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I just bought a Skywatcher telescope with a diameter of 60 mm and a focal length of 900 mm. I have a 2x Barlow lens and two eyepieces with focal lengths of 10 mm and 20 mm.
How can I calculate the magnification of my telescope, for example using the 2x Barlow and the 20mm eyepiece?
I'm fascinated by watching the sky and I can't wait to try it... Unfortunately, it's currently cloudy.
One more question; what I can see to be fascinated with this telescope?
Magnification of your telescope depends on the ratio of the focal length of your primary optics and the focal length of your objective. We can represent this with this simple formula: $$P=\frac{f_{objective}}{f_{eyepiece}}$$ where $P$ is magnification (power) and $f$ is focal length.
Your telescope has 900 mm focal length, so the magnification is $P_1=\frac{900\text{ } mm}{10\text{ }mm}= 90\text{x}$ (10 mm ocular) and $P_2=\frac{900\text{ }mm}{20\text{ }mm}= 45\text{x}$ (20 mm ocular).
A Barlow lens is used to increase the focal length by the given factor (thus 2 in this case), so directly impacts the magnification in the same way, increasing it to 180x and 90x - but gives you in return an equally smaller field-of-view and makes area-objects fainter as the same light is projected to a larger area. Note that there is a maximum useful magnification which is roughly twice the aperture (measured in mm), which is 120x in your case, so the Barlow lens will not be very helpful as it magnifies beyond reasonable due to inevitable diffraction on the clear aperture.
One usually compares them by their focal length and by their aperture - the ratio defines how bright the objects appear and thus how faint objects you can see. With an aperture of 60 mm, it has an opening ratio of f/15 which is only a moderate one.
Such telescope is particularly useful for observations of the planets in our solar system as well as star clusters and the brighter deep-sky objects. Now in summer would be an excellent time to hunt for the globular clusters like the one in Hercules (M13). Also, you can download a useful program Stellarium to find more interesting features.
