The Scope's Star Magnitude Limit

Ok so we were supposed to be talking about your telescope so let's get back to that. 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).

We can take advantage of the logarithm in the equation to simplify it, by making use of the fact that log(x²) = 2×log(x). So then: 

When you divide by a number you subtract its logarithm, so subtracting the log of Deye from DO , then substituting 7mm for Deye , we get: 

Since log(7) is about 0.8, then 5×0.8 = 4 so our equation for the gain in star magnitude is 

This represents how many more magnitudes the scope lets me see, over and above what my eye alone can see. Then to find the faintest magnitude I can see in the scope, we simply add Gmag to the faintest magnitude our eye can see, magnitude 6. This is the magnitude limit of the scope, Lmag: 

Which simplifies down to our final equation for the magnitude limit Lmag of the scope 

By the way did you notice through all this, that the magnitude limit of the scope — the faintest star I can see in the scope — depends only on the diameter of the objective? Interesting result, isn't it? 

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. 

Magnitude

Q. What is Magnitude?

A. The degree of brightness of a celestial body designated on a numerical scale, on which the brightest star has magnitude −1.4 and the faintest visible star has magnitude 6, with the scale rule such that a decrease of one unit represents an increase in apparent brightness by a factor of 2.512. Also called magnitude.

The brighter an object appears, the lower the value of its magnitude, with the brightest objects reaching negative values.
The Sun has an apparent magnitude of −27, the full moon −13, the brightest planet Venus measures −5, and Sirius, the brightest visible star in the night sky, is at −1.5

There are two types of Magnitude, they are stated below just click on the appropriate link.

1. Apparent Magnitude
2. Absolute Magnitude

Power of Aperture

How much you can see with a telescope has everything to do with the size of your aperture. Smaller telescopes (4 inches and under) will not reveal very much in the way of nebulas and galaxies. That being said, bear in mind that somewhere between 1750 - 1800, Charles Messier compiled a list of approximately 100 diffuse objects now known as the Messier Catalog.  These objects were difficult to distinguish from comets through the telescopes of the day. So don't write off that small scope.

The chart below will give you a good approximation of the effect that aperture has relative to how much can be seen. I have used stars to demonstrate the increase in visible objects and have listed the naked eye for reference. Limiting Magnitude refers to the faintest object visible with a given size of aperture. Please bear in mind that this number is also affected by local seeing conditions, the quality of the telescope and eyepiece's optics as well as the eye of the observer. More information on magnitude can be found in the glossary. 


Do Not worry about Magnitude, i will come to this topic later. Just see the chart and get to know what is written below:

Aperture inches	Aperture mm	Limiting Magnitude	Number of Visible Stars
Naked eye	Naked eye	6 (maximum)	approx 8,500
4	102	13	approx 15 million
6	152	13.5	approx 30 million
8	203	14	approx 45 million
10	254	14.5	approx 85 million
12.5	318	15	approx 130 million

Immune to Magnification

Stars are so ridiculously far away that no matter how massive or blown out of proportion they may be, to us they look like points. Magnify a point, and it's still just a point.

That means that, unlike objects that cover an area, the light from a star does not get spread out as you magnify the image. All the light from the star stays inside the point.

However... as you increase magnification, the background skyglow does get spread out, which means the background gets darker and the star stays bright. This is a nice way of increasing the contrast on stars, and sometimes making fainter stars more visible. 

Just remember, this works until you reach the maximum magnification of the scope, which is the same number as the diameter of the scope in millimeters. When you exceed that magnification (or the limits of the atmosphere), the stars start to spread out and dim down just like everything else. 

Magnification

Clearing up the myth about power (magnification).

It slices     it dices     it has 675 power!!!!

Over the years, magnification has been the most misleading specification printed on those glossy color boxes of many department store telescopes. Exaggerated claims of high power are almost always a sure sign of an inferior product. The real important specifications in no particular order are as follows:

Stability of the mount ...  does not vibrate and moves smoothly

Size of aperture ... light gathering power

Quality of optics ...  reflectivity and transmission of light

Ease of use ...  point and look. The telescope needs a good finder and, in our opinion, especially for the beginner, it MUST have a reflex sighting device of some type. (see reflex sight in glossary)

Please understand, we are not saying that power is unimportant, but it should be one of the last criteria you consider.
The amount of  power you can apply is dependent upon several factors such as:

Focal Length ... focal length is the distance that the light in the telescope travels from the objective lens or mirror to the point where that light reaches focus (the focal plane). To determine how much power a telescope can deliver with any eyepiece, simply divide the focal length of the telescope by the focal length of the eyepiece.

For example: If you have a telescope with a 1000mm focal length and a 20mm eyepiece, that eyepiece will deliver 50 power (1000/20). The same eyepiece in a 2000mm focal length telescope delivers 100 power (2000/20).

Focal Ratio ... telescopes are rated with an "f" number. As you look at the telescope market, the products are given specification like (6" f/5). This means the telescope has a 6" aperture and a focal ratio of 5. As with cameras, focal ratio simply represents the speed of the optics. The smaller the "f" number, the faster the optics. But how does this relate to magnification?
With any given eyepiece, fast focal ratios (f/3 to f/5) deliver lower power, wider fields of view and usually brighter images. Because of these brighter images and wider fields of views, fast focal ratios excel at deep sky observing and can deliver stunning wide-angle views.  Conversely, slower focal ratios (f/6 and up) deliver higher power and increasingly narrower fields of views.  Slow ratios excel at observing the moon, the planets and binary stars.