Think of your own eye as a telescope.  The aperture of that scope is 8 mm, which is the size of your pupil (aperture) when fully dark adapted.  With this eye, you might have the ability to see stars at a certain magnitude depending upon the conditions of your site.   Perhaps at home you can see magnitude 3 stars with the naked eye, where I would assume you are living either in, or close to, a nice sized city.   But if you take that same eye out to the dark country-side you will see stars somewhere in the area of 7 or 8 magnitude.  

Therefore, what you can see in any scope depends on how dark the sky is and how big of aperture you are looking through.  

There is a way to measure this.  The amount of light between any two scopes collected can be calculated by simply dividing the apertures of the two scopes and squaring them.  So when comparing, for example, a 100mm (4 inch) scope with the 8mm naked eye you will be able to accumulate 156 times the amount of light with the scope.    With a 200mm (8 inch) scope you will generate 625 times the amount of light than with the naked eye.

To understand how this relates to the stars and other celestial objects we need to understand how the magnitude system works.  

For every decrease in magnitude (i.e. from mag 1 to mag 2), it represents a 2.512 times decrease in detectible light.  Or better said, a 1st magnitude star will be 2.512 times brighter than a 2nd magnitude star.  This number is derived from a logarithm that says there is a  100 times increase of light for every 5 magnitudes (which is 2.512 to the 5th power). Therefore, a 6th magnitude star will be 100 times dimmer than a 1st magnitude star.  

So it can be charted as follows using the star Vega, at Magnitude 0, as a point of reference:

Beyond the naked eye limit, we obviously need a little bit of help.  

As I mentioned earlier, a 100mm (4 inch) telescope can see 156 times the amount of light that can be seen with the naked eye.  According to the magnitude scale, this equates to approximately 5.2 magnitudes of improvement over the naked eye.  Therefore, if you can see 7th magnitude objects with the naked eye, then you should expect to see at least 12th magnitude objects with a 100mm (4 inch) telescope.  The aforementioned 200mm (8 inch) scope will show a 7 magnitude improvement over the naked eye.  Therefore, if you can see 7th magnitude objects with the naked eye, then you should expect to see at least 14th magnitude objects with a 200mm (4 inch) telescope.

Depending upon the sky conditions, you should expect the limiting magnitude of any telescope as follows:

There are several things we can derive from this.

1.) The darker skies you can observe in, the more you can see.

2.) The more aperture you have, the more you can see (given equal optics) regardless of where you observe.

3.) Not everybody has the same eyes.  These figures can vary by as much as a magnitude depending on the sensitivity of your eyes and your experience level.

4.) This does not include objects or details detected with averted vision.  This observing technique can add perhaps a half-magnitude to the above figures.

5.) This assumes you are using equipment that does not limit the amount of light entering the eye, such as bad optics, unsteady mounts, and poorly performing eyepieces or barlows.

6.) Limiting magnitude determines if the light can be detected.  Factors such as resolving power, contrast, atmospheric stability and clarity will influence how well you see this light.

So, back to our original question...what can I see with my scope?

Stars with magnitudes in the above ranges are a given.  If you have a 4 inch scope in dark skies, you will see ~12th magnitude stars.   As for objects, that depends on the surface brightness of the object and the ability of your telescope to present that light in a way that is observable.  In other words, it may not be enough for you to merely see an 11th magnitude galaxy.  You might need more aperture or darker skies to see certain aspects of that galaxy that makes it interesting.  It's the difference between seeing M51, the Whirlpool galaxy, as a "faint fuzzy" or as a galactic spiral.

Regarding deep sky objects (DSOs), the Messier catalog of 109 objects is the most observed list of deep sky splendors.   The hardest to see of these objects might not be the ones designated with the faintest magnitude measurements.  For example, M76, the Little Dumbbell Nebula, is normally considered the faintest object, at around 10.2 magnitude or so.  But I find this object much easier to see than fainter galaxies such as M74 and M77 since their magnitudes are spread over a larger surface area.   To see the faintest of the Messier objects, you will need to have a scope/sky combination that gives you no less than ~10.5 magnitudes. That might be a 2 inch scope in dark skies, a 6 inch scope in suburban skies, or a 12 inch scope in city skies.   But to observe these objects, you will need a combination of scope and sky that will reach into the 12th magnitude range.  Only then will you be able to begin seeing certain features that the object may have.  It's no wonder that in dark skies with a 24" Dobsonian reflector, some of these objects begin to look like their photographs.

So users of even small aperture scopes should not despair.  Taken to a dark sky site, even a 3 inch telescope makes not only the entire Messier list detectable, but you can begin to see detail in any of these objects.  In the same way, users of 10 inch scopes when used in the city might have difficulty seeing even the brightest of Messier objects with a certain amount of detail.  

The difference between a 2 inch telescope and a 36 inch telescope is about 7 magnitudes. The easiest way to get these magnitudes back, without spending extra money, is to find a dark sky site.  That 2 inch scope in dark skies performs similarly to a 12 inch scope in city skies.  

For a practical experience, read "Do dark skies really make a difference?" in the Frequently Asked Questions section. 

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