For you to see a star, the light from the star has to get into your eye, and it gets in through the pupil. So the larger the pupil, the more light gets in, and the fainter are stars your eye can detect. While everyone is different, typically the pupil of the eye, when it is adapted to the dark, is about 7 mm in diameter. Click here to see how the dark-adapted pupil varies with age.
What the telescope does is to collect light over a much wider area than just the 7mm of your eye pupil. It then focuses that light down to the size of your eye pupil so you end up with much more light passing into your eye. This enables you to see much fainter stars with a telescope than you could without.
How much more light does the telescope collect? That is known as the "light grasp", and can be found quite simply as the increase in area that you gain in going from using the pupil of your eye to using the objective lens (or mirror) of the telescope.
The area of a circle is found as p/4 × D2, so the light grasp -- we'll call it GL -- is the ratio of the area of the objective to the area of the pupil of the eye, which is
Astronomers measure star brightness using "magnitudes". The magnitude scale originates from a system originally set up by ancient Greeks, where the brightest stars were stars of the first magnitude, like 'first class', and the faintest stars you could see were stars of the sixth magnitude.
When astronomers got telescopes and instruments that could measure star brightness, they found 1st magnitude stars were almost exactly 100 times the brightness of 6th magnitude stars. That works out to a factor of 2½ from one magnitude to the next. This is a relative scale, with stars like Capella, Vega, and Arcturus establishing the zero point. (Incidentally, the scale does go negative -- Sirius, for example, is magnitude -1.5)
The actual formula for finding magnitude is logarithmic because you are translating multiplying factors into a linear scale. So to find a magnitude difference I take the ratio of one brightness to another, then I take the logarithm of that ratio, then I multiply by 2.5.
Why do I multiply by 2.5? Remember that I said that a difference of 5 magnitudes (from 1 to 6) represents a brightness factor of 100. If I take the logarithm of 100 I get 2 (102 =100) so I need to multiply by 2.5 to turn that into a difference of 5 on the magnitude scale. The logic really is that simple.
So if I call the first brightness I1 and the second brightness I2 -- "I" is for "intensity" -- then the formula for the magnitude difference of I2 over I1 is
Magnitude Difference = 2.5 × log(I2/I1).
Not so hard, really.
We've already worked out the brightness increase we get from the scope as GL = (DO/Deye)², so all we need to do is take 2.5*log(GL) and we have the brightness increase of the scope in terms of magnitudes, so it's just Gmag = 2.5*log((DO/Deye)²).
Just one more thing. We can take advantage of the logarithm in the equation to simplify it. This is because of the fact that log(x²) = 2*log(x). So then 2.5*log((DO/Deye)²) = 2.5*2*log(DO/Deye), and when we set Deye to 7mm, and we get our final, simple (ok... simpler ) formula for magnitude gain of the scope:
I want to go out tonight and find the asteroid Melpomene, which is wandering through Cetus at magnitude 8.6 as I write this. It's just that I don't want to lug my heavy scope out if I can grab my smaller scope (which sits right by the front door at all times) and spot it with that. So the question is -- can I see Melpomene with my 90mm ETX?
First I go out and look at the sky. By using certain key gauge constellations, I can determine the faintest magnitude that I can see by eye -- tonight from the light-washed sky of New Berlin I can just see the 4th magnitude arc of stars in Andromeda.
I then apply the magnitude gain formula for the 90mm ETX, in the hopes that the gain will get me to better than magnitude 8.6. So the magnitude gain is
Gmag = 5 × log(DO/7) = 5 × log(90/7) = 5.5. This is how much more the scope adds to the magnitudes I can see, so that means I can see stars -- and asteroids -- at magnitude 4 + 5.5 = 9.5.
Outstanding. I can see it with the small scope. The magnitude gain formula just saved my back.
Where I use this formula the most is when I am searching for a deep sky object and want to see how the star field will look in the eyepiece. I can do that by setting my astronomy software to show star magnitudes down to the same magnitude I will be able to see in the telescope.
From my calculation above, I set the magnitude limit for "faintest" stars to 9.5 and the software shows me the star field I will see in the eyepiece. This helps me to identify the asteroid as the "star" that isn't supposed to be there.
Let's suppose I need to see what the field will look like through the viewfinder scope, so I want to find the limiting magnitude for the viewfinder. Written right on my viewfinder it says "8x25mm", so the objective of the viewfinder is 25mm, and the magnitude gain is 5 × log(25/7) = 2.8. So I would set the star magnitude limit to 6.8 and the software shows me the star field that I will see through the viewfinder.
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Updated 11 November 2011
