 35% would be experienced, since I would always compare to my dominant eye.
These tests prove that binocular summation does exist, and it is quantifiable. You should be aware, the target areas I used to record differences have a vast array of stars with LM identified and incremented by 0.01-0.02 magnitude. Testing requires a very detailed chart, experience observing to reach limiting magnitude and persistence to observe to your personal limits. It sometimes takes 5 minutes to see a star at the limits of visual magnitude. Usually, the faintest stars are observed for periods of only a few seconds at a time. Refer to the article on Binocular Limiting Magnitude.
Binocular summation, or the gain experienced by two eyes viewing is 40% gain from a single aperture. This test done with binoculars and single equivalent scopes is a bit different than a binoviewer example, but it proves the same concept.
Pogson's equation gives the percent gain in light gathering between two stars observed. M1/M2 = 2.512^(m2-m1). The gain in light gathering ratio for a magnitude difference of 0.3 mag is 2.512^(0.3) = 1.32 or 32%. For a 0.4mag gain it is 2.512^(0.4) = 1.445 or a 44.5% gain.
This may become somewhat of a deficient point for binocular vision. It is well known thhat each individual has an optimum location for detecting low contrast objects with averted vision. The location may vary with the individual, however there are studies that show location, IIRC, about 15° off center and generally at an angle from the center of vision. It shouldn't be too hard to understand that it would be near impossible to have the image fall on the optimum location in both eyes at the same time. Hawever, all of the above tests done to predict gain in limiting magnitude are done observing stars at the 5%-10% limit of vision, meaning they are only visible 5%-10% of the time viewing and they are always observed with averted vision. So while the averted vision object image cannot fall on the optimum location in both eyes at the same instant, still the eyes see more averted vision with binocular vision than without.
What Changes Occur by Binocular Vision Summation
In a telescope or binocular or binoviewer there are three changes that can be measured, light gathering or limiting magnitude, contrast (one way to measure contrast is the ability to see extended object surface brightness) and resolution. Resolution is not nearly affected by binocular summation as much as the other two.
I will use the maximum value of 40% gain. Some people do not experience that much gain, however let's just go with it and call it the maximum that might be experienced. You should be aware though that this much gain is not always experienced.
A 6" scope has a (nominal) 36" sq in. area. When you use two six inch scopes, one for each eye, you get to apply the 40% gain factor, so now the effective area of binocular summation light gathering is 36 x 1.4 = 50.4 sq inches. The sq rt of 50.4 is the effective diameter of a scope that would provide that information to one eye. That's 7.1". But two 6" scopes only provides slightly greater point source resolution than a single 6" scope. It may not provide the same resolution of a 7.1" scope.
Suppose you were to take an 8" scope and use a binoviewer. Take the 8" area = 64 sq in. Thru the binoviewer the light gets divided in half to each eye. Therefore, 32 sq in to each eye. Now you can apply the binocular vision factor to the amount of light that each eye receives. So 32 x 1.4 = 44.8. Sq rt 44.8 = 6.7". So an 8" scope with a binoviewer provides the effective light as in a 6.7" scope to one eye. The 8" scope with binoviewer is nearly equal to the effective light gathering of the two 6" scopes. The two scopes provide 12% more light than the binoviewed 8", that's about 0.1 magnitude gain. But the 8" scope as is to one eye provides even 27% more light than the two scopes, that's almost another 0.3 mag.
The 8" scope with binoviewer still provides at a minimum the resolution of an 8" scope and possibly slightly better. Binocular summation cannot increase or decrease the size of the Airy disk, but it does cancel out a slight amount of noise, so it improves your accutance. This explains why resolution is more in line with whatever size scope you started with and not the reduced effective aperture.
Binoviewer vs Binocular
Both the binoviewer and the binocular produce binocular summation. From what we understand about summation, a binocular produces gains in a range of 20% to 40% summation increase over the area of one aperture to one eye. So does a binoviewer, but a binoviewer must first split the light in half. Therefore, in a binoviewer, each eye gets light from only one half the area of the scope aperture used.
Let's assume we start with a 6" scope with binoviewer, then 150x150 = nominal 22500 mm squared for area of the lens. Half to each eye = 22500/2 = 11250 sqmm to each eye. Therefore each eye receives the equivalent from a sqrt 11250 = 106mm aperture. There's your light splitting loss to ½ the area to each eye. Now summation occurs, assumed 40% gain. Therefore binocular summation results in the neurological processing of the input as if equivalent to observing light delivered to one eye from 11250 x 1.4 = 15750 total equivalent area = sqrt 15750 = 125mm equivalent aperture.
A 150mm scope with binoviewer splits light into equivalent of two 106mm apertures then two eyes and brain result in binocular summation as if the light were delivered from an equivalent 125mm scope.
So then, what size binocular would produce the same result? A 106mm binocular. The simple shortcut to that answer is in the math above. 15750/1.4 = 11250. Sqrt 11250 = 106mm.
A 106mm binocular provides binocular summation to two eyes and the brain as if to equal what the brain could process if the light were from one single scope aperture to one eye from a 125mm scope.
The binocular is simply benefiting from summation. The binoviewer must first split the light in two, then can benefit from summation.
In general I wouldn't expect 100mm binoculars to pull in more light or resolution than any 6" scope. However, in your 6" scope, you may have eyepiece scatter, poor contrast and slight misalignment, or any small amount of some of the above. I have a 6" CR150. I also have a BT100 and 25x100 Oberwerk IF. At no time can either of the 100mm binoculars reach the resolution of the 6" scope.
A common mistake sometimes made in this math is that people will apply summation to the diameter of the aperture. For light gathering and contrast it should be applied to the area of the aperture, since light gathering is based on area of aperture. I think we all agree on that, but it's worth mentioning for all who will read this. Things will be a little different when we talk about resolution. Another common mistake people make is they add the two binocular apertures. That cannot be done because the light is not being delivered to one eye, It is being delivered to two separate eyes.
The question was asked: How does a decent quality 100mm binocular compare to a high quality 5" refractor with binoviewers? Seems like the binoculars would gather more light but the refractor would allow higher powers (150x+) since it is better corrected. However, I rarely have preferred the views at 150x+ due to atmosphere. Comments...
Here's how you compare. All of these measurements are nominal.
The 100mm binocular does not have 25% more light gathering than the NP127. The NP127 actually has a lot more light gathering than the 100mm binoculars. When used as a straight scope the NP127 has 127 squared or 16,129 mmsq area. You cannot add both apertures of the binocular. You need to look at binocular light gathering as the summation of the apertures. Binocular Summation of two apertures to two eyes provides a gain of up to 40% over the area of one aperture to one eye. So the light gathering of the 100mm binoculars is 100 squared x summation factor of 1.4 = 14000 mmsq. The NP actually has 15% greater light gathering when used as a straight scope.
But that all changes when you introduce a binoviewer.
Both the binoviewer and the binocular produce binocular summation. From what we understand about summation, a binocular gains anywhere from 20% to 40% summation increase over the area of one aperture. But a binoviewer in a scope first splits the light in half.
The NP127 is a 5" scope, then 127x127 = nominal 16129 mm squared for area of the lens. The binoviewer splits the light, half to each eye = 16129/2 = 8065 sqmm to each eye. Therefore each eye receives the equivalent from a sqrt 8065 = 90mm aperture. Now summation occurs. (It varies by object, by aspect and by person, from approximately 20% to 40% gain). Assumed maximum 40% gain, but please realize for some aspects it will be less, so this is the maximum possible. Therefore binocular summation results in the brain observing input as if delivered to one eye from 8065 x 1.4 = 11290 total equivalent area. The the diameter would be = sqrt 11290 = 106mm aperture.
The NP127 with binoviewer splits the light in two, then binocular viewing adds summation. The net affect is the NP127 used in this way delivers the equivalent of a 90mm binocular or the net summation from a 106mm scope.
So then, what size binocular would produce the same result as above? A 90mm binocular. The simple shortcut to that answer is in the math above. Since the binocular already delivers binocular summation then, 11290/1.4 = 8065. Sqrt 8065 = 90mm.
To recap, A 127mm scope with binoviewer splits light into equivalent of two 90mm apertures then two eyes and brain result in binocular summation as if the light were delivered from an equivalent 106mm scope. That is the equivalent of binocular summation received from a 90mm binocular.
So how does that all compare to a 100mm binocular?
A 100mm binocular provides binocular summation to two eyes and the brain as if to equal what the brain could process if the light were from one single scope aperture to one eye from 100*100*1.4 = 14000 Sqrt 14000 = a 118mm scope.
The binocular is simply benefiting from summation. The binoviewer must first split the light, then it can benefit from summation. But the fact of the matter is a 100mm binocular will handily beat a 127mm scope with binoviewer when used at the same low powers. However, the binoviewer has the added advantage of exercising various powers.
As far as a fine 100mm binocular versus a fine 127mm scope with binoviewer, given the same or approx the same magnification, with the exception of point source resolution, the binocular would win hands down.
In all likelyhood the 100mm binocular, being of fine quality and providing greater summation than the NP127 with binoviewer, should show greater contrast images when used at the same magnifications.
I'm not surprised at all the 100mm binoculars appear brighter and with more contrast than the NP127 with binoviewer. A 100mm binocular compares directly to a 118mm scope. BUT when using a binoviewer in a scope it would take a 141mm scope to equal the binocular summation light delivered by a 100mm binocular. A 127 scope provides only 80% of that total light, so there is somewhat of an unfair advantage to the binoculars. Although it's hard to imagine that there could be much of an unfair advantage against an NP127, this binocular has it.
Here's a very good example of what happens when you choose a binoviewer to compare to low powered viewing.
In this example, a 180 mm scope with binoviewer at 20x will yield 9 mm exit pupils. However with two 130s on a binoscope you would get a more reasonable 6.5 mm. What's the differences.
Now this would be interesting optics problem for the person who has not thought thru the binoviewer arrangement. Let's assume this person has a 6.5mm eye pupil, fairly large I would say.
Right out of the gate, the eye can only use 6.5/9 or 72% of the light delievered in the 9mm exit pupil. So the effective aperture of the 180mm scope/binoviewer is now down to 72% x 180 = 130mm. At low power of 20x it will be reduced to an effective aperture of 130mm due to exit pupil larger than eye pupil.
Ah, but that's not all. The effective aperture to each eye is now based on aperture delivered thru the 6.5mm eye pupil. Since each exit pupil in a binoviewer contains only one half the light, that is now (130x130)/2 = sqrt 8450 = 92mm. So a 180mm scope with binoviewer at a low power of 20x with 9mm exit pupils would not deliver any more light to the observer's eyes than a 20x92 binocular.
Ah, but that's not all. We haven't even adressed vignette in the binoviewer system. The system would actually deliver less light than that due to vignette of the eyepiece field stops. But we'll just ignore that say this, you don't want to be using huge scopes with binoviewers to match the light of very large exit pupil binoculars. Due to the light lost from the oversized exit pupil, you throw away a huge chunk of aperture when you try to do that.
What becomes of resolution in a binoviewer or binocular? It certainly is not halved in the binoviewer and then regains 40% from summation. Most empirical reports seem to indicate resolution may still be based on the original aperture. I have not had the benefit of using a binoviewer to test this. I will someday. But there is evidence that tends to lean towards agreeing with this. We do not get a gain of double the resolution. Some users have noted an increased acuity, meaning the ability to see the resolution at a lower power. This agrees with what has been found in vision studies in that some individuals have been measured to gain one line on a Snellen chart.
Let's talk about contrast for a minute. While the potential gain is there, the amount of gain you experience in contrast will be affected by more than just binocular vision. It will be significantly affected by exit pupil. There are volumes written on using higher power to darken sky background to bring out low surface brightness objects. Low contrast detail and resolution is affected differently than high contrast resolution. This one aspect is difficult to judge separately from binocular vision. One would need to perform a series of controlled tests, probably one with the scope binoviewed, one without the scope binoviewed and a third with an equivalent aperture single eye scope. All would need to be at the same exit pupil. That is rarely the case, so nearly all comparisons of binocular view to same single aperture view or even to effective equivalent aperture view are influenced not only by binocular summation but also by exit pupil. The same holds true for binocular scopes.
Contrast is very much dependant on the entire system. You could have significant contrast losses that are not explained here. You could be using a larger exit pupil which may have the affect of reduced apparent contrast. There are simply too many variables to know.
A telescope with binoviewer produces what I would call a false exit pupil. The exit pupil is always larger than the amount of light that it delivers (as compared to exit pupil from scope without binoviewer). That is due to the fact the beam splitter delivers half the light to each exit pupil. This mathematics is all discussed above.
Think about it. A 6" scope with binoviewer at 30x has a 5mm exit pupil. But the light delivered to that exit pupil in each side of the binoviewer is (150x150)/2 = 11250, then sqrt11250 = 106mm. The true exit pupil would be 106/30 = 3.5mm, or an exit pupil with one half the area of the false exit pupil.
The light delivered to the exit pupil in a binoviewer is not as bright as the light in an equal sized exit pupil from either a scope or binocular. Hence, it can be described as a false exit pupil.
Binoculars and binoviewers generally cannot be compared at the same size exit pupil. There would be significant difference in the magnification needed in each to get equal sized exit pupils. Increased magnification allows you to see stars of fainter magnitude in any instrument. People often mistake that gain for brightness. It is not a gain in brightness, it is a gain in reaching closer to the maximum light gathering ability of the instrument by employing higher magnification. Maximum LM in any instrument cannot be achieved until you start using magnifications that result in exit pupils less than 2mm for extremely large scopes and on the order of 1.2mm to 0.8mm for small scopes.
You will often here the binoviewer users exclaim that the binoviewer appears much brighter. At same magnification, this may not even be an equivalent comparison. Whenever you use a binoviewer to get equivalent comparison to binoculars, the scope/binoviewer combo starts out with a larger aperture to be equivalent. Therefore, when compared at the same magnification, the scope/bv will have a larger exit pupil but because the light was cut in half by the beam splitter, it is delivering less light to the exit pupil. So it may not have a brighter image. Due to potential vignette in either the binocular or the scope/BV system, the brightness of image may vary between the systems.
How does a binoviewer split light one half to each eye or the equivalent to each eye as if from half the area of the aperture, and yet the exit pupil remains constant as if each eye were receiving light from the full aperture? To help understand, think about this in terms of internal light loss.
For instance, in a binocular, two easy to understand aspects lead to not getting out all the light that gets let in. One is all the light that enters a prism does not exit the prism, but that does not change the size of the exit pupil. If the choice of material or the cut of the prism is a poor one, a dramatic difference can be noted in the amount of light that reaches the exit pupil. Of course the other is the level of quality of the coatings. In the case of poor coatings an average of say 10%-20% of the light may never exit the binocular. So, in this example it is shown there is more than one way to subtract light from the exit pupil without changing the size of the exit pupil. I don't mean to imply that sort of light loss is going on in quality binoviewer instruments. I’m just attempting to show some examples of how, similar to the beam splitting reduction of total light to each exit pupil, light can be lost and brightness reduced without changing the size of the exit pupil.
But regardless, the resolution of a binocular is always equal to or even very slightly better than just using one lens at the same magnification. This I have tested on double stars and on a USAF resolution chart.
We hear very often that regardless of aperture or magnification, if two instruments have the same size exit pupil, then the image will be equally bright. Well, this is true if both instruments have the same transmission, but often that may not be reality. This accounts for why many times we will hear binocular users exclaim this 10x50 binocular appears so much brighter than that 10x50 binocular. Exit pupil is the same, transmission is not.
Brightness is controlled by exit pupil. But also by transmission. If you have two instruments of equal aperture and equal exit pupil, but one appears brighter, it is easy to understand that the other has greater transmission losses. So if you have differences in brightness, first I would ask are you comparing equal exit pupils. Second, you must consider the affects of aperture and magnification. You need to be aware, for instance that a 40x100 with a 2.5mm exit pupil will be very much different than a 150mm scope with a 2.5mm exit at 60x. The extra increased magnification from 40x to 60x, while truly not adding anything to brightness, will have a significant affect on how deep you can see. For example, if you can see to mag 12 with the binoculars at 40x100, when using your scope at 60x100, you would be able to see to mag 13. This may be perceived as increase in brightness, but it is not. I can conceive of no condition where a scope of that size would seem brighter than a 100mm binocular when comparing equivalent use parameters. So I think this may be either a comparison of non-equal parameters or simply the perception of a brighter image.
Back to the resolution issue, I can subtract an awful lot of light from the exit pupil without changing the resolution provided to the image. Point source resolution doesn't go down dramatically with fractional light loss, not until you subtract an awful lot of light, maybe on the order of a full magnitude to 2 magnitudes. Contrast may, but resolution doesn't, at least point source resolution anyway. The loss of contrast in my binocular example, 20% to coatings and some to prisms, (really high, I know, for a good binocular, but it makes the example) will affect extended object resolution.
In a fine instrument there will not be as great a loss to illumination, but there will almost always be greater loss of illumination in a binocular than in a scope. Less expensive binoviewers will have all the same light loss issues in the prisms that a less expensive binocular has.
my perception is that I can see more detail on various objects using binoviewers as opposed to one eyed viewing.
There are a lot of questions you need to answer?
At precisely what magnifications are you comparing the views?
Changing magnification will cange the perceived view.
At what exit pupils are you comparing the views?
At what apertures are you comparing the views?
How bright are the objects to which you refer? Near the limit of the scope, or bright?
Are you talking about extended objects or point sources?
Are the eyepieces you use having any affect on the overall image?
Are the effects you see perceived, or real? can you measure it? Will a measurement show that your perception is correct?
There are so many things that can be, and often are, different when making these comparisons, that quite often they are not equal comparisons at all.
For the average person with good balanced vision, almost any time you can view an object with two eyes, you have an advantage. But that does not always mean you should take a given scope and use it to binoview an object instead of monoview.
Suppose you have a scope that has a surface brightness limit of Sb mag 15.0. If you go out to view an object that is Sb mag 15.0, you aren't going to see it in that sccope with a binoviewer, but you have a chance with monoview. Even Sb 14.6 or 14.8.
It is extremely difficult to match up equal viewing aspects with a scope in binoview vs monoview. For one thing, if you tried to compare a scope mono vs bino at the same exit pupil, the exit pupils would not have the same amount of light. If you tried to adjust the exit pupil so the sizes varied but mono and bino now had the same amount of light, then magnification would be significantly different.
For the person that experiences a lot of "noise" when mono viewing, binociular viewing could provide a significant benefit. Viewing with two eyes cancels a lot of noise. When your left eye has a floater in the way, your right eye may be providing a clearr view, canceling the noise of the left eye floater.
Resolution stands to realize gains. We know resolution improves somewhat with binocular vision. My resolution chart tests showed binocular vision improved resolution by 10%. Eye exam tests sometimes show 20%-25% improvement, not full resolution but blur detecction. But both of those are very high contrast. We don't know that contrast improves, and in fact, if light decreases contrast decreases. So what probably happens here is high contrast objects may show some improvement in resolution, but low contrast objects probably will not.
The 1.41x factor is based on scientific studies of the eye/brain. 1.41 is the maximum gain that can be achieved. If a person has eyes that vary widely, the gain will be less. If one eye provides almost no support towards binocular vision, that person will experience almost no gain at all due to binocular summation.
Assuming a 20% gain to 40% gain for two-eyed vs. one-eyed viewing,
Binoculars are equivalent to larger scopes.
A 50 mm binocular is equiv to 55mm to 59mm scope
A 70 mm binocular is equiv to 77mm to 83mm scope
A 80 mm binocular is equiv to 88mm to 95mm scope
A 100 mm binocular is equiv to 110mm to 118mm scope
Scopes are equivalent to smaller sized binoculars
125mm SCT (minus c.o. net area=115mm), is approximately equivalent to a
At 20% gain approximately equal to a 105mm binocular.
At 40% gain approximately equal to a 97mm binocular.
85mm scope is approximate equivalent to
At 20% gain approximately equal to a 78mm binocular.
At 40% gain approximately equal to a 72mm binocular.
80mm scope aperture is approximately equivalent to a
At 20% gain approximately equal to a 73mm binocular.
At 40% gain approximately equal to a 68mm binocular.
78mm scope aperture is approximately equivalent to a
At 20% gain approximately equal to a 71mm binocular.
At 40% gain approximately equal to a 66mm binocular.
When you insert a Binoviewer, the total area of the light gets split
Assuming a 20% gain to 40% gain for two-eyed vs. one-eyed viewing,
150mm Ref/BV is approximately equivalent to a
At 20% gain approximately equal to a 116mm binocular.
At 40% gain approximately equal to a 125mm binocular.
125mm SCT/BV (minus c.o. net area=115mm), is approx equivalent to a
At 20% gain approximately equal to a 89mm binocular.
At 40% gain approximately equal to a 96mm binocular.
100mm Ref/BV is approximately equivalent to a
At 20% gain approximately equal to a 78mm binocular.
At 40% gain approximately equal to a 84mm binocular.
85mm Ref/BV is approximate equivalent to a
At 20% gain approximately equal to a 66mm binocular.
At 40% gain approximately equal to a 71mm binocular.
80mm Ref/BV aperture is approximately equivalent to a
At 20% gain approximately equal to a 62mm binocular.
At 40% gain approximately equal to a 67mm binocular.
78mm scope aperture is approximately equivalent to a
At 20% gain approximately equal to a 71mm binocular.
At 40% gain approximately equal to a 66mm binocular.