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Thread: Light collecting power and aperture.

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    Default Light collecting power and aperture.



    We all know astronomical telescopes are used for their light gathering power. But how much in comparison with the naked eye? And is there a logic in it?

    It is not difficult to calculate the difference in collecting light power of a telescope in comparison with the naked eye. As we set the opening of the human eye at 7mm diameter in the dark, light gathering of a telescope with a certain aperture becomes: (D telescope in mm. divided by 7) ^2 The square is used as the surfaces of circles in square millimeters relate quadratic.

    So for instance a 70mm telescope: (70/7)^2 = 10^2 = 100

    Magnitude gain is calculated as follows: 2.512 X Log X Gathering light

    Below a table which almost speaks for itself.

    Diameter telescope in mm. Gathering light related to naked eye Multiplied gathering power with foregoing Increase in aperture Difference with foregoing magnitude Gain in accumulated Magnitudes
    Naked eye 7mm. 1 - - - -
    Binoculars 50 mm. 50 50X 7 4.3 4.3
    70 mm. 100 2X 1.4 0.7 5
    100 mm. 200 2X 1.4 0.8 5.8
    140 mm. 400 2X 1.4 0.7 6.5
    200 mm. 800 2X 1.4 0.8 7.3
    280 mm. 1600 2X 1.4 0.7 8
    400 mm. 3200 2X 1.4 0.8 8.8
    Note that doubling your light gathering power needs 1.4 increase in aperture. It is the root from 2.

    Where do we know this number from? Photographers know it. It is the same magnification number we use to diaphragm our photo-aperture with one stop (2X less light) We all know the diaphragm series 4, 5.6, 8, 11, 16, 22. This is the very same 1.4 (root from 2)

    Note that doubling your diameter gives only 0.7- 0.8 Magnitude gain. So, ads which claim 30% more gathering power are not that impressive after all. It takes a good observer to notice a 30% light gathering gain on faint stars!

    The biggest, easiest, handheld and impressive "jump" you can make in one go, is the one from naked eye to binoculars!!

    Be happy with you binoculars!!
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    Default Re: Light collecting power and aperture.

    Thanks also for doing this in magnitude increments rather than magnitudes. There's a lot of variability from one person to another in terms of individual sensitivity. Of course aging eyes can't dilate to 7mm, but again that's individual.
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    Default Re: Light collecting power and aperture.

    This is quite interesting, and if I may say "en-light-ening". :-)
    Thank you John.
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    Default Re: Light collecting power and aperture.

    Now I know why we always advise folks to get a set of binoculars, more to it than meets the eye, learned something new today, many thanks John.
    Mine are probably the most used of my astro gear, you never know when you might see a gap in the eternal cloudiness.

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    Default Re: Light collecting power and aperture.

    Very interesting. I'm glad you're good at the maths John! I'm quite pleased to know that my 72mm Evostar gathers about 100x more than the naked eye.



    100x seems like such a nice round figure lol.
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    Default Re: Light collecting power and aperture.

    Great data and information, John! Thanks for the share!
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    Default Re: Light collecting power and aperture.

    Nice sum up John! Another consideration to take into account is that viewing conditions (light pollution, seeing and transparency) limit more large apertures. Whether seeing is 1" or 5" is irrelevant when you observe with 10x50, but will certainly impact high power views with 8" SCT.
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    Default Re: Light collecting power and aperture.

    This may be different for DSOs. Stars are always focused at a point but DSOs are surfaces so we need to consider the light flux. That flux is proportional to the square of the aperture D but inversely proportional to the square of the magnification M. That magnification is not always the same for telescopes of increasing aperture because eyepieces are usually maximally 30 mm in focal length.

    So if the light flux of the human eye for a surface area is normalized to 1 and the binoculars are 7x50 for instance, then the light flux relative to the human eye will be (D/(7*M))**2 = (50/(7*7))**2 (**2 meaning squared, my Android has no other character for it). This is approximately 1.04.

    If we assume a 30 mm eyepiece and a focal ratio of 5 for all telescopes of different aperture then all of them will have the same light flux. Namely, because for a fixed focal ratio the magnification is proportional to the aperture, (D/(7*M))**2 is the same for all so there will be no difference in light flux between them. If we take D=100 as an example then F=500 and the magnification is M=500/30=16.6. So (D/(7*M))**2 is about 0.74.

    For all these scenarios the light flux on the retina is about the same. The difference is the magnification on the retina. From seeing M51 through a 24" Dob once I know that this still significant, I believe I saw color if I remember correctly. Obviously the math is completely different here.

    I hope this makes sense. I don't have this from a book so who knows if I overlooked something please let me know.

    PS At first when I wanted to post this I canceled because I thought this seems to contradict the experience of most observers. So I searched for surface brightness and found a CN discussion of 3 pages with exactly the same reasoning. I did not read the end of it but keep in mind the key assumption here is that we use a fixed 30 mm eyepiece with the telescope so the magnification is proportional to the aperture. It has to be the sensitivity of the human eye to larger objects that makes the difference.
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    Default Re: Light collecting power and aperture.

    I should add that the exit pupil is roughly the same in all cases because the light focused on the retina is the light flux integrated over the exit pupil. In the example of the 100 mm scope at F/5 and 30 mm eyepiece the exit pupil is 100/(500/30)=6.66 mm, and this is the same for all. The 7x50 binos have an exit pupil of 50/7=7.14 mm and ghe eye itself 7 so that's all roughly the same so the numbers don't change much.
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    Default Re: Light collecting power and aperture.

    The question was raised: what if someone has an eye pupil that is importantly smaller than 7mm? Let's say 2.5 mm or even 2 mm? The gain in lightgathering becomes a lot more in that case?

    The answer is yes and no. The gathering of light related to a 2 mm eye pupil becomes far more, but that doesn't lead to seeing fainter stars. Pity... The startposition of a 2mm eyepupil is at approximately at magnitude 3.2 or so. The 7mm pupil reaches magnitude 6. The first step with 50mm binoculars leads to 625X gathering power related to a 2mm eye pupil. Which means 7 magnitudes gain.

    A 7 mm. eye starts at magnitude 6 and gains 4.3. The result is magnitude 10.3 stars.
    A 2 mm. eye starts at magnitude 3.2 or so and gains 7. The result is magnitude 10.2 stars.

    I could make a new table here with different numbers in the second column. From 625 to 40,000. All other columns from 70mm and up stay the same as they are related to each other and not to the human eye.

    So, when a young observer reaches 11.8 with his 100 mm. telescope, his father and granddad will too. ( maybe more because of their techniques and experience)

    And that is the good news for seniors. The good news for our sons and daughters: keep on practicing and in the end you will pass your mom or dad. That is the way our world turns...
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