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Telescopes: basic parameters

It happens quite often to ask if a specific telescope can “show” a certain object with given magnitude. For this reason, it is useful to keep in mind a few simple relations giving the basic properties of telescopes.In this post, I explain three parameters: Limiting magnitude, resolving power, and the magnification.

Limiting magnitude is the magnitude of the faintest object one can see through a telescope. The larger the telescope aperture, the larger the light gathering power (w.r.t. the human eye), and the larger the limiting magnitude. It is given by the following relation:

Limiting Magnitude = 2.7 + 5 Log(D)

where D is the diameter of the telescope objective (lens or mirror) in mm. To have a more realistic estimate, you may subtract 0.5 from the given values. This is due to dirty optics and old coatings. For many small telescopes, you can see the numerical result in the below table.

Another important property of any telescope is its resolving power. The Rayleigh limit tells us if two stars are apart by an angle α, we can resolve them marginally if it satisfies the following relation:

α [arc second] = 1.22 λ [m] / D [m] * 206265.

where λ is the wavelength of observation, e.g., take 500 nm, and D is again the diameter of the objective.  Note that due to atmospheric turbulence, the resolving power is bound by atmospheric seeing. When seeing is good, i.e., the atmosphere is stable and has not too much turbulence, the resolution can be as low as one arc seconds. However, a typical value of 2-3 arc seconds is normal for many observing sites. Actually, this is one of the key parameters when professional astronomers try to find a good site for a new telescope. The reason the Hubble space telescope with a 2.4 m mirror captures sharpest ever images, way sharper than, e.g., 10m Keck telescopes, is due to atmospheric turbulence. For small amateur telescopes, the seeing effect can be traced with the amount of wobbling a bright star or planet shows in the eyepiece.

Magnification is defined as the ratio between the apparent angular size of an object in the telescope (through the eyepiece) , and its real angular size on the sky. It is calculated from the ratio between the objective focal length and the eyepiece focal length. In my opinion, one of the least important parameters in a telescope is its magnification. I put a rather conservative low to normal magnification, I personally use in the last column. Larger magnifications can be reached by using eyepiece with smaller focal length. However, again due to seeing effects, there is a practical limit, regardless of the size of the telescope, which is about 500x. When we use large magnification for faint or diffuse objects, not only focusing gets very hard, but also the surface brightness falls down. Hence, a large magnification is only recommended for planets and multiple stars.

I plan to discuss optical aberrations  of telescopes in a separate post.