In the most recent update I have made some changes and I also added new content, based on my additional experience. A year ago I bought a telescope and after some time I realised that the formula described in the present post is useful not only for comparing binoculars and/or spotting scopes, but also for comparing telescopes with particulars eyepieces.
I “discovered” a very interesting thing that can be helpful while deciding what kind of observing equipment to buy.
I have had 8x30 binoculars for a very long time, but recently I decided to buy something with better magnification. I had an opportunity to buy a 30x50 spotting scope for a very low price and I did it. When I compared these two observing tools after getting home I got very surprised.
In most cases while using the 30x50 spotting scope I was able to distinguish only marginally more details than with the 8x30 binoculars! Why? Because everything was much DARKER in the spotting scope and things that were not bright enough were hard to see at all, even though they were bigger.
MOST RECENT UPDATE – PART 1:
For some people it's an ever-going debate what is more important magnification (8x vs. 30x) or aperture (30 vs. 50), but the truth is that aperture on its own is meaningless – your equipment can't improve your own eyes. Looking through a telescope without any magnification would be like looking through some kind of tube. The bigger the aperture the bigger the tube. Would that improve anything? Obviously not.
The naked eye “works at your maximum aperture”, but you can't see things like the rings of Saturn with the naked eye. Well, you can't see them even in binoculars, but they are obvious already in a small telescope (magnification 70x is more than enough to see them). It's also consistent with the fact that it's better to use any kind of binoculars (any kind of magnification) instead of looking at stars with the naked eye (without any magnification).
So, to see more details you need magnification, no doubt about it. This fact alone decides that magnification is more important overall, BUT you have to increase aperture to PREVENT the view from getting too dark (bigger magnification => smaller exit pupil and darker view). And this is exactly what was missing in my 30x50 spotting scope – the magnification was good, but the aperture wasn't up to the task.
As far as binoculars are concerned the brightness is VERY important because it makes the view more “fun”. I bought and compared 8x42 and 10x42 monoculars side by side and I have to say that I can see a little more details and a few more stars (in a particular field of view) in the 10x42 monocular, HOWEVER it feels as if these “more details and more stars” can be seen as a result of “work” rather than “fun”:
1) the field of view is clearly narrower in the 10x42 monocular than in the 8x42 monocular,
2) the image is a little more shaky in the 10x42 monocular than in the 8x42 monocular,
3) the image is clearly darker in the 10x42 monocular than in the 8x42 monocular.
For most people a little more details and a few more stars (in a particular field of view) would not be enough to compensate for the less enjoyable image.
END OF MOST RECENT UPDATE – PART 1.
The observed difference in brightness can be calculated from the above numbers (“8x30” and “30x50”). Obviously we have to ASSUME that everything else is equal. The first number 8x (or 30x) represents magnification or “power” – how many times bigger the observed object appears. The second number 30 (or 50) is aperture – the diameter of each of the objective lenses (the lenses furthest from your eye), given in millimetres.
Let's start with the aperture (the second part of the final formula), because it's more natural. The bigger the aperture – the more light “goes into” the observing tool, so everything should be brighter. The amount of light “going into” the observing tool depends on the AREA of the objective lenses. Let's calculate the respective “going-in” areas:
1) …x30 “going-in” area = PI * R^2 = PI * (30/2)^2 = 3.1416 * 225 = 706.86
2) …x50 “going-in” area = PI * R^2 = PI * (50/2)^2 = 3.1416 * 625 = 1963.5
So if everything else were equal then the brightness of the …x30 observing tool would be 0.36 times of the brightness of the …x50 observing tool (706.86 / 1963.5). The value below 1 means that the …x30 observing tool would be DARKER than the …x50 observing tool. In other words the …x50 observing tool would be 2.7778 times brighter than the …x30 observing tool (1 / 0.36 = 1963.5 / 706.86).
The difference in brightness caused by magnification (“power”) is much more difficult to imagine, but one visual example should be enough to understand. Please remember that this example is NOT about focus and NOT about colours, but about the AMOUNT of light REFLECTED from the observed object that reaches the observing tool. In other words: the smaller objects reflect less light that can be magnified by the observing tool. Again, it's all about areas and again the bigger area improves the brightness, but this time the area is the biggest when the number (magnification) is the smallest!
To make my job easier I drew squares instead of circles, but the idea is clear anyway. The side of all pictures is 480 pixels long. The side of the bigger square is 60 pixels long (magnification of 8x…) and the side of the smaller square is 16 pixels long (magnification of 30x…). The area of the bigger square is 3600 square pixels (60x60) and the area of the smaller square is 256 square pixels (16x16).
As you can see on the last picture there are LESS “pixels of light” stretched (magnified) and this is why the bigger magnification is DARKER.
In reality we can't calculate any REAL areas because in real life there are no pixels to count and we don't know how far away the observed object is. All we can do is to calculate “visual” areas of the two magnifications relatively to each other. Obviously they have to be calculated as circles, not squares.
Assuming that the diameter of the whole visible area is X, then the diameter of the bigger circle (magnification of 8x…) is X/8 and the diameter of the smaller circle (magnification of 30x…) is X/30:
1) 8x… “visual” area = PI * R^2 = PI * ((X/8)/2)^2 = 3.1416 * X^2 / 16^2 = X^2 * 0.012271875
2) 30x… “visual” area = PI * R^2 = PI * ((X/30)/2)^2 = 3.1416 * X^2 / 60^2 = X^2 * 0.000872667
So if everything else were equal then the 8x… observing tool would be 14.0625 times brighter than the 30x… observing tool ((X^2 * 0.012271875) / (X^2 * 0.000872667)). This is exactly the same value as in the example above (3600/256 = 14.0625).
Now we have to combine these two results:
1) if everything else were equal then the brightness of the …x30 observing tool would be 0.36 times of the brightness of the …x50 observing tool,
2) if everything else were equal then the 8x… observing tool would be 14.0625 times brighter than the 30x… observing tool.
We have to simply multiply these two values:
0.36 * 14.0625 = 5.0625
The final value is above 1, so the 8x30 observing tool is 5.0625 times brighter than the 30x50 observing tool. This value explains why everything is much DARKER in my spotting scope and things that are not bright enough are hard to see at all, even though they are bigger than in my binoculars.
Let's simplify the overall formula as much as possible (using also the more natural order):
B&SSBC – Binoculars and Spotting Scopes Brightness Comparison
B&SSBC(A1xA2; B1xB2) = (B1/A1)^2 * (A2/B2)^2
For example:
B&SSBC(8x30; 30x50) = (30/8)^2 * (30/50)^2 = 14.0625 * 0.36 = 5.0625
MOST RECENT UPDATE – PART 2:
My formula works also for telescopes because it’s just a different (greatly simplified) way of comparing exit pupil areas. Below there is my formula with a better (general) name and less confusing “letters”.
EBC – equipment brightness comparison
M – magnification
A – aperture
EBC(M1xA1; M2xA2) = (M2/M1)^2 * (A1/A2)^2
Calculations for comparing exit pupil areas (EPA):
area = Pi * R^2
R = exit pupil/2
exit pupil = A/M
EPA = Pi*R^2 = Pi * (exit pupil/2)^2 = Pi * ((A/M)/2)^2 = Pi * (A/M)^2 * (1/2)^2 = Pi * (A/M * A/M) * 1/4 = Pi * A^2/M^2 * 1/4
EPA1/EPA2 = (Pi * A1^2/M1^2 * 1/4) / (Pi * A2^2/M2^2 * 1/4) = (A1^2/M1^2) / (A2^2/M2^2) = (A1^2/M1^2) * (M2^2/A2^2) = M2^2/M1^2 * A1^2/A2^2 = (M2/M1)^2 * (A1/A2)^2
END OF MOST RECENT UPDATE – PART 2.
The second problem with the 30x50 spotting scope is the fact that it's very hard to use by hand (without a tripod) – very distant observed objects are “jumping” all over the view because you can't keep your hands perfectly steady. For this very reason I ordered another pair of binoculars, this time 12x60 binoculars. Let's compare them to my old binoculars and my new spotting scope:
B&SSBC(12x60; 8x30) = (8/12)^2 * (60/30)^2 = 0.444444 * 4 = 1.777778
B&SSBC(12x60; 30x50) = (30/12)^2 * (60/50)^2 = 6.25 * 1.44 = 9.0000
Now, this is what I call an improvement over my old binoculars – in my new binoculars everything should be larger by 50% ((12-8)/8) AND the brightness should be better by 77%! Both these things combined should make a HUGE difference. We'll see (I will update this post as soon as the my new binoculars arrive and I verify it in reality).
Please notice that B&SSBC values can be calculated is steps:
B&SSBC(12x60; 30x50) = B&SSBC(12x60; 8x30) * B&SSBC(8x30; 30x50) = 1.777778 * 5.0625 = 9.0000
When I was searching various Internet forums about binoculars and spotting scopes I found many suggestions that the 10x50 binoculars are optimal, so let's compare them to the 12x60 binoculars I ordered:
B&SSBC(12x60; 10x50) = (10/12)^2 * (60/50)^2 = 0.6944444 * 1.44 = 1.0000
B&SSBC(10x50; 12x60) = (12/10)^2 * (50/60)^2 = 1.44 * 0.6944444 = 1.0000
EXACTLY the same brightness! So, in my opinion, I ordered the better binoculars – with the same brightness I should be able to see everything bigger by 20%. Using the 12x60 binoculars by hand (without a tripod) should be much easier than using my 30x50 spotting scope by hand, so I should have no problem at all to use them by hand.
On the Internet forums there are many comparisons between 10x50 binoculars and 15x70 binoculars, so let's verify them with my formula:
B&SSBC(10x50; 15x70) = (15/10)^2 * (50/70)^2 = 2.25 * 0.5102 = 1.148
B&SSBC(12x60; 15x70) = (15/12)^2 * (60/70)^2 = 1.5625 * 0.73469 = 1.148
So, 10x50 and 12x60 binoculars should be brighter by 14.8% than 15x70 binoculars. Not that big of a difference, but 15x70 binoculars are much bigger, especially when compared to the 10x50 binoculars AND the bigger magnification makes them harder to use by hand (without a tripod). On the other hand everything will be 50% bigger in the 15x70 binoculars when compared to the 10x50 binoculars, so it's a real toss up. HOWEVER in case of the 12x60 binoculars everything will be only 25% bigger in the 15x70 binoculars, so again the 12x60 binoculars seem like a perfect compromise to me.
Are the 10x50 and 12x60 binoculars the best as far as brightness is concerned? No, they are not! From the binoculars that are usually available there are 8x56 and 9x63 binoculars that are almost TWICE (!!!) as bright, BUT there is a catch (see my comments below calculations):
B&SSBC(8x56; 10x50) = (10/8)^2 * (56/50)^2 = 1.5625 * 1,2544 = 1.96
B&SSBC(8x56; 12x60) = (12/8)^2 * (56/60)^2 = 2.25 * 0.8711111 = 1.96
B&SSBC(9x63; 10x50) = (10/9)^2 * (63/50)^2 = 1.2345679 * 1.5876 = 1.96
B&SSBC(9x63; 12x60) = (12/9)^2 * (63/60)^2 = 1.777778 * 1.1025 = 1.96
However, there is a crucial difference: these binoculars have Exit Pupil of 7 (56/8 and 63/9), instead of 5 (50/3 and 60/12), which can be bad if you are looking at stars with artificial sources of light nearby – under such circumstances your pupils will be smaller than 7mm, which would mean that some brightness of the binoculars will be lost to your eye. What's worse your pupils are getting smaller with age, so such binoculars are good ONLY for young people. By the way, during a day even young people have pupils smaller than 7mm, but there is so much light then that it's not really important. For these very reasons the 8x56 and 9x63 binoculars are called hunting binoculars rather than astronomical binoculars.
On the side note – one of the differences between binoculars is the type of prisms. BAK4 prisms are usually better on the edges than BK7 prisms, especially when there is little light, BUT there may be some exceptions connected with some other details of the prisms. I am not a specialist, so I can't explain anything about prisms. You can Google it on the Internet.
At the end I would like to point out one thing: if you would like to build binoculars or a spotting scope with 15x magnification and the same brightness as the 10x50 and 12x60 binoculars then they would have to be 15x75 binoculars:
B&SSBC(10x50; 15x75) = (15/10)^2 * (50/75)^2 = 2.25 * 0.44444 = 1.000
B&SSBC(12x60; 15x75) = (15/12)^2 * (60/75)^2 = 1.5625 * 0.64 = 1.000
OLD UPDATE (after buying the 12x60 binoculars, but before I bought the 8x42 binoculars):
Well, my calculations have been confirmed – my new 12x60 binoculars are clearly better than my old 8x30 binoculars. I live in a block of flats in the middle of a large city where the “light pollution” is big, so my first comparison had to be based on the brightest objects of the night sky. You can check light pollution at your own location here:
https://www.lightpollutionmap.info/
I compared my binoculars when the night became truly dark, so after the end of astronomical twilight:
I compared my binoculars during the night from 29 to 30 July 2020. At my location that astronomical night lasted from 11:55 pm till 2:18 am. Below there is a screenshot of the page:
https://sunrise-sunset.org/
from July 2020 set for my location.
Please notice that at my location the “astronomical white nights” ended just 8 days earlier! What a coincidence. During the “astronomical white nights” the night-sky is never fully dark (because the Sun is too little below the horizon and the astronomical twilight never ends), so any night observations (even without any light pollution) are worse by definition.
On 30 July 2020 I could see with the naked eye two planets: Jupiter and Saturn. Please notice that usually you don't have to wait until the end of the astronomical twilight to see these planets. You can check what you can see at your own location at different times at this site:
https://stellarium-web.org/
HOWEVER this site assumes that there is little light pollution. It's still useful in the middle of the city (with big light pollution), because at least you know in what direction to look in order to check if you can see a particular star or planet.
My “balcony astronomical observatory” is terrible, BUT my new 12x60 binoculars still enabled me to see 3 moons of Jupiter without any problems! Actually there were four moons, but two of them were so close to each other (on the night sky) that I couldn't see them separately. On the screenshot below there is also the moon Amalthea, but it's NOT a Galilean moon – I had to make a big close-up to show you that there was another dot next to the moon Io – it was the moon Europa (in my 12x60 binoculars Jupiter was ONLY a DOT, but larger and brighter than its moons):
My old 8x30 binoculars were not enough to see the moons of Jupiter under such circumstances. And in my new 30x50 spotting scope I could barely see one moon of Jupiter because everything was very dark (and also because I couldn't get a totally steady vision even with a mini tripod). It proves that the most important thing is a proper combination of magnification and brightness, not magnification or brightness alone!
Please notice that using ANY kind of binoculars should enable you to see more stars than with the naked eye. It's incredible feeling, even in the middle of a big city with big light pollution – you look at the night sky only with your own eyes and you see hardly any stars, then you look through binoculars and suddenly many more stars appear! Wow!
OLD UPDATE #2:
Last night we (me and my children) went outside our city to watch the stars with lower light pollution. I picked a place with a “Zenith sky brighness” (according to the site lightpollutionmap.info) of 21.05 which was clearly better than what we have in the middle of our city (19.2). Unfortunately it was early in the night, so the mosquitoes weren't “sleeping” and they were constantly attacking us. It was hard to look at the night sky under such circumstances, but it was still worth a try because the difference in the sky brightness was incredible.
Already with the naked eye we could see much more stars AND we could also see the Milky Way (the main part of our own galaxy seen from the “inside”), but only barely. However when I looked at the Milky Way through my new binoculars I was totally amazed – the number of individual stars that I could see was simply unbelievable. My old binoculars made a big difference too, but not as stunning as the new ones.
We also saw several meteors of the Perseids – meteors from the Perseid meteor shower that is currently “active”. However, because of mosquitoes in the future I will focus on meteor showers that are visible in fall or winter: Draconids, Orionids, Leonids, Geminids, Ursids and Quadrantids.
We (me and my daughter) also saw another galaxy! Wow! It was the Andromeda Galaxy that can theoretically be seen with the naked eye as a blurred dot, but I myself had to use binoculars to see it at all, even outside the city. Please, notice that in binoculars you still can't see the “classic shape” of the galaxy, but rather a “cloud of light” that is clearly something different from the stars that are simply dots. We could see the galaxy also in my old binoculars, but the “cloud of light” was smaller. Either way it looked like this:
I had trouble to locate the galaxy on the night sky because I had been sure that I would spot it with the naked eye, so I didn't precisely memorize the stars around it. I only new the general direction at which I should be looking. It took me 15 minutes to locate the galaxy, but now I know how to find it even in the middle of my city. Yes! You can see the Andromeda Galaxy in 12x60 binoculars even in a large city with relatively big light pollution. I didn't try it with my old binoculars so I have no idea if this is possible with weak binoculars, but you can try it yourself. The only thing to do is to find a place without any bright lanterns close by, so you can see the crucial stars. It goes like this (to simulate big light pollution I used the site Stellarium but at an early hour):
1. Find the Plough / the Big Dipper / the Big Wagon.
2. Find the Ursa Minor / the Little Dipper / the Little Wagon.
3. To the right of them and a little down there is a clear triangle of stars.
4. Extend the bottom line of this triangle by a little more than 100% and a little below this line there is the Andromeda Galaxy!
Click to enlarge!
MOST RECENT UPDATE – PART 3:
Later I bought 8x42 binoculars and I LOVE them! The magnification is the same as in 8x30 binoculars, but they are 96% brighter! In other words: they are almost twice as bright!
B&SSBC(8x42; 8x30) = (8/8)^2 * (42/30)^2 = 1.96
I do not own 10x42 binoculars, but according to my formula the 8x42 binoculars are brighter by 56%:
B&SSBC(8x42; 10x42) = (10/8)^2 * (42/42)^2 = 1.5625 * 1 = 1.5625
I actually prefer the 8x42 binoculars to the 12x60 binoculars simply because they are much smaller and much lighter. In fact I consider the 8x42 binoculars as the most universal type of binoculars – they have very bright and wide view, they have good enough magnification and they are easy to handle by hand (without a tripod). Perfect!
I described many interesting things about binoculars in general here:
Most universal types of binoculars
Please notice that my formula can be even used to compare binoculars with telescopes! Let’s compare 12x60 binoculars to a telescope with 70mm aperture combined with eyepieces that give the magnifications 22x and 70x.
EBC(12x60; 22x70) = (22/12)^2 * (60/70)^2 = 2.4694
EBC(22x70; 12x60) = 1 / 2.4694 = 0.4050
EBC(12x60; 70x70) = (70/12)^2 * (60/70)^2 = 25.0000
EBC(70x70; 12x60) = 1 / 25.0000 = 0.0400
Again, the results are consistent with my experience – I could see the M13 in relatively big light pollution in my 12x60 binoculars and in my small telescope with the magnification 22x (the drop in brightness was “only” 60%), but I couldn’t see it at all in my telescope with the magnification 70x (the drop in brightness was 96%).
Pleas notice that just by increasing the magnification in ANY telescope from 22x to 70x the brightness drops by 90% (relatively to the initial brightness).
EBC(70xA; 22xA) = (22/70)^2 * (A/A)^2 = (22/70)^2 * 1 = 0.0988
Verification (“in steps”):
EBC(70x70; 22x70) * EBC(22x70; 12x60) = 0.0988 * 0.4050 = 0.0400 = EBC(70x70; 12x60)
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