Aperture vs Magnitude

Ever wonder how faint the objects you you can see with your telescope?  Here is how you can tell, and how we derive the numbers. If you don't want to see the math, jump to the end of the article and check the tables.

The light grasp of a telescope is a function of its aperture squared. That means that a 10" telescope can see four times fainter than a 5" telescope. A 12" scope can see 9 times fainter than a 4" scope. Aperture really counts when the scope gets bigger. A 24" or 40" scope can see remarkably fainter objects than a 4" scope. The comparison is their ratios squared. For a 24" scope compared to a 4" scope, you would take 24 and divide it by 4, getting 6, and square that result to get 36.

Magnitudes are set up on a logarithmic scale. Each 5 magnitudes represent 100 times the light, each magnitude being the fifth root of 100, or about 2.51188 times. This means that a magnitude 8 star is 2.51 times brighter than a magnitude 9 star. A magnitude 12 star is 2.51 times another 2.51 times brighter than a magnitude 14 star, or about 6.3 times brighter.

Comparing magnitudes with aperture becomes a logarithm challenge. Let's start with the human eye. It has a 7mm entrance pupil and can see 6th magnitude stars. We'll use that as our starting point to calculate apertures with magnitudes. If I wanted to know what aperture optic would give me another magnitude, I'd have to calculate what the aperture would be to give me 2.51188 times the light. That means that the 7mm would increase by an amount that when squared gives me 2.51. The new aperture, A, divided by 7mm would have to be the square of 2.51. This gives us (A/7)2 = 2.51. Square rooting both sides gives us A/7 = 1.585. Multiplying both sides by 7 gives us A = 11.09mm, or 0.44 inch. If you had a 0.44" aperture telescope, it would be able to detect magnitude 7 stars.

The aperture A of a telescope divided by the diameter of the human eye's entrance pupil squared will give us the amount of light it could detect as compared to the human eye. This means that (A/7)2 = K where K is the number of times brighter the scope can see. This number K can be written as a magnitude in that M2.51188 = K. Therefore, (A/7)2 = M2.51188 where M is the magnitude difference from the human eye. Converting this to a log statement we get M = log 2.51188 (A/7)2 as being the increase of magnitude over the human eye, which is 6. So, the magnitude of the star a telescope can detect is M = 6 + log 2.51188 (A/7)2. Using the drop-down rule of logs, we can change this to M = 6 + 2log 2.51188 (A/7) . If we convert the 7 from millimeters to inches, we get M = 6 + 2log 2.51188 (A/.276). If we use the quotient rule for logs, we get M = 6 + 2log 2.51188 (A) - log 2.51188 (.076). This boils down to M = 6 + 2log 2.51188 (A) - (-)2.799, or

M = 8.8 + 2log 2.51188 (A).

Some folks don't like weird log bases, so we can fix that to either base 10 or base e. This gives us the two equations M = 8.8 +2[log 10 (A) / log 10 2.51188] and M = 8.8 + 2[ln (A) / ln 2.51188] which gives us M = 8.8 + 2[log 10 (A)/.4] and M = 8.8 + 2[ ln (A)/.921] which are

M = 8.8 + 5log 10 (A)      and
M = 8.8+2.171ln(A) .

Let's try the same telescope using each formula. A 70mm scope is 10 times the aperture of the human eye, so it should give us 100 times the light grasp, which is 5 magnitudes past mag 6, so we should get magnitude 11 for an answer. Convert 70mm into inches by dividing by 25.4, that gives us 2.76 inches. In the log 10 equation we get M = 8.8 + 5log 10 (2.76) = 8.8 + 5 * 0.441 = 8.8 + 2.2 = 11.0 mag. Using the natural log equation we get M = 8.8 + 2.171* ln(2.76) = 8.8 + 2.171*1.02 = 8.8 + 2.2 = 11.0 mag. As you can see, both equations work.

If you like metrics, such as for using this equation with an 80mm refractor, then 25.4*A = D where D is the diameter of the optic in millimeters. Now we have M = 8.8 + 5log(D/25.4) = 8.8 + 5log(D) + 5log(25.4) = 8.8 + 5log(D) - 7 = M = 1.8 + 5log(D) . If you like metrics and natural logs, then it's M = 8.8 + 2.171ln(D/25.4) = 8.8 + 2.171ln(D) - 2.171ln(25.4) = 8.8 + 2.171ln(D) - 7 = M = 1.8 + 2.171ln(D) .

You don't really want to do all these calculations each time you look through a scope and want to know how deep it'll go, so here is the table: 

Aperture in inches	Magnitude Limit	Light Grasp
1	8.8	13
2	10.3	52
2.4	10.7	75
3.1	11.2	126
4	11.8	210
4.25	11.9	237
5	12.2	329
6	12.6	473
8	13.3	842
10	13.7	1316
12.5	14.2	2057
16	14.8	3370
17.5	15	4032
20	15.3	5266
22	15.5	6372
24	15.7	7583
28	16	10322
30	16.1	11849
36	16.5	17063
40	16.8	21066

From the table you can determine how large of a scope you will need to observe certain objects. For example, Pluto is between mag. 14 and mag. 15. In May when it is about mag. 14.2 you can use a 12.5" scope, but later in October when it is mag. 14.7 you will need a 16" scope. Neptune is mag. 7, so you'll need a 1" scope or more (I suggest more).

There are errors in this formulation. Reflecting surfaces absorb some of the light, refracting surfaces reflect some of the light, secondary mirrors block light, glass in eyepieces take away a little, and there are others. The errors are usually in the 0.1 or 0.2 or less magnitude range, so I usually don't worry about them. The human eye can't detect light differences lower than 11%, or 0.11, which is a mag. difference of 0.1.

Juliano and I did visual magnitude tests on our scopes a few years back, and we found that we could see deeper with point source samples than this arithmetic shows. I've seen 17.1 with my 24" scope, and if the mirror had been figured a little better, I think I could have pulled more out of it, maybe as much as 0.5 more. Also, each eye is different, and each point of light has a peak frequency that it shines at, so a 13 mag. star may look like a 15 mag. star to your eye. The central star in M57 is the perfect example. Its magnitude implies that it can be seen easily, but it is very elusive. 

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